# Upcoming Student Seminar Talks

Our student seminar for the spring term 2017 will begin on April 4th. Please remember to visit us here for newly scheduled seminars. The seminar meets at 5:00pm in Bailey 207, unless otherwise noted, with light refreshments served at 4:45pm in Bailey 204.

# Spring term 2017

### Student seminars have concluded for the term. Please check back with us in the fall.

# Past Talks

### 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

### 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.

### 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.