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**ACADEMIC YEAR 2016 - 2017**

**Fall 2016**

**September 23, 2016**

Using Computational Methods to Understand Microorganism Motility

Sarah Olson, Ph.D.

Worcester Polytechnic Institute

Microorganisms such as the bacteria E. coli and sperm are able to swim and navigate in complex environments. Developing an understanding of swimming microorganisms through computational models could lead to insight on the development of artificial micro swimmers for a variety of applications, including drug delivery. We will illustrate how vector calculus can be used to rewrite the fluid governing equations and how computational algorithms can be used to solve for the resulting fluid flow. Time permitting, we will show simulations of sperm motility and flagellar bundling of E. coli.

Sarah Olson is an Assistant Professor in the Department of Mathematical Sciences at Worcester Polytechnic Institute and is interested in mathematical biology, fluid dynamics, and numerical methods. She started her undergraduate studies at Providence College, majoring in both Mathematics and Biology, and continued at NC State for a Ph.D. and at Tulane University as a Postdoctoral Scholar. She has mentored many students on different research projects and recently received an NSF CAREER award, recognizing her integration of research and education.

**October 14, 2016**

Financial Derivatives: The Use and Pricing of Options and Futures

Professor Edward Szado

Financial derivatives are the cornerstone of risk management in financial institutions and corporate America. This discussion provides a brief overview of options and futures. I discuss their use in risk management, the impact of non-linear payoffs and their relative advantages and disadvantages. In addition, I provide a brief overview of pricing options using models based on the Black Scholes Merton option pricing model as well as a discussion of the characteristics of options and their underlying securities that determine the value and risk exposure of the options.

Dr. Szado is an Assistant Professor of Finance at Providence College, and holds pro bono positions as the Director of Research at the Institute for Global Asset and Risk Management and the Director of Research at the Center for International Securities and Derivatives Markets. Ed earned a Ph.D. in Finance from the Isenberg School of Management, University of Massachusetts – Amherst, an MBA from Tulane University and a BComm from McMaster University. He has taught at Boston University, Clark University, Providence College and the University of Massachusetts - Amherst. He is a former options trader and has worked extensively on asset allocation and risk managed investment programs. He was a founding Editor of the Alternative Investment Analyst Review (AIAR) and currently a member of the editorial board of the Journal of Alternative Investments (JAI). He is a Chartered Financial Analyst and has consulted to the Options Industry Council, the Chicago Board Options Exchange, the Chartered Financial Analyst Institute, the Alternative Investment Analyst Association and the Commodity Futures Trading Commission.

He has published more than fifteen journal articles, and coauthored four books in the areas of derivatives, alternative investments, and risk management. His research has been featured in a wide variety of media, including:
*Barron’s*,
*Bloomberg*,
*Dow Jones Newswires*, Futures Magazine,
*Hedge World*,
*Market Watch*,
*Pensions and Investments*,
*Reuters*,
*Smart Money*,
*Swiss Derivatives Review*, and the
*Wall **Street**Journal*.

**October 21, 2016**

The James Function

Professor Christopher Hammond

Connecticut College

We investigate the properties of the James Function, associated with Bill James’s so-called “log5 method,” which assigns a probability to the result of a game between two teams based on their respective winning percentages. We also introduce and study a class of functions, which we call Jamesian, that satisfy the same a priori conditions that were originally used to describe the James Function. (This talk represents joint work with Warren P. Johnson and Steven J. Miller.)

Christopher Hammond grew up in Durham, North Carolina, where he had the privilege of watching Duke basketball in its heyday. After graduating from the North Carolina School of Science and Mathematics, he earned his bachelor’s degree at the University of the South in Sewanee, Tennessee. He received his doctorate at the University of Virginia, after which he had the good fortune to obtain his current position at Connecticut College. All of his mathematical research, apart from this project, has pertained to composition operators acting on spaces of analytic functions.

**November 11, 2016**

Bayes’ Rule and the Law

Professor Leila Setayeshgar

Bayesian inference is an approach in mathematical statistics where the probability of a hypothesis is updated as more evidence and data become available. It has wide applications in many areas such as machine learning, evolutionary biology, medicine and even in the judicial system. This talk will explore how Bayesian inference can be used in a specific court case to assist jurors in the process of legal decision making, demonstrating the power of mathematics in the court room.

Dr. Leila Setayeshgar is an Assistant Professor of Mathematics at Providence College. She received her Ph.D. from the Division of Applied Mathematics at Brown University in 2012. She is passionate about teaching mathematics, and is equally enthusiastic about teaching mathematics to majors and non-majors. She teaches courses ranging from calculus to statistics and probability, and is broadly interested in doing research in probability and stochastic processes.

**November 18, 2016**

Grit and Character

Professor Jason Price

Nichols College

Do you give all that you have? The philosopher William James posited a gap between potential and its actualization. “The human individual lives usually far within his limits; he energizes below his maximum and he behaves below his optimum.” (The Powers of Men – 1907) Angela Duckworth defines Grit as perseverance and passion for long-term goals. I am involved in a study that is investigating the effect of Grit on various aspects of academic life. It is of particular interest to me that an individual’s grit score can be decomposed into two components, passion and perseverance, with both components seeming to drive success in different areas. The decomposition

Grit = Passion + Perseverance

is reminiscent of the decomposition of a character of a finite group into irreducible characters.

In this talk, I will discuss my career journey starting as an undergraduate at Providence College. I will give an introduction to the Grit study that I am involved in. I will also introduce group representations and characters. This part of the talk will pull in elements of Linear Algebra and Abstract Algebra although no knowledge of either is presupposed.

Jason Price is the Associate Dean for Liberal Arts and an Associate Professor of Mathematics at Nichols College. His research interests include number theory and educational technology. He studied at Providence College and the University of Vermont. He enjoys teaching a number of subjects, particularly Abstract Algebra, and spending time with his wife and baby girl.

****

**Spring 2017**

**March 24, 2017**

Soap Bubbles and Mathematics

Professor Frank Morgan

Williams College

A soap bubble is round because the round sphere provides the least-perimeter way to enclose given volume, as was proved mathematically by Schwarz in 1884. Similarly the familiar double bubble, which forms when two bubbles come together, is the least-perimeter way to enclose and separate the two given volumes, although we didn't prove this until 2000. If space is given a density, very popular since its appearance in Perelman's proof of the Poincaré conjecture, the question gets even more interesting. The show will include a little guessing contest with demonstrations and prizes. No prerequisites; all welcome.

Frank Morgan works in minimal surfaces and studies the behavior and structure of minimizers in various dimensions and settings. His proof with colleagues and students of the Double Bubble Conjecture is featured at the NSF Discoveries site. Morgan went to MIT and Princeton, where his thesis advisor, Fred Almgren, introduced him to minimal surfaces. He then taught for ten years at MIT, where he served for three years as Undergraduate Mathematics Chairman, received the Everett Moore Baker Award for excellence in undergraduate teaching, and held the Cecil and Ida Green Career Development Chair. Morgan also served at Williams as Mathematics Department Chair and founding director of an NSF undergraduate research project. He is currently Webster Atwell ’21 Professor of Mathematics, Emeritus, and Editor-in-Chief of Notices of the American Mathematical Society.

**April 7, 2017**

Mathematics of Options Pricing and Hedging

Professor Lucy Kimball

Bentley University

The pricing and hedging of financial options is an active area of research in mathematical finance. Classic approaches rely on assuming a binomial model or a stochastic differential equation for the evolution of the underlying stock price process. Our work is based on an extension of the binomial model to include stock price jumps that fall in a closed interval rather than just the two point distribution assumed by the classic model. This model allows for the development of a unique approach to finding an optimal hedging strategy based on real market data. In this talk I will give an introduction to financial options, some of the mathematical principles underlying classic approaches to pricing and hedging and give an overview of our work. No previous exposure to financial options is required.

Dr. Lucy Kimball is a Professor of Mathematical Sciences at Bentley University. At Bentley she served as Actuarial program Coordinator, Internship Coordinator and Math Club Advisor for many years prior to becoming Chair of the Department of Mathematical Sciences. While chair she was instrumental in developing the department’s undergraduate major in Actuarial Science and graduate degree program in Business Analytics. She is currently the Wilder Teaching Professor and Chair of the Bentley University Learning and Teaching Council.

Dr. Lucy Kimball does research in Mathematical Finance and has over thirty publications on a variety of topics including atmospheric modeling, optimization for large scale problems in electric power systems and mathematical models for pricing and hedging of financial options.

Dr. Lucy Kimball received a Bachelor of Science from the University of Massachusetts, Lowell and received a Master of Science and Doctor of Philosophy from Worcester Polytechnic Institute.

**April 21, 2017**

Tropical Mathematics: A New Dialect

Professor Catherine Buell

Fitchburg State University

If mathematics is truly the language of the universe, then surely there must be different dialects to describe the vast number of physical phenomena we observe and create; one such dialect is Tropical Mathematics. Tropical Mathematics is also known as min-plus algebra or max-plus algebra because addition is defined as x ⊕ y ≔ the maximum of x and y and multiplication is defined as x ⊗ y = x + y. Questions in Tropical Mathematics cross the disciplines of computer science, pure and applied mathematics, physics, and biology. In this talk, we will explore this new language through arithmetic, graphs, and ultimately discuss the implications of Tropical Mathematics to industrial problems and network analysis.

Dr. Catherine Buell is an assistant professor of mathematics at Fitchburg State University in Massachusetts. She has also taught at Bates College in Maine. Her research areas include algebraic group theory, visual stylometry, and undergraduate education (equity in inquiry and social justice mathematics). Her dog Donny serves as a (unofficial) math anxiety dog at the university.

**ACADEMIC YEAR 2015 - 2016**

**Fall 2015**

**September 18, 2015**

The Five-Button Door Lock -

How Computers Help Mathematicians Explore and Discover

Professor Shai Simonson

Stonehill College

Exploration, experiment, and discovery are fundamental tools for doing mathematics. The invention of computers, through simulation and visualization, has enhanced these tools. An illustration of this is the problem of calculating the number of combinations one needs to try in order to break into the common 5-button door lock. Using patterns inspired and generated by a computer algorithm, we will uncover the recursive structure of these locks.

Shai Simonson is a professor of computer science at Stonehill College, interested in discrete mathematics, theoretical computer science, and effective teaching. At various times, he has taught gym, science, mathematics, and computer science to students from first grade through graduate school. Shai's book, *Rediscovering Mathematics*, promotes teaching mathematics as an experimental science, emphasizing interactive exploration and discovery. He also co-authored a book on Java programming.

Shai was the director of ArsDigita University, http://aduni.org, a pioneer online learning site that offers free computer science lectures to students all over the world, especially in developing countries. He plays go and bridge, dabbles with poker and Scrabble, loves to hike, cycle, bowl, sing, and play disc-golf. Shai grew up in New York, spent ten years in Chicago, and now makes his home in the Boston area. He spent two years (1983 and 1999) teaching and doing research in Israel. He is married with three children.

**October 2, 2015**

Special Internship/REU Presentations

Coding a Robotic Intern to Do All Your Paperwork 101

Timothy Corwin

This 2015 summer, I had the privilege to work at one of the "Big Five" (largest English language publishers) publishing groups with their infrastructure engineering department in their historic Flatiron Building location in the heart of Silicon Alley in Manhattan, New York City. While working at Holtzbrinck (an umbrella-term for sub-companies: Macmillan, Scientific American, St. Martin's Press, Nature (NPG)/Springer, et al.) and Macmillan, I wrote VBA macros for Office tools such as Microsoft Excel and Word to help the lives of many of the tech workers who needed to generate reports of their weekly collections of big data from their global servers for analysis. In my talk, I will demonstrate some of the macros I created this summer which mined and graphed data from a list of N-servers within Excel, split combined data within cells and extracted the necessary information to convert between BST and EST times while maintaining the proper corresponding dates in American (mm/dd/yyyy) format, charted a heat map based on employee hours worked, and cross-platformed between Excel and Word to auto-generate virtual machine ticket-request forms as Word documents based on an Excel workbook. By the end of my talk, I hope to convince you that New York City is the greatest city in the world (only 2^{nd} to Providence, of course), and show you how you can write scripts for everyday Microsoft Office applications to automate just about any task.

Substitutions, Rauzy Fractals, and My Summer Research Experience

Meghan Malachi

This summer I attended a Research Experience for Undergraduates (REU) at California State University Channel Islands. Throughout the program, my team and I researched various topics in Symbolic Dynamics, a growing field in Mathematics that focuses on the study of shift spaces and modeling smooth dynamical systems using infinite sequences comprised of finitely many symbols. My team focused specifically on the classification of sequences generated by a non-erasing homomorphism, known as a substitution. We worked to classify such sequences with similar corresponding Rauzy fractals. In this talk I will provide an introduction to our work on substitution shifts and Rauzy fractal representations. I will also reflect on my overall experience at CSUCI and the many benefits of attending an REU.

**October 16, 2015**

Lions and Tigers and Graph Colorings, Oh My!

Cayla McBee, Ph.D.

Graph theory is an area of mathematics that has been used to solve a host of problems in a variety of fields such as computer science, biology and sociology. Additionally, advances in computing have allowed researchers to apply graph theory to increasingly complex problems. Some real-world problems can be analyzed in terms of graph coloring - the assignment of colors to the vertices or edges of a graph in such a way that adjacent vertices or edges incident to the same vertex have distinct colors. My talk will provide the necessary background to discuss coloring problems. We will also examine how everything from designing a zoo to coloring a map can be thought of in terms of graph coloring.

This talk will be accessible to individuals who have not had graph theory before and students are encouraged to attend.

**October 30, 2015**

Statistics and the Overtime in the National Football League

Nicholas Gorgievski, Ph.D.

Nichols College

Over the past decade, football fans, sportscasters, writers, and mathematicians all weighed in on whether the NFL "sudden death" overtime rule was fair to both teams. Many claimed that the winner of the coin toss had an advantage in winning the game. As a result of all of the debate, the NFL owners accepted a proposal to alter the overtime rule for postseason games beginning in the 2010-2011 football season and for regular season games beginning in the 2012-2013 season. However, was this rule change really necessary? In this talk, I will use statistics to examine the coin toss and its effect on the outcome of the overtime games.

Dr. Nicholas Gorgievski graduated from Providence College in 1998 with a major in Mathematics. He continued his studies at the University of Vermont and graduated with a Master of Science in Mathematics. He then earned his Ph.D. in Curriculum and Instruction specializing in Mathematics Education from the University of Connecticut. Dr. Gorgievski is currently an Associate Professor and Chair of the Mathematics Department at Nichols College in Dudley, MA. His research interests include the teaching and learning of Mathematics at the university level. He also enjoys playing tennis, watching sports, and spending time with his wife and two daughters.

** **

**Spring 2016**

**January 29, 2016**

Experiments in Mathematics: How the Computer Is Changing Everything

Professor Jill Pipher

Brown University

I'll describe some classical and important contributions of the computer to mathematics, from cryptography to the four-color theorem. Then I'll give a short tour of some modern, and surprising, interactions between mathematics and the computer. Accessible to a math-interested audience of undergraduates.

Jill Pipher is the Elisha Benjamin Andrews Professor of Mathematics at Brown University. Her primary area of research is harmonic analysis and its applications to elliptic differential equations. She also has research interests in cryptography and is a coauthor *of An Introduction to Cryptography* published by Springer. She is currently the director of the Institute for Computational and Experimental Research in Mathematics at Brown.

** **

**February 19, 2016**

Singing Along With Math

Professor T. Christine Stevens

American Mathematical Society

The opera singer Jerome Hines, who died in 2003, sang at the New York Metropolitan Opera for over forty years. He was also a math major who retained a lifelong interest in mathematics. In the 1950's he published several papers that were based on work that he had done as a student. I'll focus on the first of these papers, which describes a new method for finding the roots of an equation. I'll also discuss Hines' mathematical background and why he kept working on mathematics, even after he became a successful opera singer.

Professor Chris Stevens leads the Division of Meetings and Professional Services at the American Mathematical Society (AMS) here in Providence. Before joining the AMS in 2014, she was a professor of mathematics and computer science at Saint Louis University. Her research interests include topological groups and the history of mathematics, and she is a big opera fan.

**March 18, 2016**

Failures and Successes of Spatial Imagination: *Hamlet* and Early Modern Mathematics

Travis D. Williams, Associate Professor of English

University of Rhode Island

Shakespeare's most famous play and contemporaneous mathematics asked its audiences to exercise their spatial imaginations to "see" what was not literally there. This paper will consider how both literature and mathematics call upon the same imaginative resources, and what these similarities allow us to understand about artistic and mathematical creativity. My goal is to allow mathematics and literature to speak to and about one another, outside the traditional models of source and allusion, foreground and background, or text and context.

Travis D. Williams is Associate Professor of English at the University of Rhode Island. He has published on Shakespeare, Montaigne, the humanist dialogue and mathematics, the rhetoric of mathematical notation, and the printing history of the earliest English-language arithmetic books, among other topics. He is writing a book provisionally entitled *Literature, Mathematics, and the Writing Arts in the Age of Shakespeare*.

This talk is sponsored by the Mathematics & Computer Science Department and the English Department.

**April 15, 2016**

Confessions of a Mathematical Bibliophile

Professor Bruce Burdick

Roger Williams University

My book on early printed mathematics in the Americas is now seven years old. I will tell some stories about the research that I did for the book and about related projects that I undertook after the book went to the press. Some objects of interest will be displayed.

Bruce Burdick is a professor of mathematics at Roger Williams University in Rhode Island. He has also taught mathematics at schools in Ohio, Maine, and Hawaii.

He is the author of Mathematical Works Printed in the Americas: 1554-1700 and has authored or co-authored twelve mathematical papers. (His one co-authored paper gives him an Erdős number of six.) His writing credentials include as well two science fiction stories that were published in Analog Science Fiction/Science Fact.

An avid traveler, he has visited all fifty states of the U.S., nine out of the ten provinces of Canada, and recently chalked up his sixty-fourth country.

His book collection comprises many thousands of volumes. One unusual feature of the collection is that it includes over two thousand almanacs from around the world, whose publication dates represent a span of time from the seventeenth century to the present.

When not on the road, Bruce Burdick lives in Providence, Rhode Island where he enjoys the rich music and culinary scenes.

**ACADEMIC YEAR 2014 - 2015**

**Fall 2014**

**September 17, 2014**

Special REU Presentations

Introduction to Numerical Monoids and My PURE Math 2014 Experience

Meghan Malachi

This summer I spent five weeks as a Pacific Undergraduate Research Experience (PURE) Math intern at the University of Hawai'i at Hilo. I was introduced to the Factorization Theory of Numerical Monoids. Numerical monoids are co-finite, additive subsets of the set of natural numbers. Factorization theory of numerical monoids has recently become a pivotal as well as lucrative topic in mathematical research, as numerical monoids are non-unique factorization domains; there are several invariants for measuring how many ways to factor a particular element in a numerical monoid. These non-unique factorizations provided a basis for my team's research on irreducibility and primality within numerical monoids. This talk will provide an introduction to numerical monoids, a presentation of my research team's ultimate findings on primality, and a brief description of my overall experience with PURE Math this summer.

A Semester of Mathematics in Budapest

Tucker Kibbee

"Clearly, the Hungarian educational system has been the most successful for pure mathematics; it's a model that ought to be studied very carefully because it works." - Donald Knuth, *Two Year College Mathematics Journal, Vol. 13*

This past spring, I studied abroad in a math program called Budapest Semesters in Mathematics (BSM). The talk will involve a complete (and hopefully not too nostalgic) overview of the program and my stay in Budapest, including courses, professors, food, drink, and fun. I will have veteran tips for the aspiring BSM-er as well as some funny stories and some not-so-funny stories that should entertain those who have returned from their study abroad and instruct those yet to embark. BSM is a unique program in a unique city, and in my talk I will try to do justice to the cultural and educational experience that was my semester in Budapest.

**September 24, 2014**

Special REU Presentations

Computational Modeling of the Thyroid Hormones Homeostasis and Its Manipulation by Chemicals

Kai Bartlette

This summer, I attended an REU program in Modeling and Industrial Applied Mathematics at North Carolina State University. During my 11-week stay, my research team focused on developing a model to show how the drug propylthioracil (PTU) inhibits hormone production in the thyroid gland. Propylthiouracil is usually a treatment for hyperthyroidism. To describe the interaction between propylthiouracil and these hormones, a thyroid hormone production model was created adapted from Ekerot et al. comparing thyroid hormones in dogs. A physiologically-based pharmacokinetic (PBPK) model was then created to provide a basis for concentration equations. After solving the differential equations for drug concentrations in each organ, a dynamic model for thyroid hormone production was created by connecting the thyroid model to the PBPK model.

The Plus-Minus Weighted Davenport Constant

Benjamin Wright

This 2014 summer, I was part of a team investigating the plus-minus weighted Davenport constant of finite abelian groups in an REU program at Fairfield University. The plus-minus weighted Davenport constant of a finite abelian group,
*G*, is the least natural number,
*k, *such that every sequence of length
*k* in
*G* has plus-minus weighted zero sum subsequence. In this talk, I will introduce the general bounds on the plus-minus weighted Davenport constant, describe the experience and process of working on the problem, and present the results of our work at Fairfield. There will certainly be no time for humorous anecdotes in this talk.

**October 8, 2014**

Mathematics and Planet Earth

Professor Catherine Roberts

College of the Holy Cross

What role do mathematicians play when it comes to understanding nature? Weather prediction, climate change, understanding how people interact with the environment, how diseases spread and how populations interact are just a few examples of how mathematics can help us understand the quantitative elements of complex problems. Professor Roberts will describe the steps used in mathematical modeling and then explain how they were used to help the Grand Canyon National Park figure out how to schedule white-water rafting trips down the Colorado River.

Catherine grew up on Cape Cod and has her Ph.D. in applied mathematics from Northwestern University. She is professor and chair of the mathematics and computer science department at College of the Holy Cross. She is editor-in-chief of a scientific journal called Natural Resource Modeling and serves on other editorial boards, such as on the MAA's Monthly.

**October 22, 2014**

The Polar Planimeter

Lynette J. Boos

In 1854 the Swiss mathematician Jakob Amsler built a cleverly designed instrument for computing the area of an arbitrary two-dimensional shape. We will discuss the history of this instrument and consider the mathematics behind its function.

**November 19, 2014**

The Mathematics that George Washington Studied

Professor Frederick Rickey

U.S.M.A. at West Point

As a teenager, George Washington compiled two notebooks (179 pages) about the mathematics he studied: decimal arithmetic, geometry, logarithms and trigonometry, and surveying. We will examine an example of each of these, pointing out (when we can) where the explanations and problems came from, as well as a few errors in the manuscript.

These notebooks have had a hard life. For half of the nineteenth century they were on loan to biographers who treated them casually and dispersed some pages. We shall describe some of our success in locating these missing pages. Once they were sold to the government they were disbound and the pages reordered. This troublesome issue will also be described.

V. Frederick Rickey, a logician turned historian, retired in July 2011 because he could not get any work done while working. Now he is so busy with historical research that he still can't get everything done. After earning three degrees from the University of Notre Dame (Ph.D. 1968) he went to Bowling Green State University where he rose through the professorial ranks to become a Distinguished Teaching Professor Emeritus. In 1993 he received one of the first MAA Haimo Awards for Distinguished College or University Teaching of Mathematics. He "retired" in 1998 and joined the faculty at West Point (where the cadets called him "Sir" far too often).

He has been on leave six times, including as Visiting Mathematician at the MAA Headquarters. While there he was involved in the founding of Math Horizons, built its first gopher and web pages, and wrote a successful NSF grant for an Institute for the History of Mathematics and Its Use in Teaching (IHMT).

He has broad interests in the history of mathematics and is especially interested in the development of the calculus. He loves teaching and enjoys giving lectures to mathematicians and mathematics students about the history of their field (something he continues to do in retirement).

**Spring 2015**

**February 4, 2015**

Some Facts and Some Open Problems and Conjectures on Rational Systems

Emmanouil Drymonis, Ph.D.

A difference equation is an equation where future values of a system depend on present and past values. This talk is about the dynamics of a special class of systems of two difference equations that we call Rational Systems in the Plane. We present some facts and some open problems and conjectures on rational systems. We are primarily interested in the boundedness nature of solutions, the periodic character of the equation, the global stability behavior of the equilibrium points, in invariants, and in convergence to periodic solutions. We believe that the rational systems that we study are genuine examples which provide prototypes for the development of the basic theory of nonlinear difference equations. Applications of difference equations and of systems of difference equations arise in such diverse disciplines as biology, meteorology, economics, numerical solutions of complex systems, and others. The field of difference equations is a fertile research area with many conjectures and open problems which have not been answered yet. In this area of research there is an abundance of projects for undergraduate and graduate students.

**February 11, 2015**

Indexing Large Text Collections Using the Vector Space Model

Kevin C. O'Kane, Professor Emeritus

University of Northern Iowa

From the multimillion document collections of research reports, memos, and email at large corporations, to the nearly endless mishmash of Internet web pages, tweets, texts and cat photos, someone, somewhere wants to search them. To accomplish this, we need efficient ways to identify, organize, store, search, and retrieve content. One of the most widely used approaches to this problem is called the
*vector space model*. It works by constructing, from the indexing vocabulary itself, a multidimensional hyperspace whose axes (possibly thousands) correspond to the terms of the indexing vocabulary. A point along an axis indicates the importance of a term. Each document in the collection is then expressed as a vector whose components correspond to the terms of the vocabulary, the values of which indicate the importance of the corresponding term in the document. Thus, based on its component vocabulary, a document vector describes a point in the hyperspace. Queries are likewise rendered as vectors which also define points in the hyperspace. Documents that lie within an adjustable, multidimensional envelope from the queries are retrieved and ranked according to their distances from the query points. The first use of this model was in the SMART System (Salton 1988, 1992) and it has been the basis of many implementations since.

The talk will discuss the model with examples drawn from its application to a collection of 293,000 medical abstracts (sorry, no cat pictures). Topics will include:

- Zipf's Law,
- Word frequency analysis, stop list generation, and word stemming,
- Term weighting: Inverse Document Frequency weights, and discrimination coefficients,
- Similarity functions,
- Construction of document-term, term-document, term-term, and document-document matrices,
- Synonym and phrase identification,
- Term and document clustering,
- Database implications: SQL or NoSQL?
- Document retrieval.

Salton, G., (1988)
*Automatic Text Processing*, Addison-Wesley, Reading.

Salton, G., (1992) The state of retrieval system evaluation,
*Information Processing & Management*, Vol. 28, No. 4, pp. 441-449.

Dr. O'Kane is former professor and head of Computer Science at the University of Northern Iowa and, prior to that, professor and head of Computer Science at the University of Alabama. He also taught at the University of Tennessee and the Ohio State University College of Medicine. He is the author of about 50 publications in the areas of information retrieval and medical informatics. He received his S.B. in chemistry from Boston College and his Ph.D. in computer science from the Pennsylvania State University. He is the author of an open source IS&R workbench as well as an open source compiler and interpreter for the Mumps language.

**February 18, 2015**

Seeing the Imperceptible: Problems in Information Visualization

Professor Michael Gousie

Wheaton College

With the proliferation of data, there are growing challenges in finding the information that is truly useful. Information Visualization is a branch of computer science that seeks to create visual representations of abstract data. These applications often include options for the viewer to interact with the system, thereby, hopefully, uncovering some golden nuggets of information. In this talk, we will look at several information visualization systems that show large data sets in unique graphical ways. These systems range from applications freely available on the web to more specialized programs developed at Wheaton College. In all cases, the systems allow users to see data and/or patterns in the data that are imperceptible using traditional methods.

Michael Gousie is a Professor of Computer Science at Wheaton College, MA. He earned his BA at Providence College and his MS at the University of New Hampshire. Being confused as to where he was headed, he went back to Providence to teach for a few years. He then received his Ph.D. at Rensselaer Polytechnic Institute, working on problems involving 3D data in Geographic Information Systems (GIS). He continues to work on GIS problems, but these now often include an information visualization component. More recently, he has done extensive interdisciplinary work developing visualization software for various kinds of data sets.

**March 4, 2015**

Linear Algebra and Forensics

Donna Beers, Ph.D.

Simmons College

Have you taken a selfie? Or have you ever doctored a photo of yourself so that you look taller or thinner or free of blemishes or other imperfections? With the ready availability of digital technology, we have all become photographers; and, free, online photo editing tools allow us to alter our pictures to suit our pleasure. The downside of this, however, is that when we look at photos, whether in tabloids or on the Web, we wonder: Is this a fake? For, just as you can alter pictures to make yourself look more attractive, so, too, politicians, advertisers, and others with particular agendas are using digital imaging tools to manipulate photos to create false impressions. Determining whether a photo has been doctored is just one of the many questions that photo forensics tries to answer.

The good news is that linear algebra offers powerful tools for carrying out digital photo forensics. In this talk we will explore two applications: detection of explicit images through use of the RGB color model and image reconstruction from compressed data files using the Inverse Discrete Cosine Transform.

An algebraist, Donna Beers, who is Professor of Mathematics at Simmons College, teaches a range of courses including linear and modern algebra, discrete mathematics, calculus and real analysis, and mathematics for elementary school teachers. Her research publications are in the areas of group theory and group algebras of infinite abelian groups. Her scholarly interests also include inquiry into student learning to strengthen teaching. Promoting undergraduate research is a vital aspect of her teaching and she has supervised numerous independent studies and internships. To honor the accomplishments of Simmons College students in mathematics, she initiated and serves as adviser to the Massachusetts Theta Chapter of Pi Mu Epsilon. In October 2012, she was honored as the inaugural recipient of the Toby Sloane Award for Student-Centeredness in Teaching.

**March 18, 2015**

Symmetry Breaking in Graphs

Kathleen McKeon, Ph.D.

Connecticut College

Suppose you have
*n* indistinguishable keys on a circular ring. How many keys do you need to mark with different colors so you can distinguish them from one another? Translating this puzzle to a graph theory problem leads to the following question. Given a graph G, what is the minimum number of colors that must be assigned to its vertices in order to destroy all the symmetries of the graph? This is called the distinguishing number of G, D(G). We'll look at the distinguishing numbers of different families of graphs and the relationship between the symmetry group and the distinguishing number of the graph.

Dr. McKeon is a Professor of Mathematics at Connecticut College, where she has taught since receiving her Masters and Ph.D. from Michigan State University. Dr. McKeon is a graph theorist and was first introduced to graph theory in her undergraduate abstract algebra course at Worcester Polytechnic Institute, where she earned her B.S. Her research in graph theory has included topics in enumeration, combinatorial generation, and assorted graph labeling and coloring problems. During her career at Connecticut College, she has served as Mathematics Department chair and associate director of the Holleran Center for Community Action and Public Policy. Most recently, Dr. McKeon and the other two female faculty in the department collaborated with a group of students to start a student chapter of the Association for Women in Mathematics.

**April 22, 2015**

Taxicabs and Sums of Two Cubes: An Excursion in Mathematics

Professor Joseph H. Silverman

Brown University

Some numbers, such as

9 = 1^{3} + 2^{3} and 370 = 3^{3} + 7^{3},

can be written as a sum of two cubes. Are there numbers that can be written as a sum of two cubes in two (or more) essentially different ways? This elementary question will lead us into beautiful areas of mathematics where number theory, geometry, algebra, calculus, and even internet security, interact in surprising ways. The talk will be accessible to undergraduates at all levels.

Dr. Joseph H. Silverman is Royce Professor of Mathematics at Brown University and a leading expert in the theory of elliptic curves. He received his Ph.D. at Harvard University in 1982 and has authored eight books and more than 100 research articles. Among his honors are Sloan and Guggenheim fellowships, a Steele Prize for Mathematical Exposition from the American Mathematical Society (AMS), a Distinguished Teaching Award from the Mathematical Association of America, and being named a Fellow of the AMS. Professor Silverman is also an active researcher in cryptography and a co-inventor of the NTRU cryptosystem. He has supervised 31 Ph.D. students and is an avid teacher and mathematics education advocate.

**ACADEMIC YEAR 2013 - 2014**

**Fall 2013**

**September 18, 2013**

Special REU Presentations

The Whirlpool Hash Function: Generalizations, Applications and Connections

Laura Wells

In the field of cryptography, it is essential to anticipate attacks that will result in insecure electronic communication. Hash functions are used for password storage, message integrity verification, pseudorandom number generation and non-repudiation in digital security. The Whirlpool hash function was developed in 2003 and endorsed by NESSIE, an international organization for selecting crytopgraphic functions for widespread use. In our research project, we generalized the standard version of Whirlpool and studied its algebraic properties. Knowing the algebraic structure of the function is particularly relevant in the development of future cryptographic systems that will meet the ever-increasing demand for higher security.

Summer Program for Women in Mathematics

June-August 2013

Mary Alice Sallah

This summer, I was invited to stay at George Washington University, taking four graduate level courses, which focused on different areas of study in the field of mathematics. Over the five week program, my classmates and I attended a series of guest lectures, completed nightly homework assignments, created individualized presentations for each course, and shadowed former participants at a variety of math-based careers. The program provided an excellent overview of the vast academic and nonacademic career possibilities available after undergraduate study, helping each participant find their path in the field of mathematics.

Using the Power Series Method to Study Delay Differential Equations with Chaos

Benjamin Weidenaar

The Power Series Method (PSM) which is similar to Automatic Differentiation (AD) uses Maclaurin polynomials and Cauchy products of polynomials to solve initial value (IV) ordinary differential equations (ODEs). There is extensive literature on the power of these two methods and their differences in solving IV ODEs. In this talk (paper) we use PSM (for the first time as far as we know) to solve delay differential equations (DDEs). We show how PSM has to be modified to handle delays and demonstrate the efficacy and robustness of the algorithm on some examples. In particular, we apply the PSM to the pursuit problem with and without delay and compare with other methods. We show that PSM preserves the properties of the pursuit problem more accurately. We discuss some of the known theory of chaos in DDEs. We demonstrate chaos for the pursuit problem through the time delay.

**September 25, 2013**

Special REU Presentations

Empirically Evaluating and Quantifying the Effects of Inspections and Testing on Security Vulnerabilities

Patrick Hilley

Currently developers and project managers do not have adequate empirical evidence to make informed, objective choices about the most effective and the most cost-efficient quality assurance methods for identifying security vulnerabilities. The software security engineering community is sorely in need of objective, quantitative information that will allow developers to make informed choices among available practices for vulnerability prevention and removal. Decades of software engineering research has produced evidence of the effectiveness of fault and failure prevention and removal practices, though essentially none exists for vulnerabilities.

We have learned from years of empirical study about general fault/failure detection that different approaches (i.e., testing and inspection approaches) exhibit different characteristics. Understanding which approach is more effective for identifying the type of defects that a developer expects to be present in his system provides great benefit when choosing the most appropriate quality assurance techniques for use on a project. There is a need for this type of objective, empirical evidence about the effects of various practices for vulnerability prevention and removal.

Example REU Projects on this topic include: (1) mining bug repositories to identify code segments that contain security vulnerabilities then tracing code history to determine which type of testing was applied, and (2) evaluating the effects of different testing techniques on identification of security vulnerabilities.

CUDA-Accelerated Brownian Dynamics Simulations

Justin Dufresne

Brownian Dynamics is a method of conducting molecular simulations which does not explicitly model particles, such as water, in which we are not interested. While Brownian Dynamics is much less time-consuming than other molecular simulation techniques, it is still prohibitively slow for large data sets. Parallel computing involves structuring programs in order to process many computations all at the same time rather than serially (that is, one after the other). CUDA, an extension of the C programming language, exposes the GPU's computation power to a program that would otherwise be run on a CPU, in order to process many mathematical operations in parallel. This summer, we hypothesized that it would be possible to greatly speed up Brownian Dynamics simulations by doing most of the calculations in parallel on a GPU using CUDA. In this presentation, we will discuss the theory and methods behind the parallel programming techniques we used, as well as the unique challenge we faced in applying these methods to Brownian Dynamics simulations.

An Inquiry-Based Approach to Teaching Parameterization

Nicole DeMatteo

Within Mathematics Education, there is often a disconnect between the actions of the instructors and the students within the classroom, manifested with students passively watching the instructor "do mathematics." A teaching approach called IBL (Inquiry-Based Learning), which encourages students to collaborate and become actively engaged in their learning, has been shown in education research to be effective for sparking students' curiosity and helping them to develop a deeper understanding of the subject. Our project seeks to develop an activity, based on research that employs IBL to teach parameterization of curves in space to a multivariable calculus class.

**October 9, 2013**

A Bayesian Approach to Detecting Change Points in Climatic Records

Eric Ruggieri, Ph.D.

College of the Holy Cross

Given distinct climatic periods in the various facets of the Earth's climate system, many attempts have been made to determine the exact timing of 'change points' or regime boundaries. However, identification of change points is not always a simple task. A time series containing
*N *data points has approximately N^{k}* *distinct placements of
*k *change points, rendering brute force enumeration futile as the length of the time series increases. Moreover, how certain are we that any one placement of change points is superior to the rest? In this talk, I'll describe a Bayesian Change Point algorithm which provides uncertainty estimates both in the number and location of change points through an efficient probabilistic solution to the multiple change point problem. To illustrate the algorithm, I'll talk about its application to the NOAA/NCDC annual global surface temperature anomalies time series which has often been cited as evidence of global warming, as well as a much longer 5 million year record of global ice volume.

Dr. Eric Ruggieri earned his B.A. in Mathematics and Computer Science from Providence College ('05) and his Sc.M. and Ph.D. in Applied Mathematics from Brown University. For the past three years, Dr. Ruggieri was an Assistant Professor of Statistics at Duquesne University before joining the faculty at Holy Cross this fall.

**October 30, 2013**

Newton's Method is Actually Very Cool!

Lisa Humphreys, Ph.D.

Rhode Island College

Most students see Newton's Method in first semester calculus, but few see the full potential of the elementary technique. Come learn how this can be extended and the interesting mathematical questions that arise. See how multivariable calculus, linear algebra, differential equations, and advanced calculus all come into play. We will even apply Newton's Method to a model of the famous Tacoma Narrow's Suspension Bridge.

Lisa Humphreys earned a Ph.D. in mathematics from the University of Connecticut. Her research has primarily been in the area of nonlinear differential equations. Most of that work is centered around models of suspension bridges. Early in her career she focused on proving the existence of solutions in the area of degree theory and then using variational methods and mountain pass techniques to find numerical solutions. After a successful honor's project with an undergraduate at RIC, she changed gears (or bifurcated!) and began studying continuation methods. She has published numerous results both numerical and theoretical including several in
*Nonlinear Analysis*,
*The American Mathematics Monthly* and
*The College Mathematics Journal*. In recent years, Lisa has been active in professional development for in-service mathematics teachers. She has given many conference presentations and professional development workshops for teachers as well as dozens of lessons directly to elementary and middle school students.

**November 13, 2013**

An Introduction to
*p*-adic Numbers

Daniel Ford

Boston University

We begin by reviewing the construction of the real numbers, highlighting a rather naive and uninteresting choice involved in their realization. We can then overcome this naivety and construct the
*p*-adic numbers themselves. The
*p*-adic numbers which are also rational can be thought of as a "base
*p* expansion", as opposed to the typical fractions of integers. The implications of
*p*-adic numbers include the amazing proof of Fermat's Last Theorem developed by Sir Andrew Wiles in 1995. While Wiles' proof lays outside the realm of understanding for most mathematicians, an interesting question called the Local-Global Principle is well within reach, and we will discuss some of its marvelous answers.

Daniel Ford graduated from Providence College as a mathematics major in 2010. He is currently a fourth-year graduate student and Teaching Fellow at Boston University. He is studying number theory, which in its most basic form studies properties of the integers and extensions thereof. Beyond his doctoral studies, he works with a great program called PROMYS for Teachers, which is a 6-week summer course in elementary number theory for high school mathematics teachers. The program is steeped in history and strives to combine higher mathematics with the basic high school education.

**Spring 2014**

**February 12, 2014**

Games, Competition, Cooperation, and Evolution

Professor Charles Hadlock

Bentley College

In a talk suitable for students at all levels, I will first give a brief introduction to game theory, illustrated through some common but challenging situations. I will then discuss the fascinating interplay between competition and cooperation in the framework of the famous prisoners' dilemma game and other complex situations that often defy intuition. I will also show how similar situations arise in business, biology, and many other fields, where mathematics is a vital tool for increased understanding.

Charlie Hadlock graduated from Providence College in 1967 and received his Ph.D. in 1970 from the University of Illinois. At Providence College he played on the tennis team and was a member of the debating society. His introduction to game theory came from watching floormates manipulate the elevator in Aquinas Hall to avoid capture after dropping various noisemakers down the elevator shaft to harass the resident prefects. A few years after graduation, he was elected as the first alumni representative on the Providence College Corporation and participated actively in the decision to go coed. The broad arts and sciences education he got at Providence College has served him very well, enabling him to apply mathematical perspectives to real world problems in many different fields. He has had one career with the consulting firm of Arthur D. Little, specializing in environmental and risk issues, and another as a mathematics professor, teaching at various times at Amherst, Bowdoin, Wellesley, and MIT, as well as in Bogota, Colombia. Professor Hadlock has spent the last 20 years at Bentley University, where he has at times chaired both the mathematics and finance departments and served as dean of the undergraduate college. He has written four books published by the MAA.

**February 26, 2014**

Newton's Method: Complex Numerics and Complex Dynamics

Professor Paul Blanchard

Boston University

Newton's method is an iterative root-finding algorithm that is both simple and surprisingly efficient. We start with an initial guess for the root and apply the algorithm repeatedly until we obtain the desired approximation. Unfortunately, a random initial guess does not always lead to a root. In this talk, we use the theory of complex dynamics along with some computer graphics to explain the difficulties that might arise, and we suggest ways to avoid these pitfalls. As the story unfolds, we encounter both chaos and fractals.

Professor Paul Blanchard grew up in Sutton, Massachusetts, spent his undergraduate years at Brown University, and received his Ph.D. from Yale University. He has taught college mathematics for more than thirty years, mostly at Boston University. In 2001, he won the Northeastern Section of the Mathematical Association of America's Award for Distinguished Teaching of Mathematics, and in 2011, the conference "Differential Equations Across the Collegiate Curriculum" was held to celebrate his 60^{th} birthday. He is a Fellow of the American Mathematical Society.

Professor Blanchard's main area of mathematical research is complex analytic dynamical systems and the related point sets---Julia sets and the Mandelbrot set. For many of the last twenty years, his efforts have focused on modernizing the traditional sophomore-level differential equations course. When he becomes exhausted fixing the errors made by his two coauthors, he heads for the golf course to enjoy a different type of frustration.

**March 26, 2014**

Smooth Centers

Su-Jeong Kang

Since the announcement in 1950, the Hodge conjecture has been modified and generalized a few times. In this talk, I will discuss several versions of the Hodge conjecture and the smooth center that we define to understand algebraic cycles on a singular variety. If there is time left, I will also discuss our conjecture on the smooth center of K-groups.

**April 2, 2014**

Sums and Products

Professor Carl Pomerance

Dartmouth College

What could be simpler than to study sums and products of integers? Well maybe it is not so simple since there is a major unsolved problem: For any positive epsilon and arbitrarily large numbers N, is there a set of N positive integers where the number of pairwise sums is at most N^{2-epsilon} and likewise, the number of pairwise products is at most N^{2-epsilon}?

Erdos and Szemeredi conjecture no. This talk is directed at another problem concerning sums and products, namely how dense can a set of positive integers be if it contains none of its pairwise sums and products? For example, take the numbers that are 2 or 3 mod 5, a set with density 2/5. Can you do better? This talk reports on recent joint work with P. Kurlberg and J. C. Lagarias.

Carl Pomerance received his B.A. from Brown University in 1966 and his Ph.D. from Harvard University in 1972 under the direction of John Tate. Currently he is the John G. Kemeny Parents Professor of Mathematics at Dartmouth College, after previous positions at the University of Georgia and Bell Labs. A number theorist, Pomerance specializes in analytic, combinatorial, and computational number theory, with applications in the field of cryptology. He considers the late Paul Erdos as his greatest influence.

**April 23, 2014**

Moving Big Data around the Globe

Adam Villa

Transmitting data via the Internet is a routine and common task for users today. The amount of data being transmitted by the average user has dramatically increased over the past few years. Transferring a gigabyte of data in an entire day was normal; however users are now transmitting multiple gigabytes in a single hour. With the influx of "Big Data", a user has the propensity to transfer even larger quantities. When transferring data sets of this magnitude on shared networks, the performance of the entire system will be impacted. This talk will discuss the issues and challenges inherent with transferring Big Data over shared networks. Various transfer techniques for moving large amounts of data will be examined and methods for estimating their performance will also be discussed. There are many complexities to long distance and long duration transfers that need to be considered. We will see that moving big data around the globe is not trivial.

**ACADEMIC YEAR 2012 - 2013**

**Fall 2012**

**September 26****, 2012**

A Geometric Vision of Relations on the Grothendieck-Teichmuller Group

Sheldon Joyner, Ph.D.

Brandeis University

One approach to the study of the absolute Galois group of the field of rational numbers is to consider its embedding into a group first defined by Drinfel'd known as the Grothendieck-Teichmuller group, GT. Here I will discuss an approach to understand the relations on this group using paths in specific moduli spaces. Simple relations on paths translate into relations on GT, via an action of groups of homotopy classes of paths on sections of certain bundles with connection on the relevant moduli spaces, by means of parallel transport along paths.

Sheldon Joyner is a South African number theorist. After completing his MSc at Stellenbosch University in South Africa, he came to the States in 2000, where he studied first at the University of Arizona and later at Purdue University. He spent three years in a postdoctoral fellowship at the University of Western Ontario, before returning to the States to lecture at Brandeis University.

**October 9****, 2012**

The Gauss Bonnet Theorem

Patrick Boland, Ph.D.

University of Michigan

The geometry of surfaces is a classical topic in mathematics. During the nineteenth century, a beautiful formula relating the curvature (a geometric notion) and the genus (a topological notion) of a surface was discovered. This Gauss Bonnet theorem has become a prototype for a wide variety of formulas that relate seemingly disparate areas of math. The talk will introduce the Euler characteristic and curvature of a surface in three dimensional space. We will discuss the Gauss Bonnet theorem in the context of several examples and outline a proof of the formula.

As time permits, we will talk about applications and generalizations of the theorem.

Patrick Boland is a 2003 graduate from Providence College. He obtained a Ph.D. in mathematics from the University of Massachusetts in 2010. He has been affiliated with the University of Michigan for the last three years as a postdoctoral researcher. His main research areas are geometry and topology.

**October 17****, 2012**

Delay Differential Equations

Jeffrey Hoag

When modeling a physical situation with a differential equation, it may be appropriate to introduce a time delay. This will be illustrated by some simple and not-so-simple examples with a focus on mathematical questions that arise when a delay is included in a differential equation.

**October 31****, 2012**

Fibonacci in Nature

Karam Habchi

Providence College Class 2012 Alumnus

Mathematics & Biology Double-Major

The Fibonacci sequence is the numbers in the following integer sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, etc. The first two numbers in the sequence are 1 and 1 (alternatively, 0 and 1) and each subsequent number is the sum of the previous two numbers. In mathematical terms, the sequence is defined by F_{n} = F_{n-1 }+ F_{n-2} where F_{0 }= 0 and F_{1 }= 1. The sequence is named after Leonardo of Pisa who was also known as Fibonacci. He first introduced the sequence in his book on arithmetic,
*Liber Abaci, *which was written in 1202 (although it wasn't named the Fibonacci sequence until the 19^{th} century). In the book, Fibonacci used the sequence to describe the growth of an idealized rabbit population and centuries later it was discovered that the Fibonacci sequence could describe many patterns that occur biologically in nature. We will explore some of these naturally occurring patterns, which include bee ancestry, nautilus shells, phyllotaxis (the arrangement of plant leaves), and plant seeding. We will also discuss the relationship between the Fibonacci sequence and the golden ratio, a number that has fascinated intellectuals since 500 BC, and how the golden ratio influences plant biology and its possible connection to human biology as well.

**November 7, 2012**

Who Has the Power in the Electoral College?

You Might Be Surprised

Professor Tommy Ratliff

Wheaton College

The Presidential election this year has again demonstrated the influence of the Electoral College in directing extraordinary attention to the so-called battleground states while the campaigns effectively neglect the rest of the nation. The conventional wisdom is that smaller states, like Rhode Island, would object to the elimination of the Electoral College because the inclusion of the two Senate seats in their Electoral College vote increase their power disproportionately compared to their population. However, if we examine the Electoral College as a two-tiered voting system and compare the power of individual voters across states, we will see that the conventional wisdom actually has it wrong. This talk will explore how power is measured in a two-tier system, review several proposals for equalizing the voters' power in the Electoral College, and determine the impact of the proposals on the election this year.

Tommy Ratliff is a professor of mathematics at Wheaton College in Norton, Massachusetts. His current research is in voting theory, with a focus on issues related to electing groups of candidates as in committee elections. He completed his Ph.D. at Northwestern in algebraic topology and held visiting positions at Kenyon College and St. Olaf College before joining the faculty at Wheaton in 1996.

He has been active in the Northeastern Section of the MAA, including serving as Chair of the Section 2005-2007, and he is currently serving as the Governor for the Section.

**November 28****, 2012**

Beyond P and NP - Computational Complexity Hierachy

Frank Ford

The holy grail of theoretical computer science is the solution of the P=NP question. Researchers have defined many other classes of problems using time and space and even types of proofs. This talk will describe some of these classes and discuss their relationships. A common picture of these classes is shown below. If time permits, we will try to discuss the class IP which is often defined in terms of Arthur and Merlin trying to verify proofs.

**Spring 2013**

**February 6****, 2013**

Saving the World One Equation at a Time

A Brief Introduction to the Mathematics of Missile Defense

Professor Jessica Libertini

University of Rhode Island

Since officially withdrawing from the Anti-Ballistic Missile Defense Treaty in 2002, the United States has worked to answer the technical challenge of defending ourselves, our friends and allies, and our deployed forces around the world from a ballistic missile attack. The mission starts rather simply - a missile is traveling on a ballistic (and therefore predictable) trajectory, and we want to stop it. So, why is it so hard? After a brief hands-on demonstration of the complications of the missile defense problem, we will explore additional challenges, such as: what are the chances we will be able to see the threat and determine its path; how can we get there in time and still hit it hard enough to knock it out of commission; where should I put my sensors and interceptors to maximize the likelihood of success; how many of each kind of asset do I need to maximize success while constraining cost? Even more fundamentally, how do I quantify success? Come see how lessons from undergraduate mathematics courses, such as calculus, probability, optimization, and geometry, can be applied to address this very real and important issue.

Dr. Jessica Libertini earned her B.S. and M.S. in Mechanical Engineering (Johns Hopkins and Rensselaer Polytechnic) and her Sc.M. and Ph.D. in Applied Mathematics from Brown. She spent 9 years working for General Dynamics on projects ranging from submarines to satellites, including several years as a representative to the U.S. Missile Defense National Team where she provided technical analysis to decision makers in Congress, the Office of the Secretary of Defense, and the Office of the President of the United States. In academia, Dr. Libertini has held postdoctoral positions with the U.S. Military Academy (West Point) and the National Research Council. She currently holds a faculty position at the University of Rhode Island. Most importantly, Dr. Libertini enjoys talking with students of all ages, sharing the value of mathematics as it applies to the problems facing our complex world today.

**February 13****, 2013**

*p*-adic Analogues of the Mandelbrot Set

Jacqueline Anderson

Brown University, Ph.D. in May 2013

Providence College Class 2008 Alumna

The Mandelbrot set is the set of complex numbers c such that set {0, f(0), f(f(0)), ...} is bounded when f(z)=z^2+c. It is well-known for its fascinating geometric and dynamical properties. I will begin by describing some of these properties. I will then briefly introduce the p-adic numbers, a number system that is built from the rational numbers much in the same way as the real numbers are built from the rationals, but with very different properties. Finally, I will define a p-adic analogue of the Mandelbrot set and I will outline some of my research exploring the size and structure of these sets and how they compare to the classical Mandelbrot set.

**March 11****, 2013**

Instabilities in Smectic Liquid Crystals under the Magnetic Field

Sookyung Joo, Ph.D.

Old Dominion University

Liquid crystal phases form when a material has a degree of positional or orientational ordering yet stays in a liquid state. The increasing demand for video displays has led to the development of devices employing smectic liquid crystals. We present mathematical theories of liquid crystals and consider the minimizers of the energies and solutions of the governing equations as a way to describe the influence of the temperature or applied ﬁelds. We characterize the critical ﬁeld and obtain the asymptotic expression of unstable modes using the Γ-convergence theory. If a magnetic/electric ﬁeld is applied in the direction parallel to the smectic layers, an instability occurs above a threshold magnetic ﬁeld. When the ﬁeld reaches this critical threshold, periodic layer undulations are observed. We study this phenomenon analytically by considering the minimizer of the second variation of the Landau-de Gennes free energy at the undeformed state and by carrying out a bifurcation analysis of the nontrivial solution curve. We also perform numerical simulations to illustrate the results of our analysis. Numerical simulations in 2D and 3D near and well above the critical ﬁeld will be presented.

Dr. Sookyung Joo earned her B.S. and M.S. in Mathematics from Ewha Womans' University in Korea and her Ph.D. in Mathematics from Purdue. She continued her work at IMA, University of Minnesota and UC Santa Barbara as a postdoctoral position. She has worked at Old Dominion University as an assistant professor since 2010. Her research interest is partial differential equations motivated from material science such as liquid crystal and superconductivity.

**April 3****, 2013**

The Cayley Bacharach Theorem through the Ages

Bernadette Boyle, Ph.D.

Sacred Heart University

Algebraic Geometry is an area of mathematics which uses knowledge of both algebra and geometry to study mathematical concepts. One classic theorem in algebraic geometry is the Cayley Bacharach Theorem which gives us information about the intersection points of simple curves. In this talk we will learn about this theorem and see how it has developed and been generalized by mathematicians over hundreds of years - there will be plenty of pictures!

Dr. Bernadette Boyle graduated from Providence College in 2007 with a major in Mathematics (and minor in Theology). She then continued her studies at the University of Notre Dame in South Bend IN. She graduated with her Ph.D. in Mathematics with a specialty in Commutative Algebra and Algebraic Geometry in 2012. She is now an Assistant Professor at Sacred Heart University in Fairfield CT. Outside of math, she enjoys running, sailing, and ultimate frisbee.

**April 17****, 2013**

An Episode in American Mathematics:

The Statistical Contributions of C.H. Kummell

Jim Tattersall and Asta Shomberg

According to the historian of statistical methods, Stephen Stigler, the years 1770 to 1850 were ones of great interest in statistics in Europe, whereas statistical research in the United States did not reach its height of development before the latter half of the nineteenth century, when the westward migration of the American populace increased the need for more reliable maps and hydrographic charts. The establishment of the United States Coast Survey, the Lake Survey, and the Nautical Almanac in 1807, 1841, and 1849, respectively, set off the rapid advancement of interest in statistics and the development of statistical methods, which was pioneered by Robert Adrain, Benjamin Peirce, his son Charles Sanders Peirce, Simon Newcomb, and Erastus Lyman De Forest. Stigler described their statistical accomplishments, but admitted that his account was incomplete for he had omitted C.H. Kummell's and R.J. Adcock's work. We note Adcock's contribution and focus on the achievements of Charles Hugo Kummell, a statistician for the Lake Survey and the U.S. Coast and Geodetic Survey. We describe his research into laws of errors of observations and his contributions to the development of the least squares method. In addition, we offer some details on his life and give an account of his efforts to promote the subject of statistics while a member of the Philosophical Society of Washington.

**ACADEMIC YEAR 2011 - 2012**

**Fall 2011**

**September 16, 2011**

Maker Breaker Games on Graphs and Matroids

Jenny McNulty

The University of Montana

A Maker-Breaker game, often called a tic-tac-toe game, is a 2 person game in which Maker tries to "make" something and Breaker who goes second wants to prevent Breaker from doing this. In Tic-Tac-Toe, the first person (let's say X) is trying to "make" a line, that is get three X's on a horizontal, vertical and diagonal line. The second player (O) has played enough times to know she can never "make" a line, the best she can do is to prevent X from making a line, her strategy is to "break" the lines and prevent X from winning. Unlike traditional tic-tac-toe, we'll say X wins if he makes a line and O wins if she breaks all the lines. We'll consider games of the sort on graphs and matroids. In these games we'll change goal for Maker that is he will not be always trying to make a line, but instead some other object (like a forest, cycle, fixing set, proper coloring etc.) This is an introductory talk which will involve hands-on game playing. All terminology mentioned above will be explained. This is joint work with my undergraduate and graduate students.

Jenny McNulty is a graduate of Providence College. After receiving her BA in Chemistry and Mathematics at PC, she went on to SUNY Stony Brook where she obtained an MA in Mathematics and then to the University of North Carolina. At UNC, she completed her Ph.D. in matroid theory. In 1993, Dr. McNulty started her career at The University of Montana, currently holding the position of Associate Dean. At UM, Dr. McNulty has taught numerous courses, supervised many students, and organized numerous seminars and outreach programs; her favorite activity among all of these is sharing her love of mathematics. Her research area is Combinatorics, with an emphasis on Matroid Theory; she is currently writing a book, designed especially for undergraduates, on this subject. While Montana is far away from her native Long Island home, she has learned to love her new home and is an avid kayaker and ice hockey player.

**October 19, 2011**

Perception of Pain and Discomfort from Two Types of Orthodontic Braces

Asta Shomberg

Orthodontic procedures are often accompanied by pain and discomfort. Since the orthodontic treatment causes some degree of suffering for the patients, it is important for orthodontists to explore all possibilities of avoiding, reducing or alleviating pain in orthodontics.

Pain is a subjective response, which is difficult to measure. It shows large individual variations, which depend upon patient's age, gender, individual pain threshold, cultural differences, present emotional state, etc. In spite of this, patients' responses to different types of arch-wire or separators, two types of orthodontic appliances, have been investigated and described in the literature.

The aim of the present study was to evaluate the clinical properties of two types of orthodontic brackets with regard to pain and discomfort experienced by patients. Nonparametric statistical tests were employed to compare pain perception levels in two groups of orthodontic patients. The results showed that there is a statistically significant difference in pain levels due to the difference of bracket type. Further objectives of the study were to examine the association between perception of pain and gender, teeth discrepancy index, size of arch-wire and presence of extractions.

**November 9, 2011**

Ecological Modeling with Graph Theory

Cayla McBee

Graphs serve as mathematical models for many real-world applications occurring in a number of fields including computer science, ecology, and sociology. In this talk I will cover a basic introduction to graph theory and examine specific applications of graph theory to the field of ecology. Recent studies in ecology have used graphs to model the spread of disease through a population. We will look at how graphs with different properties can be used to predict disease dynamics on human contact networks.

**November 30, 2011**

What Is Computable and What Kinds of Machines Are Needed for the Computation?

Frank Ford

Some people define computer science as the study of algorithms and the machines that compute them. Jennifer Wing, President's Professor of Computer Science at Carnegie Mellon University expressed it as "Computer science is the study of computation—what can be computed and how to compute it."

Alan Turing approached computability by constructing Turing Machines to abstract what was needed to do computation. Noam Chomsky approached a different problem, understanding the parsing of languages, by creating a hierarchy of languages each of which can be described using different kinds of rewriting rules. The two ideas converged and became equivalent and now form the basis of formal language theory in computer science.

This talk will give an introduction to this area. Turing machines, pushdown automata and finite-state transition machines will be matched with recursively enumerable, context-free languages and regular languages. These concepts appear in many courses including Operating Systems, Networks and Artificial Intelligence. Regular expressions even occur when viewing some handy commands in UNIX.

**Spring 2012**

**January 25, 2012**

The Fractal Geometry of the Mandelbrot Set

Robert L. Devaney

Boston University

In this lecture we describe several folk theorems concerning the Mandelbrot set. While this set is extremely complicated from a geometric point of view, we will show that, as long as you know how to add and how to count, you can understand this geometry completely. We will encounter many famous mathematical objects in the Mandelbrot set, like the Farey tree and the Fibonacci sequence. And we will find many soon-to-be-famous objects as well, like the "Devaney" sequence. There might even be a joke or two in the talk.

A native of Methuen, Massachusetts, Robert L. Devaney is currently Professor of Mathematics at Boston University. He received his undergraduate degree from the College of the Holy Cross in 1969 and his PhD from the University of California at Berkeley in 1973 under the direction of Stephen Smale. He taught at Northwestern University and Tufts University before coming to Boston University in 1980.

His main area of research is dynamical systems, primarily complex analytic dynamics, but also including more general ideas about chaotic dynamical systems.

He is the author of over one hundred research papers in the field of dynamical systems as well as a dozen pedagogical papers in this field. He is also the (co)-author or editor of fourteen books in this area of mathematics.

In 2012 he will become President-elect of the Mathematical Association of America. Then, in 2013-14, he will serve as the President of the MAA.

Professor Devaney has delivered over 1,500 invited lectures on dynamical systems and related topics in all 50 states in the US and in over 30 countries on six continents worldwide. He has also been the "Chaos Consultant" for several theaters' presentations of Tom Stoppard's play Arcadia. And, in 2007, he was the mathematical consultant for the Kevin Spacey movie called Twenty One.

For the last twenty years, Professor Devaney has been the principal organizer and speaker at the Boston University Math Field Days. These events bring over 1,000 high school students and their teachers from all around New England to the campus of Boston University for a day of activities aimed at acquainting them with what's new and exciting in mathematics.

In 1994 he received the Award for Distinguished University Teaching from the Northeastern section of the Mathematical Association of America. In 1995 he was the recipient of the Deborah and Franklin Tepper Haimo Award for Distinguished University Teaching at the annual meeting of the Mathematical Association of America. In 1996, he was awarded the Boston University Scholar/Teacher of the Year Award. In 2002 he received the National Science Foundation Director's Award for Distinguished Teaching Scholars. In 2002, he also received the ICTCM Award for Excellence and Innovation with the Use of Technology in Collegiate Mathematics. In 2003, he was the recipient of Boston University's Metcalf Award for Teaching Excellence. In 2004 he was named the Carnegie/CASE Massachusetts Professor of the Year. In 2005 he received the Trevor Evans Award from the Mathematical Association of America for an article entitled Chaos Rules published in Math Horizons. In 2009 he was inducted into the Massachusetts Mathematics Educators Hall of Fame. And in 2010 he was named the Feld Family Professor of Teaching Excellence at Boston University.

**February 15, 2012**

One versus Several Complex Variables

Lynette Boos

What is complex analysis with several variables? Is it just like one complex variable but with more indices? In fact, many theorems that hold in one variable do not hold in several variables, and vice versa. In this talk we will present some results in several complex variables. Along the way, we will explore some of the differences between one and several complex variables and hopefully convince you that complex analysis with two variables is more than twice the fun!

This talk will be accessible to anyone who stayed awake during most of Calculus III.

**March 7, 2012**

Meredith Burr, Ph.D.

Rhode Island College

Providence College Class 2005 Alumna

The Central Limit Theorem explains that the normal or "bell" curve occurs so frequently in nature because the normal random variable is an attractor. More precisely, the theorem says that the sum of a large number of independent and identically distributed random variables with finite variance tends to a normal distribution. But what happens if the variance is infinite or if the random variables have an infinite mean? In this talk, we will look at a group of central limit theorems with non-normal attractors. We will look at these central limit theorems for random variables as well as for stochastic processes. Emphasis will be on visual displays as well as on examples of the anomalous behavior in nature. Some applications of the limit theorems will also be discussed.

**March 28, 2012**

Tangent Lines: A Multifaceted Concept

Peter J. Byers

Over the last few decades, math education has increasingly emphasized making connections between different topics. Thus, a key part of improving teaching is improving connections and deciding which connections are most important. In a Calculus or Pre-Calculus class, tangent lines can be connected with each of the following topics:

- Tangent lines to circles, as taught in 10th grade Geometry

- High school velocity and rate problems

- Graphing calculators

- Double roots of polynomials

- Continuous functions and limits

- Derivative as a number and as a function

- Exponential functions (including problems in interest, population growth, radioactive decay, etc.)

- Advanced topics such as inverses, linear approximation, and the chain rule for compositions of functions

This presentation will consider what changes can be made in order to teach these connections more effectively, and particularly how tangent lines to exponential functions can be treated with geometric (non-calculus based) techniques like scaling.

**April 18, 2012**

The Hodge Conjecture

Su-Jeong Kang

In 2000, the Clay Mathematics Institute published a list of seven important unsolved mathematics problems. Because of their importance the problems on the Clay Institute's list were titled the Millennium Prize Problems and the Hodge conjecture was one of the seven problems singled out for recognition. As of today, six of the problems, including the Hodge conjecture, remain unsolved with the Poincaré conjecture being the only one of the Millennium Prize Problem to have been solved.

The Hodge conjecture, originally formulated by the Scottish mathematician William V.D. Hodge and later modified by several other mathematicians, including Alexander Grothendieck, *the *central figure in modern algebraic geometry, postulates the existence of a surprisingly intimate relationship between the geometry of complex algebraic varieties and the topology of complex algebraic manifolds. The conjecture has been shown to be true in several special instances but a general proof has eluded mathematicians for the last 60 years.

In this talk, I will introduce the historical development of this challenging, but beautiful problem. Several elementary examples will be discussed and we will prove the Hodge conjecture in these instances. If time permits, I will also discuss the progress of my project to generalize the Hodge conjecture to *bad* objects.

This talk is intended for a general audience, especially for those people who find great beauty in great mathematics!

**ACADEMIC YEAR 2010 - 2011**

**Fall 2010**

**September 22, 2010**

Numbers, Numbers, Numbers: From Nicomachus to Ramanujan

J.J. Tattersall

Two manuscripts, the* Introduction to Arithmetic*, by Nicomachus of Gerasa and
*Mathematics Useful for Understanding Plato* by Theon of Smyrna were written in the second century A.D. They were the main sources of knowledge of formal Greek arithmetic in the Middle Ages. The books are philosophical in nature, contain few original results and no formal proofs. They abound, however, in intriguing number theoretic observations. We extend some of the results found in these ancient works and introduce several types of numbers that lend themselves naturally to undergraduate research, in particular, happy, sad, polite, Demlo, Niven, decimal columbian, and highly composite numbers.

Jim Tattersall received his undergraduate degree in mathematics from the University of Virginia, a Master's degree in mathematics from the University of Massachusetts, and a Ph.D. degree in mathematics from the University of Oklahoma. On a number of occasions he has been a visiting scholar at the Department of Pure Mathematics and Mathematical Statistics at Cambridge University and a visiting fellow at Wolfson College, Cambridge. He spent the summer of 1991 as a visiting mathematician at the American Mathematical Society. In 1995‑1996, he spent eighteen months as a visiting professor at the U.S. Military Academy at West Point. He was given awards for distinguished service (1992) and distinguished college teaching (1997) from the Northeastern Section of the Mathematical Association of America (MAA). He is former President of the Canadian Society for History and Philosophy of Mathematics, and Associate Secretary of the MAA.

**October 6, 2010**

Estimation of False Discovery Rate under Parametric Assumptions with Application to DNA Microarrays

Asta Shomberg

Multiple hypothesis testing is one of many techniques used to analyze gene expression data. DNA microarray technology allows us to measure expression levels for several thousand genes simultaneously. This results in testing a few thousand hypotheses. The compound error measures, such as per comparison error rate, or family wise error rate, become too strict when the number of hypothesis tests is large. Consequently the hypothesis tests loose power. Thus, we need an alternative compound error measure which control the error incurred and maintains a good power of the test.

We shall discuss a compound error measure, called the false discovery rate, which takes into account the proportion of erroneous rejections (false positives). Our approach will be twofold: (1) we will consider the maximum likelihood estimator of false discovery rate for a fixed rejection region; and (2) we will propose a rejection rule based on maximum likelihood estimator of the false discovery rate.

**October 20, 2010**

On the Stability of Approximate Inertial Manifolds for Perturbed Ordinary Differential Equations

Joseph L. Shomberg

An ordinary differential equation (ODE) containing a perturbation term may produce a family of attracting sets to which solutions of the equation converge. In this talk we will investigate the continuity, or stability, of a family of attractors obtained from a nonlinear equation under perturbation; so the central question in this talk is:

*Do small changes in the perturbation parameter correspond to small changes in the attractor?*

Since approximate inertial manifolds (AIMs) converge to an important attractor, the inertial manifold, it is important to know how AIMs react to changes in the underlying equation. We investigate the stability of two families of AIMs obtained from an ODE undergoing two separate changes: (i) a singular perturbation of "hyperbolic relation" type, and (ii) a regular perturbation acting as a vanishing coefficient on the damping term.

A numerical procedure based on a Lyapunov-Perron fixed-point method is used to construct the families of AIMs. In particular, AIMs are constructed for various choices of the perturbation parameter. We then use a statistical measure called mean absolute deviation to determine the variability, hence the stability, of the AIMs as the parameter changes; that is, as it tends to zero.

**November 3, 2010**

1955

C. Joanna Su

1955 was an important year for homological algebra.

In 1955, Peter Hilton traveled to Europe to work with Karol Borsuk and Beno Eckmann. As a result of the trip, they established the homotopy theory of modules, which was the birth of homological algebra. This intriguing idea produced a natural and parallel analog to the existing homotopy theory in algebraic topology and opened up a new branch of mathematics.

However, the study of the subject became unfashionable in the late 1960s after Hilton and other mathematicians encountered several significant complications. One of the most critical obstacles required the discovery of an exact sequence which was parallel to and carried the same characteristics as the homotopy exact sequence of a fibration in topology. While the concept of a fiber map in module theory seemed very natural and straightforward, proving the existence of this sequence was impossible at the time and prevented further connections between algebraic topology and homological algebra.

This problem remained an unsolved mystery for nearly five decades. In this talk we will discuss its solution.

**November 17, 2010**

P versus NP - the Holy Grail of Computer Science

Frank Ford

In 1971, Steven Cook stated the P versus NP problem formally in
*Proceeding of the Third ACM Symposium on the Theory of Computing, *but the roots of this question can be traced to Turing, Church and Gödel. The theory of computational complexity stems from two articles published in 1965. A handwritten letter of 1956 from Gödel to Von Neumann presents the problem. P can be viewed as the class of Yes/No questions which can be solved in a number of steps that is bounded by cx^{n} where x is a measure of the length of the codification of the problem and c and n are constants. NP is the class of problems which have a verifier program which takes a proposed solution to a problem and determines if it is a solution in cx^{n} for a problem of size x as above. Many problems have been shown to be NP-complete. These are problems that are so "hard" that if they are in P, then every NP problem is in P. Some practical problems that are in the NP class include Protein folding and placing of transistors on a silicon chip. We will discuss the history and research directions for this problem and the current opinion of the "solution" announced in August 2010 by Vinay Deolalikar of HP Research Labs, Palo Alto.

** **

**Spring 2011**

**February 2, 2011**

Drilling a Square Hole and Other Mathematical Approximations

Lynette J. Boos

We will start by discussing whether or not it is possible to drill a square hole, what shape of drill this would require, and how closely the hole would approximate a perfect square. Then we will talk about approximation problems in more detail and finish with a result from our research in complex analysis. This result gives necessary and sufficient conditions for a continuous function to be approximated by polynomials in the setting of a Riemann surface, which is a generalization of the complex plane.

This talk will be open and accessible to everyone!

**February 16, 2011**

Topics in Combinatorial Phylogenetics

Cayla McBee

Combinatorial phylogenetics is an area of mathematical biology that uses genetic data available from presently extant organisms to determine their evolutionary relatedness. Determining these historical relationships is important to various areas of research such as evolutionary biology, conservation genetics and epidemiology. I will provide an overview of some widely used evolutionary models focusing on group-based substitution models and how they are used with Hadamard conjugation. I will also mention a surprising link between substitution models and algebraic combinatorics.

This talk assumes no prior knowledge of combinatorial phylogenetics and will be accessible to everyone.

**March 2, 2011**

The Fourth Dimension and the People You Meet There

Thomas Banchoff

Brown University

The Fourth Dimension of space has fascinated not only geometers but also teachers, writers, and artists including Edwin Abbott Abbott ("Flatland"), Madeleine L'Engle ("A Wrinkle in Time") and Salvador Dali ("Corpus Hypercubicus"). How did they use the fourth dimension in their work, and what new insights can we gain from modern computer graphics and the Internet?

Thomas Banchoff is a Professor of Mathematics at Brown University. His specialization is in Geometry and Topology. Professor Banchoff received his Ph.D. from the University of California at Berkeley under the supervision of Shiing-Shen Chern. Before Professor Banchoff joined the Brown faculty in 1967, he taught at Harvard University and the University of Amsterdam. He received a Teacher-of-the-Year Award from Brown University and a Haimo Award for Excellence in College or University Teaching from the Mathematical Association of America (MAA). He also served a term as President of the MAA and he is an artist/member of the Providence Art Club.

**March 30, 2011**

Blow-ups of Algebraic Varieties

Su-Jeong Kang

In study of algebraic varieties (curves, surfaces, etc.), we often encounter non-smooth varieties, such as a curve with a cusp. Such kind of varieties are called singular varieties, and a non-smooth point on a singular variety is called a singularity of the variety. For example, the cuspidal curve is a singular curve and the cusp is its singularity. A singular variety is not easy to handle, because many typical tools are not applicable to the singular point; we cannot draw a tangent line at the cusp and so a linearization is not possible at that point. Luckily we have a way to resolve the singularity algebraically, through the process of, so-called, blow-ups. This is one of the most fundamental and important construction in algebraic geometry. In this talk, I will introduce blow-ups by using simple examples. If time permits, I will talk about how this process helped me to get a new example holding the Hodge conjecture.

This talk is for general audience, and all examples will be completely worked out to give a concrete description.

**April 13, 2011**

Second-Order Difference Equations: Some Recent Results and Open Problems

Jeffrey Hoag

Although non-linear difference equations are often associated with chaos, this talk will focus on some results and unsolved problems that involve other phenomena, including global stability of solutions and periodic solutions.

There will be simple illustrative examples and models from population dynamics.