Student Seminar, Spring 2018

 

Upcoming Student Seminar Talks

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We will continue meeting during common hour on Thursdays in Bailey 207, with light lunch served at 12:30pm in Bailey 204, unless otherwise noted.

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.

Past talks

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 d2=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 curvature of 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.

Please view a list of seminars from previous years here.

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