Please view the seminars for the current term here.

# Spring term 2018

**Music Theory and Number Theory**

**Professor George Todd, ****Union College**

Thursday, May 31st, 1:00pm, Bailey 207

For two millennia, many musicians in the western world tuned their instruments using some form of Pythagorean tuning, a tuning system generally attributed to Pythagoras. From the Renaissance to the present, various tuning systems were developed to deal with the shortcomings of Pythagorean tuning, including meantone, well-tempered,and equal tempered tunings. In this talk, we will discuss these tuning systems and what they have to do with the number theory topics of continued fractions, diophantine approximation, and irrationality. There are no mathematical prerequisites for this talk.

### Counting from Infinity

**A film by George Csicsery
Thursday, May 24th, 12:55pm, Bailey 207**

In April 2013, a lecturer at the University of New Hampshire submitted a paper to the

Annals of Mathematics. Within weeks word spread: a little-known mathematician, with no permanent job, working in complete isolation, had made an important breakthrough toward solving the Twin Prime Conjecture.Yitang Zhang’s techniques for bounding the gaps between primes soon led to rapid progress by the Polymath Group, and a further innovation by James Maynard.

### Social choice and topology

**Sarah Yeakel, Visiting Assistant Professor, University of Maryland**

### Friday, April 27th, 1:00pm, Bailey 207

A town has decided to put a statue in the local park and the time has come to vote on placement and size. The governor wants a fair democratic process with everyone in town getting an equal say, but is it possible to come to a decision? Beno Eckmann found such a question and realized that he had solved it 50 years earlier, using topology. We’ll discuss Eckmann’s arguments and their implications for the governor, with a gentle introduction to groups.

### The Euclidean Algorithm and Irrational Numbers

**Professor Emeritus Julius Barbanel, ****Union College**

### Thursday, April 19th, 1:00pm, Bailey 201

The Euclidean Algorithm is a procedure for determining the

greatest common divisor of two positive integers. Irrational numbers

are real numbers that cannot be expressed as the ratio of two

integers. These two ideas do not seem to be related. We shall

explore a rather surprising historical connection between them. This

exploration will include a quick tour of ancient Greek mathematics.

### Counting from Infinity

**A film by George Csicsery
Tuesday, February 27th, 5:00pm, Bailey 201, with light refreshments at 4:45pm in Bailey 204**

In April 2013, a lecturer at the University of New Hampshire submitted a paper to the

Annals of Mathematics. Within weeks word spread: a little-known mathematician, with no permanent job, working in complete isolation, had made an important breakthrough toward solving the Twin Prime Conjecture.Yitang Zhang’s techniques for bounding the gaps between primes soon led to rapid progress by the Polymath Group, and a further innovation by James Maynard.

Thursday, January 25th, we will be co-sponsoring with the English Department as we invite **Math and Science Writer Evelyn Lamb: Interdisciplinary Events**

Visualizing Hyperbolic Geometry

Thurs., January 25, at 4:00 in Reamer Auditorium

For two thousand years, mathematicians tried to prove that Euclidean geometry, the geometry you probably learned in high school, was all there was. But it’s not! In the early nineteenth century, János Bolyai and Nikolai Lobachevsky independently discovered that by tweaking one of Euclid’s postulates, geometry can look totally different. We will explore the rich world of hyperbolic geometry, one of the new and beautiful systems of geometry that results from this tweak. Dinner at Wold House to follow, sponsored by Wold House.

STEM Writing for Fun and Profit

Fri., January 26, at 1:00 in Olin Auditorium. Refreshments served.

Whether you are interested in a career as a math and science communicator at a magazine or museum or just need to be able to explain technical aspects of your latest project to your coworkers, communication is essential in all sorts of math and science careers. We will talk about the joys and challenges of embarking on a writing career with a STEM background and how to communicate more effectively in any STEM career.

Evelyn Lamb, Ph.D., is a freelance math and science writer based in Salt Lake City. She has written for outlets including Scientific American, Slate, Nature News, Nautilus, and Smithsonian. She writes the blog Roots of Unity for Scientific American and contributes to the Blog on Math Blogs for the American Mathematical Society. Her work has appeared in the Best Writing on Mathematics anthology. Her visit is sponsored by the Departments of English, Mathematics, Mechanical Engineering, and Physics and Astronomy, together with Writing Programs and the Office of Minerva Programs.

### Generalizations of Collatz Functions

**Jason Turner
Undergraduate – Union College**

Thursday, January 18th, 1:00pm, Bailey 207

For a positive integer x, let T(x) = 3x+1 if x is odd and T(x) = x/2 if x is even. The Collatz Conjecture, attributed to German mathematician Lothar Collatz, states that regardless of the initial value x, every iteration sequence will eventually reach 1. Despite its concise statement, the conjecture has yet to be solved in over 75 years, although many have tried. It is widely believed that the Collatz conjecture will be proven within some more general result. We will look at several generalizations of the original Collatz function T, as well as briefly discuss the experience of participating in a math REU.

### Arrow’s Impossibility Theorem

**Professor Alan Taylor
Union College**

Thursday, January 11th, 1:00pm, Bailey 207

Kenneth Arrow won the 1972 Nobel Memorial Prize in Economic Sciences in part because of his proof, in 1951, of what is now known as Arrow’s Impossibility Theorem. This result has been described as saying that democracy is impossible, and while this is almost certainly an overstatement, Arrow’s Theorem is indeed striking. Remarkably, the proof is neither long nor complicated, and it requires absolutely no mathematical background. We’ll describe Arrow’s Theorem and give an outline of the proof that omits only a number of details that are relatively straightforward.

### Using Theory and Data for Better Decisions

**Dr. Nicholas Mattei
Research Staff Member in the Cognitive Computing Group the IBM TJ Watson Research Laboratory**

Thursday, November 2nd, 1:00pm, Bailey 207

Modern technology enables computers and (by proxy) humans to communicate at distances and speeds previously unimaginable, connecting large numbers of agents over time and space. These groups of agents must make collective decisions, subject to constraints and preferences, in important settings including: item selection; resource or task allocation; and cost distribution. In CS, these topics fall into algorithmic game theory (AGT) computational social choice (ComSoc). Results in these areas have impact within AI, decision theory, optimization, recommender systems, data mining, and machine learning.

Many of the key theoretical results in these areas are grounded on worst case assumptions about agent behavior or the availability of resources. Transitioning these theoretical results into practice requires data driven analysis and experiment. I’ll discuss my work that focus on applying theoretical results and data from PrefLib to real world decision-making including a novel, strategy-proof mechanism for selecting a small subset of winners amongst a group of peers.

Nicholas Mattei is a Research Staff Member in the Cognitive Computing Group the IBM TJ Watson Research Laboratory. His research is in artificial intelligence (AI) and its applications; largely motivated by problems that require a blend of techniques to develop systems and algorithms that support decision making for autonomous agents and/or humans. Most of his projects and leverage theory, data, and experiment to create novel algorithms, mechanisms, and systems that enable and support individual and group decision-making. He is the founder and maintainer of PrefLib: A Library for Preferences; the associated PrefLib:Tools available on Github; and is the founder/co-chair for the Exploring Beyond the Worst Case in Computational Social Choice (2014 – 2017) held at AAMAS.Nicholas was formerly a senior researcher working with Prof. Toby Walsh in the AI & Algorithmic Decision Theory Group at Data61 (formerly known as the Optimisation Group at NICTA). He was/is also an adjunct lecturer in the School of Computer Science and Engineering (CSE) and member of the Algorithms Group at the University of New South Wales. He previously worked as a programmer and embedded electronics designer for nano-satellites at NASA Ames Research Center. He received his Ph.D from the University of Kentucky under the supervision of Prof. Judy Goldsmith in 2012.

### Geometric Constructions in Number Theory

**Professor Jeff Hatley, Union College**

Thursday, October 26th, 1:00pm, Bailey 207

We often think of mathematics as being divided into very different and distinct subfields, e.g. algebra, analysis, geometry. But, many of the greatest advances in mathematics have been made by building bridges between these seemingly disparate topics. This talk will describe one such bridge, where problems in number theory (i.e. concerning integers, or “whole numbers”) are best understood from the perspective of geometry.

### Mathematics of Gerrymandering

**Professor Jeff Jauregui, Union College**

Friday, October 20th, 1:00pm, Bailey 207

Gerrymandering is the manipulation of political boundary lines for the benefit (or to the detriment) of a particular group, a practice nearly as old as the United States. In a case that is currently before the U.S. Supreme Court, the legality of partisan gerrymandering is being challenged. One of the key elements for the plaintiffs is the “efficiency gap,” a purported mathematical measure of gerrymandering that was developed by Stephanopoulos and McGhee in 2015. We will discuss the efficiency gap, including its value and its limitations, and other aspects of how math can be applied to help combat gerrymandering.

### “Another world” for mathematics

**Professor Kimmo Rosenthal, ****Union College**

Thursday, October 12th, 1:00pm, Bailey 207

“There is another world, but it is in this one.” One possible interpretation of this enigmatic quote by Paul Eluard is that “another world” is the invisible world of the mind. Mathematics does involve entering another world. In Math 199 students enter a “world” where they acquire the foundation on which to build their mathematical knowledge through the means of classical logic and the theory of sets and functions. But, is there “another world” for mathematics?

This raises other questions: Is there still a place for “math for math’s sake”? Does lack of broad general interest make something have less value? The quote below from Murnane suggests that what truly matters is the act of (intellectual) exploration itself. Perhaps abstract mathematics should be viewed more as a rigorous and aesthetic intellectual art form, to be valued for its beauty alone without paying heed to the modern-day shibboleths of “relevance” and “applicability”.

We will briefly discuss intuitionistic logic (as opposed to classical logic) and then some very basic ideas behind topos theory (topos is a Greek word meaning “place”); hence a topos is a “place” for doing mathematics. Finally, we will consider the idea of infinitesimally small numbers in this context. Does “not non-zero” have to mean zero? Could we have a notion of a number d with d≠0 and yet d

^{2}=0? We will end with a very short proof (without using limits) of the product rule for derivatives, showing why there is no f’g’ term (a proof that is valid, but only in “another world”).“An explorer’s task is to postulate the existence of a land beyond the known land. Whether or not he finds that land and brings back news of it is unimportant. He may choose to lose himself in it forever and add one more to the sum of unexplored lands.” Gerald Murnane, The Plains

### What Can Symmetries Do for You? Shapes of Spaces

**Professor Megan M. Kerr, ****Wellesley College**

Thursday, September 28th, 1:00pm, Bailey 207

Differential Geometry is the study of shapes. In practice, it is about the interplay of multivariable calculus and linear algebra, with applications to a broad array of problems, from understanding the shape of the universe to understanding the shape of red blood cells. The

curvatureof a surface measures the shape; for example, the curvature of a small round sphere is greater than that of a big round sphere — a sphere with a very large radius looks flat (zero curvature). There are infinitely many ways to bend and stretch a surface without making holes or creases, and so, a (topological) space can take on infinitely many shapes.We will consider a special class of spaces with a high degree of symmetry. Happily, these symmetries arise naturally. We will explore what happens when we vary the shape of a given manifold, controlling the variations so that the symmetries — or most of them — remain. A little Linear Algebra makes the geometry a piece of cake, and likewise a little geometry provides a fresh perspective on the algebra.

### Partially Ordered Sets and Noetherian Rings

**Cory Colbert, Gaius Charles Bolin Fellow in Mathematics at Williams College**

Thursday, September 21st, 1:00pm, Bailey 207

A partially ordered set is a set with order relations. If R is a commutative ring, we can look at its set of prime ideals and form a partially ordered set with respect to subset inclusion. We are concerned with the reverse direction: if X is a partially ordered set, when can we build a commutative ring R such that its set of prime ideals equals X? We’ll start from the beginning, so this talk will be accessible to everyone.

### The Isoperimetric Inequality

**Professor Christina Tonnesen-Friedman, Union College**

Friday, September 15th, 12:55pm, Bailey 207

Among all closed non-self-intersecting planar curves of a fixed length, the circle encloses the maximum area. This statement seems rather obvious, but the proof is not quite as simple as one might think.

We will go through two proofs; a very intuitive geometric one (with a flaw) and a calculus proof.

The talk assumes a little bit of multivariable calculus, but is otherwise self-contained.

## Spring term 2017

### Trigonometric functions and numerical analysis at an 18th-century Indian royal court.

**Professor Kim Plofker, Union College**

Tuesday, May 23rd, 5:00pm

How do you find a trigonometric function value if you don’t have a calculator? Every mathematician before about the mid-20th century had to deal with this issue. Some of them came up with ingenious computational solutions that anticipated techniques in modern numerical analysis. We will look at some of these developments in Islamic and Indian mathematics from the 15th to the 18th centuries.

### What Art Galleries, Prisons, and Zoos Have in Common

**Professor Ellen Gasparovic, Union College**

Tuesday, May 16th, 5:00pm

In 1973, Victor Klee asked his friend, Václav Chvátal, how many guards would be necessary to securely keep watch over an art gallery with

nwalls. This seemingly simple question spawned an extensive class of problems in computational geometry known asart gallery problems. In this talk, we will answer Klee’s question by proving the original Art Gallery Theorem and consider several extensions and related questions, such as the Prison Yard Problem and the Zookeeper’s Problem.

**Movie Night, Tuesday, May 9th**

### N is a Number: A Portrait of Paul Erdos

A man with no home and no job, Paul Erdös was the most prolific mathematician who ever lived. Born in Hungary in 1913, Erdös wrote and co-authored over 1,500 papers and pioneered several fields in theoretical mathematics. At the age of 83 he still spent most of his time on the road, going from math meeting to math meeting, continually working on problems. He died on September 20, 1996 while attending such a meeting in Warsaw, Poland.

The film opens at Cambridge University’s 1991 honorary doctorate ceremony, where Erdös received an award he says he would gladly trade for a “nice new proof.” For Erdös, the meaning of life is “to prove and conjecture.”

The structure of N is a Number is based on Erdös’s 50 years of perpetual wandering, “like a bumblebee,” carrying news and mathematical information from university to university. Erdös established himself as a serious mathematician at the age of 20 when he devised a more elegant proof for Chebyshev’s theorem, i.e., that there is always a prime number between any number and its double.

N is a Number is a one-hour 16mm documentary filmed over a four-year period in four countries between 1988 and 1991. The film was produced, directed and edited by George Paul Csicsery.

### Trains Arrive on Average at

Lambda Times per Hour: The Poisson Process and its Application in Finance

**Professor Leon Tatevossian, **Adjunct Instructor,

Mathematics in Finance, Courant Institute, NYU and Financial Engineering, Industrial Engineering & Operations Research, Columbia U.

**Tuesday, May 2nd, 5:00pm**

The Poisson process is a powerful tool for modeling repeated occurrences of an “event” (e.g. trains arriving at a station, customers visiting a website to make a purchase). Given a precise notion of “arrival rate” and other plausible assumptions we can derive the probability of any specific “outcome” over a selected time period.

A natural concept in mathematical finance is modeling the occurrence of defaults among a collection (“basket”) of companies. If from the “market” (meaning from observed prices of financial instruments) we could extract a company-by-company arrival rate of defaults then we might try to “stitch together” that information to get the default picture for the entire basket. But … what additional concept might we need?

### Winning losing games

**Professor Alden Gassert, Western New England University**

Tuesday, April 25th, 5:00pm

Games of chance are prevalent in our society in both leisure (lottery, casinos) and business (stock market). In most cases these games are heavily biased against the players, yet that does not deter people from playing. Many people make their livelihood creating systems to win despite the odds. In this talk, I introduce a set of biased games and ask the classic gambler’s question: “is there a winning strategy?” You may already be able to guess the answer, but the process we use to answer the question is both unusual and surprising. With this talk, I hope to demonstrate a snippet of why mathematics is such a intriguing and wonderful field. The first half of the talk will involve some simple probabilities and will be accessible to (and hopefully enjoyable for) all listeners. In the second half of the talk, I will go deeper into the mathematics involved in our problem to make our results rigorous. Some knowledge of linear algebra will be helpful for this part, but it is not required.

### The Role of Orthogonal Decomposition in Collective Decision-Making

**Professor William S. Zwicker, Union College**

Tuesday, April 18th, 5:00pm

Suppose several teachers are assessing the level of preparation of their common students, with the goal of splitting them into one group ready to tackle more abstract and challenging concepts, and a second group needing more review. Each teacher submits a recommended division, and these views will be aggregated in some way, into a single collective decision as to the best split. This context seems quite different from that of an election in which voters submit ballots, each of which ranks candidates for President, and the collective decision identifies the winner. However, in both cases we are aggregating several input binary relations of a specified type into a single binary relation of a possibly different type.

We’ll discuss two “universal” rules for aggregating binary relations, each of which generates a surprising diversity of well-known aggregation rules as special cases. Differences between general rules may arise from an orthogonal decomposition that separates input information into two components, with one rule using both components and the other discarding one of them. We’ll discuss two decompositions, related to the two types of collective decisions mentioned above, and to a single voting rule proposed by John Kemeny.

### Exploring Environmental, Sociocultural and Economic Sustainability in China

**Jing Jin, Grace Kernohan, ****Bella Li, Raquel Paramo (Class of 2017), ****Union College**

Tuesday, April 11th, 5:00pm

Economic growth in China has been extremely fast in recent decades. It has brought both positive and negative social influences, and also negative environmental impacts. We investigate the inter-linkages of the environmental, sociocultural and economic dimensions, and analyze the data at province level to understand the correlations. We construct composite indicators to measure the sustainability using the principal component analysis and factor analysis. The minimum (socially sustainable) and maximum (environmentally sustainable) economic growth levels are determined from the sustainability window analysis. The environmental Kuznets curve is evaluated to shed light on the effectiveness of economic growth and government policies in terms of improving environmental conditions. The energy and environment efficiencies are examined using the data envelopment analysis.

### Infinity, Mathematics, and Literature

**Professor Kimmo Rosenthal, Union College**

Tuesday, April 4th, __12:45pm-1:45pm__

The Argentinian writer Jorge Luis Borges said “There is one concept that corrupts and deranges the others. I speak not of Evil, whose limited domain is Ethics; I refer to the Infinite.”

In his essay Literary Infinity: The Aleph, the French scholar Maurice Blanchot posited that “the very experience of literature is perhaps fundamentally close to the paradoxes and sophisms of the evil infinite.”

This talk will be a bricolage of ideas intercalating a mathematical approach to infinity with possible connections between mathematics and writing. Is the contemplation of infinity (or, more generally, doing mathematics) related to the task of literature which, according to Blanchot, is to express the inexpressible? We will be considering Borges’ famous story, The Library of Babel, which has inspired a mathematical text by William Bloch on the “unimaginable mathematics” of this library. This book raises issues regarding infinity, combinatorics, randomness and information theory, topology and more. We will also consider the concept of sets of measure zero in the context of Borges’ The Book of Sand. Can a book with countably many infinitely thin pages be invisible if held sideways?

This talk is intended to be accessible to a wide audience, prerequisites being an interest in both mathematics and literature.

“An explorer’s task is to postulate the existence of a land beyond the known land. Whether or not he finds that land and brings back news of it is unimportant. He may choose to lose himself in it forever and add one more to the sum of unexplored lands.” Gerald Murnane, The Plains

## Winter term 2017

### Making Honesty the Best Policy (Mathematically)

**Professor Alan Taylor, Union College**

Tuesday, February 28th, 5:00pm

In an auction, should you bid what you really think the lamp is worth? More generally, are there auction rules for which this kind of honesty is the best policy? In a divorce, should you reveal the extent to which you want (or don’t want) the dog? More generally, are there fair-division procedures for which this kind of honesty is the best policy? In an election, should you vote for your favorite candidate? Again, more generally, are there voting rules for which this kind of honesty is the best policy? We’ll see what mathematics has to say about each of these questions.

### From Counting to Topology and Back

**Professor Marco Varisco, University at Albany**

Tuesday, February 21st, 5:00pm

We all know how to count, but what does it really mean? We’ll first answer this basic question, at least from a mathematical point of view, and then we’ll explore a natural geometric generalization. This will lead to an informal description of “homotopy groups,” a fundamental concept in a branch of geometry called algebraic topology. At the end we’ll see how this generalization brings us back, surprisingly, to the original issue of counting.

### Hidden Figures Movie Showing

Join us for a movie, dinner and discussion, with Professor Emeritus Twitty Styles, who had Dorothy Vaughan as a math teacher in high school! – the bus leaves for Bowtie Cinema at 6:20pm following discussion. Sponsored by the Math Club, Association for Women in Mathematics, and the Math Department.

### How Unstable is Democracy? Condorcet Cycles of Order 2

**Professor William Zwicker*, Union College *including joint work with Professor Davide Cervone, Union College**

Tuesday, January 31st, 5:00pm

The Marquis de Condorcet (1743 – 1794) demonstrated a fundamental flaw in majority rule – it’s possible that every conceivable decision get toppled, by majority vote, in favor of a different decision. Three computer scientists recently produced an example showing that the situation is worse: even a form of stability weaker than Condorcet’s fails in some cases. We provide improved (smaller) examples of the phenomenon, and describe the mathematics underlying their construction.

### Self-Similarity and Fractal dimensions

**Professor William Wylie,** Syracuse University

Tuesday, January 24th, 5:00pm

Fractals are often beautiful pictures constructed from repeating patterns on infinitely smaller scales. They are used to model objects in nature, for example in movies and video games. We will explore some self-similar fractals with the goal of understanding how “solid” they really are. Are they two dimensional? one-dimensional? Something in between?

### Experimenting With Mixtures: Fun With Fish Patties, Lattes, and Rocket Fuel

**Professor Roger Hoerl,** Union College

Monday, January 16th, 5:00pm

Much experimentation and statistical modeling is performed with variables that can be varied independently, such as temperature, pressure, flow rate, and so on. However, in some cases the key variables impacting the outcome of interest are the proportions of these ingredients in a mixture. For example, the taste of a Starbucks latte is primarily a function of the proportion of espresso, steamed milk, and other potential ingredients, such as flavorings. This is also true for many pharmaceuticals, chemicals, foods, gasoline, and even rocket fuels. Since the proportions of ingredients in a mixture must sum to 1.0, there is a linear constraint on the independent variables. This dramatically changes the geometric shape of the experimental region, as well as the statistical models that can be fit to the resulting data. Therefore, alternative designs and models must be employed. This presentation will illustrate viable approaches to such problems, and in particular present a recently-published strategy for problems that incorporate both mixture variables and also process variables that are not ingredients, and therefore can be varied independently.

### A Brief Introduction to Elliptic Curves

**Professor Harris Daniels,** Amherst College

Tuesday, January 10th, 5:00pm

Elliptic curves are some of the most fascinating and closely studied objects in the history of mathematics. Their origins date all the way back to Diophantus of Alexandria’s Arithmetica, and yet there are exciting aspects and applications of elliptic curves still being discovered today. The goal of this talk will be to introduce elliptic curves in the context of diophantine equations and discuss some of the things that make them so interesting.

If time permits, I will discuss some open problems and ongoing research concerning elliptic curves.

## Fall term 2016

### Commuting pairs of matrices

**Professor Leila Khatami,** Union College

Monday, November 7th, 5:00pm

One of the first things that we learn about multiplication of matrices is that it is not commutative. This means that for square matrices A and B we normally do not expect AB to be equal to BA. Having said that, there are many pairs of matrices that do in fact commute. In this talk we discuss some of the properties and characteristics of such pairs of matrices.

### Hard Problems

**a feature documentary by George Csicsery**

Tuesday, November 1st, 5:00pm

Hard Problemsis a feature documentary about the extraordinarily gifted students who represented the United States in 2006 at the world’s toughest math competition—the International Mathematical Olympiad (IMO). It is the story of six American high school students who competed with 500 others from 90 countries in Ljubljana, Slovenia. The film shows the dedication and perseverance of these remarkably talented students, the rigorous preparation they undertake, their individuality, and the joy they get out of solving challenging problems. Above all, it captures the spirit of math competitions at the highest level.

While aiming to inspire and entertain,Hard Problemsprovides an insightful and thoughtful look at the process that produces successful teams, and ultimately, great mathematicians of the future.

### A Look at Permutations and Their Partitions

**Professor William Adamczak,** Siena College

Tuesday, October 25th, 5:00pm

A collection of permutations known as roller coaster permutations was recently introduced by Ahmed and Snevily. Roller coaster permutations are described as permutations that maximize the total switches from ascending to descending (or visa versa) for a permutation and all of its subpermutations simultaneously. More basically, this counts the greatest number of ups and downs or increases and decreases for the permutation and all possible subpermutations. These permutations have connections to sorting algorithms and forbidden permutations. In this talk we will introduce these permutations as well as some initial results on their rich structure and a result from work as part of an undergrad research project with a student over the summer.

### Measure theory

**Professor Jeff Jauregui,** Union College

Tuesday, October 11th, 5:00pm

If you have a bounded subset of the real line, how do you define its “size”, or “measure”? For an interval $[a,b]$, the measure ought to be just the length of the interval, $b-a$. But what about more complicated sets that have lots of gaps, like the subset of rational numbers in $[a,b]$, or the famous Cantor set? In this talk I will give an introduction to the subject of measure theory, an important branch of analysis, that has far-reaching applications to probability, differential equations, quantum mechanics, and much more.

### Detecting Breast Masses and the Location of the Prostate

**Professor Jue Wang,** Union College

Monday, October 3rd, 5:00pm

I will present a fast Enclosure Transform to localize complex objects of interest from speckle imagery. This approach explores spatial constraints on regional features from a sparse image feature representation. Unrelated, broken ridge features surrounding an object are organized collaboratively, giving rise to the enclosureness of the object. Three enclosure likelihood measures are constructed, consisting of the enclosure force, potential energy, and encloser count. In the transform domain, the local maxima manifest the locations of interest objects, for which only the intrinsic dimension is known a priori. I will demonstrate two medical applications in detecting (1) suspicious breast masses in screening breast ultrasound, and (2) the location of the prostate in trans-abdominal ultrasound for verification of patient positioning in radiotherapy treatment of prostate cancer.

### A Generalization of Zeckendorf’s Theorem via Circumscribed $m$-gons

**Professor Pamela Harris,** Williams College

Tuesday, September 27th, 5:00pm

Zeckendorf’s theorem states that every positive integer can be uniquely decomposed as a sum of nonconsecutive Fibonacci numbers, where the Fibonacci numbers satisfy the recurrence relation $F_n=F_{n-1}+F_{n-2}$ for $n\geq 3$, $F_1=1$ and $F_2=2$. The decomposition of positive integers as sums of nonconsecutive Fibonacci numbers have been shown to have the following properties: the distribution of the number of summands in such decompositions converges to a Gaussian, and the gaps between summands converges to geometric decay.

In this talk, we extend these results by creating an infinite family of integer sequences called the $m$-gonal sequences arising from a geometric construction using circumscribed $m$-gons. We provide a notion of a legal decomposition, and prove that the decompositions exist and are unique. We then examine the distribution of the number of summands used in the $m$-gonal decompositions and prove that it displays Gaussian behavior and, in addition, we show that there is geometric decay in the distribution of gaps. We end with some open questions in this area.

**A Joint Mathematics-Computer Science Seminar**

### To Vote or Not to Vote? Computer-Generated Proofs and Paradoxes in Voting Theory

**Dominik Peters,** Oxford University

Tuesday, September 20th, 5:00pm

The mathematical theory of voting seeks to find good procedures that can be used to make group decision when group members have different preferences. Unfortunately, the field is riddled with “impossibility results” which show that certain desirable properties cannot be enjoyed by any voting rule. For example, French economist Hervé Moulin proved in 1988 that every “sensible” voting rule suffers from the “No-Show Paradox”: there will always be a situation where a voter is better off staying at home — adding her honest vote to the ballot box leads the voting rule to select a worse outcome according to her preferences! In this work, we try to understand better when no-show paradoxes occur. Moulin’s method shows that they must occur when there are at least 25 voters — but maybe there is a rule that doesn’t suffer from the paradox when there are fewer voters? We answer this question precisely using computers and powerful SAT solvers, obtaining computer-generated but human-readable proofs of our claims.

Joint work with Felix Brandt and Christian Geist (Munich).

Please view a list of seminars from previous years here.

## Spring 2016

### Positive Curvature on Teardrops and Footballs

by **Professor Christina Tonnesen-Friedman,** Union College

Monday, May 23, 2016, 5:00pm

In this talk we will look at variations of the unit sphere with mild singularities at the north and south pole (giving us so-called teardrops and footballs). Using cylindrical coordinates we will be able to find explicit examples with positive curvature.

The talk is intended to be accessible to anyone with a background in multivariable calculus (MTH 115 + MTH 117 or IMP120/121).

### The Joy of Abstraction (with a gentle approach to category theory)

by **Professor Kimmo Rosenthal,** Union College

Tuesday, May 17, 2016, 5:00pm

Can the old dictum “art for art’s sake” be replaced by “math for math’s sake”? In this day and age, when relevance, applicability, and connections with other disciplines are touted as paramount, is there still a place for purely abstract mathematics viewed more as a rigorous and aesthetic intellectual art form, to be valued for its beauty alone? Why does something new, especially if it is abstract and does not conform to the accepted point of view, provoke “fear and trembling”, if not outright hostility? Georg Cantor, the founder of set theory, was called “a corrupter of youth” and his work “a disease”, and category theory was labelled as “abstract nonsense”. This seminar talk will be a bricolage of historical observations, personal anecdotes and, of course, some mathematics with a modest and hopefully accessible introduction to category theory. Math 199 students may find it interesting that there is a “more abstract” approach to 199 than they have encountered, with questions such as “how can you describe a one-element set without discussing elements”?

“Mathematics, like music, can dispense with the universe.”Jorge Luis Borges

“There is another world, and it is in this one.”Paul Eluard

“An explorer’s task is to postulate the existence of a land beyond the known land. Whether or not he finds that land and brings back news of it is unimportant. He may choose to lose himself in it forever and add one more to the sum of unexplored lands.”Gerald Murnane

### Hindu-Muslim Relations, Courtly Patronage, and Related Numerical Methods for Computing $\sin(1^\circ)$ in Early Modern Central/South Asia

by **Prof. Kim Plofker**, Union College

Tuesday, May 3rd, 2016, 5pm

Before the development of power series expressions for trigonometric functions and electronic devices for evaluating them, the computation of accurate sine values required a great deal of mathematical creativity and ingenuity as well as hard work. This story, with elements of interest for both mathematicians and non-mathematicians, begins with the invention of trigonometry in the Hellenistic world and continues through the skillful blending of classical geometry with numerical methods in medieval Muslim mathematics, up to an account of a recently (re)discovered Sanskrit manuscript from the early 18th century. As our discussion will show, this manuscript reveals an intriguing instance of scientific transmission and mathematical synthesis at the court of the learned Maharaja Sawai Jai Singh in western India.

### The P vs. NP Problem (video from HRUMC 2016)

by **Professor Scott Aaronson,** MIT

April 25, 2016, 5:00pm

I’ll discuss the status of the famous P = NP problem in 2016, offering a personal perspective on what it’s about, why it’s important, why many experts conjecture that P ≠ NP is both true and provable, why proving P≠ NP is so hard, the landscape of related problems, and crucially, what progress has been made in the last half-century. I’ll say something about diagonalization and circuit lower bounds; the relativization, algebrization, and natural proofs barriers; and the recent works of Ryan Williams and Ketan Mulmuley, which (in different ways) hint at a duality between impossibility proofs and algorithms.

### Repeating decimal expansions

by **Professor Emeritus Karl Zimmermann,** Union College

April 19, 2016, 5:00pm

It is well known that the decimal expansion of a rational number is either purely repeating or eventually repeating. In other words, there is some point in the expansion at which the rest is made up of the same finite string of digits repeated over and over again. In this talk, we’ll begin by using long division to explain why this repetition must occur. We’ll

then call on a theorem from elementary number theory* to determine the length of the repeating strings when the denominator of the rational number is prime. Finally, we’ll state the corresponding results for non-prime denominators.* Number Theory is NOT a prerequisite for this talk!!

### Limb coordination in crayfish swimming: understanding the role of long range connections

by **Professor Lucy Spardy,** Skidmore College

April 11, 2016, 5:00 pm

During forward swimming, pairs of crayfish swimmerets (limbs) exhibit a robust pattern, moving rhythmically in a back to front metachronal wave with neighbors delayed by approximately 25% of the period. We study the mechanism responsible for this coordinated limb behavior using a model which represents the underlying neural circuitry as a chain of coupled oscillators. Previous modeling efforts have only considered the effects of nearest neighbor coupling, ignoring the presence of longer range connections in the system.

In this talk, I’ll address how long-range coupling affects this mechanism, using an oscillator chain whose architecture reflects the known neural circuitry in the swimmeret system. I’ll present analytical arguments and numerical simulations that indicate that long-range coupling tends to speed up the metachronal wave. Combined with insights from a computational fluid dynamics model, we suggest that this may maximize swimming efficiency.

### Topology, Statistics, and Tumors in Mice

by **Professor David Damiano,** College of the Holy Cross

Tuesday, April 5, 2016 5:00 pm

Mouse models are used to test the delivery of drugs to tumors. A primary method of drug delivery involves using targeted antibodies that bind to antigens on the surface of tumor cells. In order to track the delivery of antibody to tumor, antibodies are labeled by a radioactive tracer. In this talk we will describe a novel method of analyzing the uptake of radio-labeled antibodies by tumor that combines topology and statistics, specifically persistent homology and hypothesis testing. Along the way, we will present background in each area and discuss the application of this method to SPECT (Single-photon emission computed tomography) images from two pre-clinical studies of antibody targeting of tumors.

## Winter 2016

### Taking the Long View: The Life of Shiing-shen Chern

a film by **George Paul Csicsery,** 2011

Monday, February 29, 2016 5:00 pm

### The Chain Rule: Variations on a Theme

by **Professor Brenda Johnson,** Union College

Tuesday, February 23, 2016 5:00 pm

The chain rule is a standard topic in first-term calculus courses. It tells us how

to express the first derivative of a composition, $f\circ g$, in terms of the first derivatives of $f$ and $g$. In this talk, we’ll explore different ways in which this standard material can be generalized and interpreted in other mathematical contexts.

### Insights from Geometry in Science and Engineering

by **Peter Spaeth,** GE Global Research

Tuesday, February 16, 2016 5:00 pm

Mathematics plays a fundamental role in our ability to draw non-obvious conclusions about the world around us. I will emphasize the part geometry plays in some examples from satellite motion planning and population dynamics. Along the way we will introduce symplectic geometry, a branch of mathematics and physics where 2-dimensional area is the fundamental concept.

### Chaos on the Circle

by **Professor Emeritus Susan Niefield,** Union College

Monday, February 8, 2016 5:00 pm

Take a point $P$ on a circle in the plane and double the angle $\theta$ to get a point $f(P)$. Repeating this process gives a set $\{P,f(P),f(f(P)),\dots \}$ of points, called the

orbit of $P$ under $f$. What kind of orbits can we find? Of course, that depends on the starting point $P$. Some orbits are finite, while others are not. Among the infinite ones, there are even orbits that hit almost every point on the circle.This map is an example of a

chaotic dynamical system. After presenting a definition ofchaos, we will show that points on a circle can be represented as binary sequences, and use this representation to prove that the angle-doubling map $f$ is chaotic.

### Geometric Perspectives on Fair Division

by **Professor Emeritus Julius Barbanel,** Union College

Tuesday, January 26, 2016 5:00 pm

Suppose that you and I are given a cake that we wish to divide between the two of us. We may value pieces of cake differently. For example, maybe I like the chocolate filling best, but you like the icing best, so a piece of cake that I value as half the cake, you might value as one third of the cake. What would constitute a “good” division of the cake? We shall consider two sorts of criteria, namely “fairness” and “efficiency.” Fairness involves such questions as “Did I get at least half?” or “Did I get at least as much as you?” Efficiency involves questions such as “Is this the best way to divide the cake or is there a way to divide the cake that would make each of us happier?” We will see that there are interesting mathematical ideas and revealing geometric pictures associated with these issues. We will also consider what happens when we wish to divide the cake among more than two people.

### Topological Data Analysis

by **Professor Elizabeth Munch,** University at Albany

Tuesday, January 19, 2016 5:00 pm

Because we have fully entered the age of big data, it is important that we find ways to explore, summarize, and answer questions with this data. However, sometimes the problem isn’t just that the data is big, but that it is complicated. Often, that means it is even too complicated for the standard methods to be useful. In this talk, we will discuss a new collection of tools available from the field known collectively as Topological Data Analysis. We use tools from algebraic topology, linear algebra, and graph theory to ask and answer questions about our data and to provide understanding for the domain sciences from which the data arise.

### The Congruent Number Problem

by **Professor Jeffrey Hatley,** Union College

Monday, January 11, 2016 5:00 pm

Abstract: The Congruent Number Problem asks the following simple-sounding question: which rational numbers occur as the area of a right triangle with sides of rational length? For example, the familiar (3,4,5) right triangle has area 6, so 6 is a congruent number; but Fermat showed in the year 1640 that 1 is not a congruent number. Our investigation of this question will lead to a surprising and beautiful interaction between algebra and geometry, bringing us to the forefront of modern number theory and a math problem with a million dollar prize.

## Fall 2015

### The Basel Problem

by **Professor Paul Friedman**, Union College

Monday, November 2, 2015 5:00 pm

In 1644, Pietro Mengoli posed the following problem:

Find the numerical value of the sum of the reciprocals of the squares, that is, evaluate

$1+\frac 14+\frac 19+\frac 1{16}+\cdots = \sum_{k=1}^\infty \frac 1{k^2}$.

At the time, mathematicians were able to show that this sum was indeed finite (unlike the

harmonic series, $1+1/2+1/3+1/4+\cdots$). However, it was not until 90 years later that Euler was able

to evaluate this sum.In this talk, we will present (one of) Euler’s computations that this sum is, remarkably, $\pi^2/6$.

In essence, his work relies on a study of the sine function, writing it both as an infinite polynomial

(a Taylor series, as in Math 110, 113, and BC Calculus) and as an infinite product. The result follows somewhat

naturally from comparing these two considerations.After that, time permitting, we will discuss another proof of the same result that uses

integration techniques from Math 117.

### Julia Robinson and Hilbert’s Tenth Problem (Movie)

a film by **George Csicsery**

Monday, October 26, 2015 5:00 pm

In 1900 the prominent German mathematician David Hilbert published twenty three problems in mathematics as the most important mathematical problems of the 20th century. Some of these problems were very influential for 20th century mathematics, and a lot of them have been solved since then. The tenth Hilbert’s problem asks whether there is an algorithm for deciding if a polynomial equation with integer coefficients has an integer solution. This problem was resolved in the negative in 1970 as the result of the work of several mathematicians in the span of 21 years.

Julia Robinson and Hilbert’s Tenth Problemfeatures one of these prominent mathematicians, Julia Robinson,and her quest to solve this problem. Robinson is a pioneer among American women in mathematics and rose to prominence in a field that was historically dominated by men. She was the first woman elected to the mathematical section of the National Academy of Sciences, and the first woman to become president of the American Mathematical Society. In the film, Robinson’s major contribution to the solution of Hilbert’s 10th problem triggers a tour of the 20th century mathematics that moves from Paris in 1900, through the United States, to the Soviet Union and back.

### Words Within Words – A Recursive Approach to Catalan Numbers and Fine Numbers

by **Professor David Vella**, Skidmore College

Tuesday, October 20, 2015 5:00 pm

In this talk, I present a new recursive formula for the Catalan numbers. The proof uses Dyck words, one of the many objects known to be counted by the Catalan numbers. Dyck words are strings of two distinct symbols – such as $A$ and $B$ – such that no initial segment contains more $B$’s than $A$’s. Thus, $AABABB$ is a Dyck word of length 6, but $ABBABA$ is not, because the initial segment $ABB$ has too many $B$’s. It is well known that the number of Dyck words of length $2n$ is given by $C_n$, the $n$th Catalan number. We obtain our recursion by “factoring” Dyck words into smaller Dyck words which appear inside them. While this proof is a very satisfying combinatorial proof of our recursion, we actually present a second proof, using generating functions, because this approach can be modified to yield a new formula that expresses the $n$th Fine number in terms of the Catalan numbers.

### Quantum Entanglement and the Correlation Numerical Range

by **Professor Jon Bannon**, Siena College

Tuesday, October 13, 2015 5:00 pm

(Joint work with Eli Bashwinger and Mohammad Javaheri) One of the fundamental open problems of quantum information theory (Tsirelson’s problem) roughly asks whether the quantum correlations between two separated systems described by quantum mechanics agree with those described by quantum field theory. If the answer is `yes’, then certain known estimates of the Bell inequalities describing quantum entanglement are known to be tight, whereas if the answer is `no’, then there must exist quantum correlations that cannot be described using finite-dimensional approximations. Tsirelson’s problem is essentially equivalent to a question about a certain generalized numerical range for matrices. The three-dimensional version of this equivalent problem originally required sophisticated computational algebraic geometry to solve, but our recent solution to an elegant related problem posed by Don Hadwin and Deguang Han yields a reasonably easy “bare-hands” solution of the three-dimensional case. We discuss this elegant related problem and the bare-hands solution, and if there is time left over we can discuss what our findings point to regarding quantum correlations.

### Complex Calculus

by **Professor Caner Koca**, City University of New York

Friday, October 9, 2015 5:00 pm

About a century after the invention of Calculus by Newton and Leibniz, mathematicians such as Euler, Gauss, Riemann and Cauchy discovered and developed a complex-number version of all the key ideas in calculus, such as differentiation and integration. In this new theory, one looks at

complex-differentiablefunctions from complex numbers to complex numbers, and study their properties. This analogy, though very formal, can sometimes lead to really unexpected, surprising and slightly disturbing facts! For example, the complex sine and cosine functions turn out to be unbounded, or thecomplexexponential function isperiodic. In this talk, we will see the basics of the theory of complex functions, and outline some of the similarities and differences between real and complex calculus. I will give special emphasis on how one can visualize and graph some of these complex functions, especially of the so-calledmulti-valued complex functions, which in turn give rise to some of the fascinating examples of Riemann surfaces.

### The Elegance of Line: Ruled Surfaces and the Dynamics of the Olivier Models

by **Professor Davide Cervone**, Union College

Wednesday, September 30, 2015 5:00 pm

Union College owns a rare collection of 19th-century mathematical models designed by Théodore Olivier. Nine of the more than 40 models are housed in a display case in Bailey Hall, outside the Mathematics Department office. One of the remarkable qualities of these models is that they are manipulatives; that is, they have parts that move to help illustrate families of surfaces and how they interact. Due to the age and delicacy of the models, however, we no longer are allowed to adjust them, but we can still appreciate their elegant design. In this talk, we will discuss the differential geometry that underlies and motivates the Olivier models, and use interactive computer simulations to demonstrate their transformations, bringing these wonderful objects to life once again.

### 3D printing and Mathematica

by **Professor Jeff Jauregui**, Union College

Monday, September 21, 2015 5:00 pm

I will show how you can easily design mathematical models for 3D printing using the software package Mathematica. Many examples, printed at Union’s very own Collaborative Design Studio, will be shown. No previous experience with Mathematica is assumed.

### What I thought about this summer

by **Professor Bill Zwicker**, Union College

Tuesday, September 15, 2015 5:00 pm

When three hundred mathematicians are competing for fifty research grants, who wins? In many places (including Catalonia, where my Spanish co-author Josep Freixas lives) the value of a researcher’s

Hirsch Citation Index–HCI– plays a role. TheHCIassigns a numerical value to your research record, based on productivityP(the number of research papers you published) and impactI(the number of other papers that cite yours). A record that strikes anappropriate balancebetweenPandIyields a higherHCI, improving your chances for a grant.Fifty-five years before Hirsch suggested his index, John Nash proposed a solution for the

two person bargaining problem: how should two peopleAandBreach an agreement that will determine the amount of utility (money? pleasure?) each gets? Nash’s solution strikes anappropriate balancebetween the amount of utilityAgets from the bargain, and the amountBgets.Our talk will focus on the key idea that Josep Freixas and I had this summer, of developing an improved version of

HCIby importing Nash’s more sophisticated notion ofappropriate balanceinto Hirsch’s context.

## Spring 2015

### Dimensions: a film

by **Étienne Ghys, Jos Leys and Aurélien Alvarez**

May 26, 2015 5:00 pm

Physicists tell us that we live in a 4-dimensional space-time continuum. But how can our 3-diminutional minds imagine a four (or higher) dimensional space? This film, inspired by ideas from ancient and modern mathematicians, as well as works of artists and writers, uses computer graphics to help us achieve this goal!

“Dimensions” is a collaborative project done by of Étienne Ghys, a mathematician from École Normale Supérieure in Lyon, Jos Leys, an engineer turned computer graphics enthusiast, and Aurélien Alvarez, a graduate student also from École Normale Supérieure in Lyon. We are going to watch the first half of this 2-hour film together. The second part of the film, which touches on deeper aspects of the topic, is available online in 8 different languages and with subtitles in more than 20 languages. See http://www.dimensions-math.org.

### Equidistant sets

by **Professor Mark Huibregtse,** Skidmore College

May 18, 2015, 5:00pm

Given two subsets A and B of the plane, the

equidistant setdetermined by A and B is the locus of points equidistant from the two sets (which are called thefocal sets). For example, the equidistant set determined by two points P, Q in the plane is the perpendicular bisector of the segment PQ. In the summer of 2014, a student (Adam Winchell) and I did a summer research project on equidistant sets. We were motivated by an article that had recently appeared in the January 2014 issue of the American Mathematical Monthly:On Equidistant Sets and Generalized Conics: The Old and the New, by Mario Ponce and Patricio Santibàñez. In the talk, I will report on the questions we considered and the answers we found. Along the way we will consider several (hopefully) interesting examples.

### Infinity (and then some)

by **Professor Kimmo Rosenthal**, Union College

April 27, 5:00pm

There is one concept that corrupts and deranges the others. I speak not of Evil, whose limited domain is Ethics; I refer to the Infinite.-Jorge Luis BorgesThe concept of the infinite has been both intriguing and bedeviling since the days of the Greeks with their concept of apeiron. The infinite has always had metaphysical and religious overtones and has influenced many thinkers, as well as artists and writers. It was not until the second half of the nineteenth century that the mathematician Georg Cantor laid the foundations for a rigorous mathematical study of infinity (he also met with strong opposition from many quarters, which we shall discuss). This talk will be an olio of remarks about infinity, some historical, including a discussion of the life of Georg Cantor, some mathematical (can an infinite set be “small”?), and some literary, (remarks about stories by Borges such as The Book of Sand and The Aleph) – and then some. One is almost tempted to posit that the discussion could veer in infinitely many (countable, of course) directions.

### The three ancient Greek construction problems: some history and some solutions

by **Professor Julius Barbanel**, Union College

April 20, 2015, 5:00pm

Ancient Greek mathematicians struggled with three geometric construction problems:

- Trisect an arbitrary angle
- Square the circle (i.e., given a circle, construct a square having the same area)
- Double the cube (i.e., given a cube, construct another cube having twice the volume)
They were unable to solve any of these problems using the traditional Euclidean ruler-and-compass tools. However, they did solve all three using so-called “mechanical methods”. We shall discuss the history of these three problems and present two mechanical solutions.

### Continuity, Smoothness and Infinitesimals

by **Professor Stewart Shapiro**, Ohio State University

April 14, 2015, 5:00pm

The purpose of this talk is to present, in outline, the basics of an interesting theory of the continuous, due to Anders Kock and William Lawvere. It sustains some intuitions about continuity that are lost in the now orthodox Dedekind-Cantor framework. Smooth infinitesimal analysis requires intuitionistic logic–it is inconsistent with the law of excluded middle. I will explain the basis of this logic, and then the underlying principles of the mathematical theory. The talk will be self-contained.

### Commuting Polynomials and Fermat’s Little Theorem

by **Professor Karl Zimmerman**, Union College

February 24, 2015, 5:00pm

Fermat’s Little Theorem (FLT) is a beautiful and very useful theorem in elementary number theory. It can be stated as follows:

FLT: Let p be a prime and d any integer. Then d

^{ p}−d is divisible by p.On the other hand, polynomials f and g are said to commute under composition provided

(f∘g)(x)=(g∘f)(x), that is, if f(g(x))=g(f(x)). At first glance, these topics don’t seem to be related, but in this talk I’ll use an elementary proposition about commuting polynomials along with some important concepts from Math 199 to give a proof of Fermat’s Little Theorem.

## Winter 2015

### Between the Folds: A Film Documentary

by **Vanessa Gould**

January 27, 2015, 5:00 pm

Origami may seem an unlikely medium for understanding and explaining the world. But around the globe, several fine artists and theoretical scientists are abandoning more conventional career paths to forge lives as modern-day paper folders. Through origami, these offbeat and provocative minds are reshaping ideas of creativity and revealing the relationship between art and science.

BETWEEN THE FOLDS chronicles 10 of their stories. Featuring interviews with and insights into the practice of these intrepid paper folders, the film opens with three of the world’s foremost origami artists: a former sculptor in France who folds caricatures in paper rivaling the figures of Daumier and Picasso; a hyper-realist who walked away from a successful physics career to challenge the physics of a folded square instead; and an artisanal papermaker who folds impressionistic creations from the very same medium he makes from scratch.The film then moves to less conventional artists, exploring concepts of minimalism, deconstruction, process and empiricism. Abstract artists emerge with a greater emphasis on concept, chopping at the fundamental roots of realism, which have long dominated traditional origami. The film also features advanced mathematicians and a remarkable scientist who received a MacArthur Genius Award for his computational origami research.

While debates ebb and flow on issues of folding technique, symbolism and purpose, this unique film shows how closely art and science are intertwined. The medium of paper folding—a simple blank, uncut square—emerges as a resounding metaphor for the creative potential for transformation in all of us.

“A gorgeous cinematic experience. I was so captivated by the documentary that halfway through I felt intense admiration for humanity, the same tingling I feel when listening to music so exquisite it’s almost painful.” — Karen A. Frenkel, Talking Science

“This film is much more than a loving look at the fascinating, brilliant characters who devote their lives to folding paper. It reveals origami itself as richer and more intricate than you could imagine… and by the end, you find yourself convinced that the mystery of folding could be one of the universe’s deep secrets.” — Chris Anderson, Curator, TED Conference

### Optimize Care in Osteoporosis Using GIS and UMOT

by **Professor Jue Wang**, Union College

January 19, 2015, 5:00 pm

Osteoporosis is a major public health threat for 44 million Americans. The estimated national direct cost for osteoporosis and associated fractures is $19 billion per year ($52 million each day) – and the cost is rising. It is predicted that the cost for osteoporosis and low-energy fractures over the next two decades will total $474 billion. In this talk, I will first use the Geographic Information Systems (GIS) to build maps for osteoporosis rates for population densities by age, gender, race, ethnicity, and other related factors. We will examine the correlations between the osteoporosis rates and affecting factors. Then I will show the feasibility of using ultrasound modulated optical tomography (UMOT) to detect and assess osteoporosis. Monte Carlo simulations and physical experiments are performed. This study could improve care for patients with osteoporosis and for prevention.

### On Prime Numbers and Arithmetic Progressions

by **Professor Alan Taylor**, Union College

January 12, 2015, 5:00 pm

The number-theoretic study of the interplay between prime numbers and arithmetic progressions goes back to some early work of Lagrange and Waring in the 18th century. Questions that have since arisen include the following:

- Does every long arithmetic progression contain a large set of primes?
- Does every large set of primes contain a long arithmetic progression?
The answer to the first (Dirichlet’s theorem) is one of the deepest results of the 19th century. The answer to the second (the Green-Tao theorem) is one of the deepest results of the 21st century. We’ll discuss these along with the original theorem (and proof) of Lagrange and Waring.

## Fall 2014

### Continued Fractions and Other Mathematical Adventures

by **Professor Mohammad Javaheri**, Siena College

November 11, 2014, 5:00 pm

Abstract: Continued fractions are dated back to as early as Euclid’s elements, but they were first phrased by John Wallis. Euler, Lagrange, and Galois are among mathematicians who studied continued fractions. In this talk, we view continued fractions from the perspective of semigroup actions. Lagrange, for example, showed that quadratic irrationals have periodic continued fractions. Our main objective is to discuss a related density result that describes pairs of linear fractional transformations whose generated semigroup has dense fixed point sets. Along the way, we discuss topics such as rational approximations, golden ratio and Fibonacci numbers, linear fractional transformations, discrete dynamical systems, and chaos.

### Before Ebola There Was AIDS: Evaluating the Connection between Gender Based Violence and HIV/AIDS

by **Professor Roger Hoerl**, Union College

November 3, 2014, 5:00 pm

Abstract: While the media has been focusing primarily on the recent outbreak of the Ebola virus, HIV/AIDS continues to infect and kill millions of people around the globe. Fortunately, organizations such as the World Health Organization, the Gates Foundation, and several others, are providing billions of dollars in funding to combat the spread of HIV, and save the lives of those infected. In order to effectively deploy these funds for greatest impact, many agencies utilize mathematical models that predict progression of the pandemic. One of these models is the Modes of Transmission Model (MoT), developed for UNAIDS, the global HIV/AIDS consortium led by the United Nations. Recent Math graduate Keilah Creedon focused her senior thesis on studying this model, and in particular, addressing one of its limitations; inability to account for the impact of gender based violence. Significant research has been published that demonstrates the negative impact of violence against women in spreading HIV. Keilah modified the MoT model to account for such violence, and to determine the relative impact of violence against women versus other factors contributing to the spread of the disease. Further, she conducted a sensitivity analysis to quantify the uncertainty of the model outputs to uncertainty in model inputs. When Keilah presented her thesis at the 2014 Steinmetz Symposium, she did not yet have results. I will therefore review her original research design, and then share the results we obtained from her modified model.

### Beyond Complex: the Quaternions

by **Professor Jeffrey Jauregui**, Union College

October 28, 2014, 5:00 pm

Abstract: The real numbers are very familiar to us, and the complex (real + imaginary) numbers slightly less so. But what if you attempted to explore beyond the imaginary number

by introducing an independent imaginary number,i? This problem puzzled William Hamilton in the mid-19th century until he was suddenly struck with the solution (which he famously carved into the Broome bridge in Dublin), thereby inventing a mathematical system known as the quaternions. Some natural questions are: What do you gain by passing beyond the complex numbers to the quaternions? What do you lose? How far can you push this construction? Why were quaternions such a hot topic in mathematics and physics for several decades? We will answer these questions and much more.j

### Shocking Secrets of Sasakian Geometry Revealed

by **Professor Ralph Gomez**, Swarthmore College

October 14, 2014, 5:00 pm

Abstract: In this talk we shall take a casual stroll through an odd-dimensional geometry known as Sasakian geometry. This geometry was initiated by Shigeo Sasaki in the 1960′s and has gained an immense amount of attention particularly in the last two decades. One reason for this attention is that Sasakian geometry can be viewed as a tool that can be used to construct special spaces, which are called “Einstein”. We will discuss these ideas by together building up a good example. If time permits we shall discuss more recent results in the area.

### LaTeX, Beamer, Inkscape and Sage: Integrating free open-source software resources for writing and presenting math

by **Professor Kim Plofker**, Union College

October 7, 2014, 5:00 pm

Abstract: Students (and professors) writing or presenting papers in the mathematical sciences often struggle with adapting familiar and accessible software tools to the demands of math-intensive output. Proprietary software designed for such tasks often requires considerable expense and effort to maintain. Is there another way? This talk will discuss the advantages, disadvantages and tasks of creating a toolkit of free open-source software applications for combining math text and visuals in a variety of formats.

### Amalgamate!

by **Alex Clain ’15 & Professor William Zwicker**, Union College

September 29, 2014, 5:00 pm

Abstract: Amalgamation takes place in diverse contexts: • Students will be divided into one group ready for advanced material, and another needing more review. We need to amalgamate the varying views of their former teachers into one such division. • Different types of evidence (genetic, physiological, fossil record, …) suggest varying stories about which species are descended from which. We need to amalgamate into one “phylogenic tree.” • Several friends at a restaurant will share a bottle of wine. Varying preferences (red, white, rosé) must be amalgamated into one choice. Many others! The actual methods used to amalgamate in one context look, initially, quite different from those used in another. As we’ll see, these differences in context are key to showing that a long list of some very well known methods are actually all restrictions of a single rule, first suggested by John Kemeny. In his summer research, Alex proved that a particularly famous amalgamation rule – the Borda count – can be added to the list.

### Imagining the Universe

by **Professor Brad Henry **, Siena College

September 23, 2014, 5:00 pm

Abstract: Is it possible to choose a star in the sky, travel to that star, and discover that you have arrived back in our own solar system? What might such a trip tell you about the shape of the universe? The concept of “shape” has fascinated scientists, mathematicians, and daydreamers for millennia. In the past, seafarers looked at the ocean horizon and wondered what lay beyond. Today stargazers dream of a spacecraft that will allow them to explore the depths of space. And all the while, mathematicians build sophisticated theories to try to explain our physical universe without ever leaving their desks. In this talk, we will seek to understand possible mathematical models of the universe by studying special knotted loops in space called Legendrian knots.

### Generating functions, the Fibonacci numbers, and the Golden Ratio

by **Professor Kathryn Lesh**, Union College

September 15, 2014, 5:00 pm

Abstract: The Fibonacci numbers are given bu the sequence 1, 1, 2, 3, 5, 8, 13, …, where each number is obtained by adding the two numbers before it. The Golden Mean is the number (1+√5)/2, and has a long and illustrious history in mathematics, art, and architecture, among other fields. We’ll use generating functions to derive a connection between the Fibonacci numbers and the Golden Mean, and look at some interesting consequences.

## Spring 2014

PBS Documentary: The Infinite Secrets of Archimedes

May 26, 2014, 5:00 pm

Primes, Primes, and More Primes

Prof. Allison Pacelli, Williams College & Union Alumni 1997

May 19, 2014, 5:00 pm

Big Data Analytics: The Good, the Bad, and the Ugly

Prof. Roger Hoerl, Union College

May 12, 2014, 5:00 pm

The Best Metric on a Sphere

Prof. Christina Tønnesen-Friedman, Union College

May 6, 2014, 5:00 pm

Exercises In Style: The Binomial Coefficients

Prof. John McCleary Vassar College

April 29, 2014, 5:00 pm

The birthday paradox and its consequences for encryption parameters

Prof. Kathryn Lesh, Union College

April 22, 2014, 5:00 pm

Playing With And Classifying Pythagorean Triples

Prof. Paul Friedman, Union College

April 15, 2014, 5:00 pm

When do two square matrices “commute”?

Prof. Leila Khatami, Union College

April 7, 2014, 5:00 pm

## Winter 2014

Movie Night

Union Undergraduate Mathematics in Conjunction with Union Math Club

Introduction by Prof. Karl Zimmermann

March 3, 2014, 5:00 pm, Golub House

Allocating Indivisible Items in Categorized Domains

Prof. Lirong Xia, Rensselaer Polytechnic Institute

February 25, 2014, 5:00 pm

The Rise of Abstraction: From Cantor to Bourbaki to Category Theory

Prof. Kimmo Rosenthal, Union College

February 18, 2014, 5:00 pm

Groups, Graphs, and Geometry: What Negative Curvature Can Do for You

Susan Beckhardt, Union College Alumni & Graduate Student at SUNY Albany

February 10, 2014, 5:00 pm

Randomization and Disposal of a Prize

Andrew Mackenzie, Union College Alumni and Graduate Student at University of Rochester

February 4, 2014, 5:00 pm

## Fall 2013

Taking the Long View: The Life of Shiing-shen Chern,

November 12, 2013, 5 pm

How to cut a set into two pieces

Prof. William Zwicker, Union College

November 4, 2013, 5 pm

The Euclidean Algorithm and Irrational Numbers

Prof. Julius Barbanel, Union College

October 29, 2013, 5 pm

Mathematical Morsels from The Simpsons and Futurama

Prof. Sarah Greenwald (’91), Appalachian State University

October 21, 2013, 5 pm, Olin 115

Inferring Causation without Randomization: A matched design to assess the number of embryos to transfer during in vitro fertilization

Prof. Cassandra Wolos Pattanayak, Wellesley College

October 15, 2013, 5 pm

Symplectic Geometry and the Physics of Motion

Fatima Mahmood, University of Rochester

October 7, 2013, 5 pm

Mathematics and Poetry

Prof. Kim Plofker, Union College

October 1, 2013, 5 pm

The hypercube and hypersphere: breaking them down and building them up

Prof. Davide Cervone, Union College

September 24, 2013, 5 pm

## Spring 2013

Data Mining: Fool’s Gold? Or the Mother Lode?

Prof. Richard De Veaux, Williams College

May 27, 2013, 4:30 pm

Knots (and the Universe?)

Prof. Cynthia Curtis, The College of New Jersey

May 20, 2013, 4:30 pm

Fermat’s Last Theorem

A Horizon Documentary

May 14, 2013, 4:45 pm

Geometry of the Real Projective Plane

Prof. Christina Tønnesen-Friedman, Union College

April 30, 2013, 4:45 pm

A Game of Commuting Matrices

Prof. Leila Khatami, Union College

April 22, 2013, 4:30 pm

Morphisms in Musical Analysis

Prof. Thomas Fiore, University of Michigan-Dearborn

April 17, 2013, 4:30 pm

From Pigeonhole Principles to the Erdös-Szekeres Theorem

Prof. Alan Taylor, Union College

April 11, 2013, 12:45 pm

## Winter 2013

N is a Number: A Portrait of Paul Erdös,

March 5, 2013, 4:45 pm

Number Crunching Before Computers

Prof. Kim Plofker, Union College

February 25, 2013, 4:30 pm

Higher Order Condorcet Cycles

Prof. William S. Zwicker, Union College

February 18, 2013, 4:30 pm

Mathematics and Literature: Borges’ Library of Babel

Prof. Kimmo Rosenthal, Union College

February 14, 2013, 12:50 pm

What is the shape of a hanging cable?

Prof. Susan Niefield, Union College

January 22, 2013, 4:45 pm

Knots and Links in Graphs

Prof. Brenda Johnson, Union College

January 14, 2013, 4:30 pm

## Fall 2012

How Hard Can It Be to Predict An Election?

Prof. Roger Hoerl, Union College

November 6, 2012, 4:45 pm, Bailey Hall 207

Skin Cancer Recognition Using Image Processing

Prof. Tomáš Kouřim, Czech Technical University in Prague, Faculty of Nuclear Sciences and Physical Engineering, Department of Mathematics

October 23, 2012, 4:45 pm, Bailey Hall 207

Louis Bachelier and the Origins of Stochastic Finance

Andrew Barnes, GE Global Research

October 15, 2012, 5 pm, Bailey Hall 207

Identification Numbers and Check Digit Schemes

Prof. Paul Friedman, Union College

October 11, 2012, 12:50 pm, Bailey Hall 201

Who Discovered Integral Calculus? Archimedes: The Method of Exhaustion and the Mechanical Method

Prof. Julius Barbanel, Union College

October 2, 2012, 4:45 pm, Bailey Hall 207

Student Summer Research

Keilah Creedon (’14) and Amer Khraisat (’13), Union College

September 24, 2012, 5 pm, Bailey Hall 207

An Introduction to the Seminar and to Polygonal Numbers

Prof. Karl Zimmermann, Union College

September 17, 2012, 5:00 pm, Bailey Hall 207

## Spring 2012

So You Want to Be a Grad Student

Susan Beckhardt and Dan Stevenson, SUNY Albany

May 28, 2012, 4:45 pm, Bailey Hall 207

When lines and circles are all the same!

Prof. Pedram Safari, Harvard University

May 22, 2012, 4:45 pm, Bailey Hall 207

Modeling Leatherback Sea Turtle Populations: How Can Counting Be So Difficult?

Prof. Sheila Miller, New York City College of Technology

April 30, 2012, 4:45 pm, Bailey Hall 207

Taking the Long View: The Life of Shiing-shen Chern (2011)

Produced and directed by George Paul Csicsery,

April 16, 2012, 4:45 pm, Bailey Hall 207

Wheels and Thunderbolts: Second-Order Indeterminate Equations in Medieval Indian Mathematics

Prof. Kim Plofker, Union College

April 3, 2012, 4 pm, Bailey Hall 207

## Winter 2012

Mathematics in Literature

Is that really a math course?

Prof. Donald R. Wilken, Department of Mathematics, University of Albany

March 5, 2012, 4:30 pm, Bailey Hall 207

Voting with Rubber Bands, Weights, and Strings

Prof. William S Zwicker, Union College

February 20, 2012, 4:30 pm, Bailey Hall 207

Introduction to Geodesics and the basic concepts of Projective Geometry

Prof. Stefan Rosemann, Mathematisches Institut, Friedrich-Schiller-Universität, Jena, Germany

February 13, 2012, 4:30 pm, Bailey Hall 207

Commuting Quadratic Polynomials

Prof. Karl Zimmermann, Union College

February 7, 2012, 5 pm, Bailey Hall 207

Geometric Perspectives on Fair Division

Prof. Julius Barbanel, Union College

January 24, 2012, 5 pm, Bailey Hall 207

Least-Squares Curve-Fitting Techniques For Improved Medicine Development

Prof. Charles Bergeron, Rensselaer Polytechnic Institute

January 10, 2012, 5 pm, Bailey Hall 207

## Fall 2011

Is 3828001 prime? Primality testing: a key ingredient for encryption

Prof. Kathryn Lesh, Union College

November 7, 2011, 5 pm, Bailey Hall 207

The Isoperimetric Inequality

Prof. Christina Tønnesen-Friedman, Union College

November 1, 2011, 4:45 pm, Bailey Hall 207

The Theory of Relations

Prof. Kimmo Rosenthal, Union College

October 24, 2011, 5 pm, Bailey Hall 207

The Two Envelope Problem

Prof. Chris Hardin, Union College

October 17, 2011, 5 pm, Bailey Hall 207

Fibonacci Relationships Geometrically

Prof. Vincent Ferlini, Keene State College

October 11, 2011, 4:45 pm, Bailey Hall 207

Which is Larger e^{π} or π^{e}?

Prof. Susan Niefield, Union College

October 4, 2011, 4:45 pm, Bailey Hall 207

Movie Day

N is a number, A Portrait of Paul Erdös,

September 26, 2011, 5:00 pm, Bailey Hall 207

Interlude: Newton method, complex numbers and dynamics

Prof. Julien Keller, Centre de Mathmatiques et Informatique, Universit de Provence, France

September 19, 2011, 5:00 pm, Bailey Hall 207

## Spring 2011

Einstein, Extremal, and Quasi-Einstein metrics on Spheres, Teardrops, and Footballs

Prof. Christina Tønnesen-Friedman, Union College

May 30, 2011, 4:30 pm, Bailey Hall 207

The path from Euclid’s geometry to differential geometry

Prof. Hal Sadofsky, University of Oregon

May 23, 2011, 4:30 pm, Bailey 207

Influence of Elliptical Arcs on Half-Pipe Performances

Peter Bonventre, Steven Neier, Erik Skorina, 2011 Union College Mathematical Contest in Modeling team

May 16, 2011, 4:30 pm, Bailey Hall 207

Using Group Theory to Solve Chemical Problems

Prof. Janet Anderson, Union College

May 9, 2011, 4:30 pm, Bailey Hall 207

Constructions With Straightedge and Compass: Doubling the Cube

Prof. Karl Zimmermann, Union College

April 26, 2011, 4:30 pm, Bailey Hall 100

Mathematics and the History of Mathematics in Western and Non-Western Traditions

Prof. Kim Plofker, Union College

April 18, 2011, 4:30 pm, Bailey Hall 207

The Mathematics of Stellar Wind Bubbles

Prof. Francis Wilkin, Union College, Department of Physics and Astronomy

April 11, 2011, 4:30 pm, Bailey Hall 207

## Winter 2011

Catalan Numbers Everywhere

Prof. Sam Hsiao, Bard College

January 17, 2011, 5:00 pm, Bailey Hall 207

A Proof that π Is Irrational

Prof. Paul Friedman, Union College

January 10, 2011, 4:30 pm, Bailey Hall 207

## Fall 2010

A Brief History of Set Theory

Prof. Alan Taylor, Union College

November 1, 2010, 4:30 pm, Bailey Hall 207

The Chain Rule — What more can we say?

Prof. Brenda Johnson, Union College

October 25, 2010, 4:30 pm, Bailey Hall 207

Adjoints- The Power of Abstraction

Prof. Kimmo Rosenthal, Union College

October 18, 2010, 4:30 pm, Bailey Hall 207

Typesetting with LaTeX

Prof. Chris Hardin, Union College

October 4, 2010, 4:30 pm, Bailey Hall 207

The Geometry of Influence: Weighted Voting and Hyper-ellipsoids

Prof. William S. Zwicker, Union College

September 27, 2010, 4:15 pm, Bailey Hall 207

The Principle of Competitive Exclusion in Population Biology

Prof. Hubert Noussi, Union College

September 13, 2010, 4:30 pm, Bailey Hall 207

Some Antinomies in Epistemic Game Theory and the Modal Logics of Knowledge, Belief, and Rationality

Prof. Herbert Gintis, Santa Fe Institute and Central European University

September 10, 2010, 4:00 pm, Bailey Hall 207

## Spring 2010

What can you carry in your knapsack?

Prof. Kathryn Lesh, Union College

May 17, 2010, 4:30 pm, Bailey Hall 207

Spirals in Plants?

Prof. Christophe Golé, Smith College

May 3, 2010, 4:30 pm, Bailey Hall 207

The Euclidean Algorithm and Irrational Numbers

Prof. Julius Barbanel, Union College

April 26, 2010, 4:30 pm, Bailey Hall 207

How Sweet It Is: An Analysis of the “Sweet Spot” of a Baseball Bat

Peter Bonventre, Steven Neier, and Pengfei Zhang, Union College

April 20, 2010, 4:00 pm, Bailey Hall 207

SCUBA Diving: An example of where some math common sense can go a long way!

Prof. Jennifer Blue, Union College

April 12, 2010, 4:30 pm, Bailey Hall 207

Introduction to Perturbation Theory

Prof. Pablo Suarez, Union College

April 6, 2010, 4:00 pm, Bailey Hall 207

## Winter 2010

The Logic of Knowledge

Prof. Chris Hardin, Union College

March 2, 2010, 4:00 pm, Bailey Hall 201

The Kervaire Invariant of Immersions

Prof. Mike Hill, University of Virginia

February 22, 2010, 4:30 pm, Bailey Hall 207

Embarrassing Moments in the History of Calculus

Prof. Kim Plofker, Union College

February 15, 2010, 4:30 pm, Bailey Hall 207

Commuting Polynomials and Generalized Odd Polynomials

Prof. Karl Zimmermann, Union College

February 8, 2010, 4:30 pm, Bailey Hall 207

Connect the Dots: Geometric Representations of Graphs

Prof. Alice M. Dean, Skidmore College

February 2, 2010, 4:00 pm, Bailey Hall 201

Reconstruction of Ultrasound Imaging Information

Prof. Jue Wang, Union College

January 25, 2010, 4:30 pm, Bailey Hall 207

Competition of Microorganisms in the Chemostat

Prof. Hubert Noussi, Union College

January 19, 2010, 4:00 pm, Bailey Hall 201

Intuitionistic Logic, Pointless Topology, and other Curiosities

Prof. Kimmo Rosenthal, Union College

January 11, 2010, 4:30 pm, Bailey Hall 207

Please view the seminars for the current term here.