Student Seminar, Winter 2019

 

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.

Winter term 2019

 

Unexpected Groups

Jetjaroen Klangwang, Union College

Thursday, February 28th, 1:00pm, Bailey 207, with reception at 12:30pm in Bailey 204

Exercise : Prove that the set of real matrices
$$
G = \left\{ \begin{bmatrix}
a & a \\ a& a
\end{bmatrix}
\Big| \; a \neq 0 \right\}
$$
forms a group under the usual matrix multiplication.
 
This exercise guarantees an unease if we think that the question asks us to show that the given set is a of the multiplicative group of non-singular matrices.
 
The purpose of this talk is to give some examples of such groups, and to answer the question: Which sets of singular matrices form groups under matrix multiplication?

 

Past Talks

 

Using linear algebra to understand knots

Cynthia Curtis, Professor of Mathematics at The College of New Jersey – Union College Class of 1987

Thursday, February 14th, 1:00pm, Bailey 207, with reception at 12:30pm in Bailey 204

Given two single-variable functions, we are allowed to take their composite to produce a new function again of a single variable. In this talk, we will ask: “In what other contexts does ‘composition of functions’ make sense?”. We will slowly broaden our definitions of “function” and “composition”, starting with the types of functions that appear in the Calculus sequence, and moving to include well-behaved geometric figures. This will lead us to the abstract concept of an operad. We will give several examples, as well as an interpretation of what these new objects can do for us.

Generalizing composition of functions and Operads

Peter Bonventre, Union College Class of 2011

Friday, February 8th, 1:00pm, Bailey 207, with reception at 12:30pm in Bailey 204

Knots are prevalent in nature, and the study of knotting is important in diverse areas such as DNA, bonding of molecules, and statistical mechanics. Understanding knots has been fundamental within mathematics to our ability to understand three-dimensional spaces. In this talk we use linear algebra to generate polynomials, which help decide whether two given knots are different. This is a surprisingly hard question! The polynomials can also help us know when to look for hidden symmetries in the knots. The first knot polynomial we introduce was found by James Waddell Alexander II in 1923. We then discuss new polynomials arising from research with undergraduates Vincent Longo, Alyssa Springstead, and Hoang Cao at The College of New Jersey.

The Joy of Abstraction

Kimmo Rosenthal, Union College

Thursday, January 31st, 1:00pm, Bailey 207, with reception at 12:30pm in Bailey 204

The imagination is the only genius. It is intrepid and eager and the extreme of its achievement lies in abstraction Wallace Stevens.

It may seem incongruous for the epigraph to a math talk to be from one of the great American poets. However, while the ubiquity and utility of mathematics is widely acknowledged, its burnish of aestheticism is much less so. 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 an intellectual art form? Abstraction has always appealed to me and indeed guided me. Why does it often provoke outright hostility? We shall follow the path of abstraction from the set theory of Cantor (called a “corrupter of youth”) to point-set topology, followed by the mysterious emergence of Bourbaki (the mathematician who never existed), and finally category theory, which earned the epithet of “abstract nonsense”. Of course, there will be some mathematics along the way, reasonably modest in scope.

Hall’s Marriage Theorem

Alan Taylor, Union College

Thursday, January 24th, 1:00pm, Bailey 207, with reception at 12:30pm in Bailey 204

Suppose we have a collection of women and a collection of men, and each woman finds some of the men acceptable (and the rest not). When is it possible to match each woman with a man she considers acceptable, subject to the obvious constraint that the matching be one to one? The answer to this metaphorical question is a beautiful result in finite combinatorics known as Hall’s marriage theorem. We will discuss Hall’s theorem, sketch a proof of it, and consider a couple of natural questions it suggests, all with the hope of providing an illustration of how research gets done in mathematics.

 

Summer Opportunities for Math Students

Thursday, January 17th, 1:00pm, Bailey 207, with reception at 12:30pm in Bailey 204

This week’s seminar will focus on ways in which you can put your mathematical skills to use over the summer.

Julia Greene ’19 will speak about the Teaching Experiences for Undergraduates Program, Professor Jeff Hatley will speak about Research Experiences for Undergraduates, and Keri Willis of the Becker Career Center will speak about summer internships.

Turning the lights out, mathematician style

Leila Khatami, Union College

Thursday, November 1st, 1:00pm, Bailey 207, with reception at 12:30pm in Bailey 204

“LIGHTS OUT!” is a single player game played on a 5 by 5 grid where each cell has a button that can be turned on or off. Pressing a button toggles the light in the cell and its neighboring cells. The game starts with some cells turned on and some turned off. The goal of the game is to turn all cells off. The game was originally introduced in 1995 as a handheld electronic came. Nowadays, the original game, as well as many of its variants, are readily accessible in app stores and elsewhere. It is not obvious (or even true!) that all starting configurations of the game are “solvable”. In this talk, we use mathematical tools to see if a game is “solvable”. We also briefly discuss ways to find the most efficient solutions for solvable games.

 

Solving the General Cubic Equation

$ax^3$+$bx^2$+$cx+d=0$

Paul Friedman, Union College

Thursday, October 25th, 1:00pm, Bailey 207, with reception at 12:30pm in Bailey 204

The solution to the general quadratic equation,

$ax^2+bx+c=0$,
is well-known to most high school students:


It was also known to many “ancient” cultures … some dating back to 2000 BC! However, the solution to the general cubic equation, $ax^3+bx^2+cx+d=0$, is not as well known, and it was not found until the 1500s.

In this talk, we will look at how the Renaissance mathematicians Scipione del Ferro, Tartaglia, and Cardano, solved the cubic equation, though we will do so using modern language and notation. As a cute consequence, we will be able to derive some remarkable identities, such as:

 

Fair Division of a Graph: Envy Freeness up to one Good, or Two

William Zwicker, Union College

Thursday, Oct 18th, 1:00pm, Bailey 207, with reception at 12:30pm in Bailey 204

Countries A and B are dividing up a disputed island with several cities, linked by roads:

    • Each city must go entirely to A, or entirely to B
    • A city may be worth more to one country than to the other
    • You must be able to drive among A’s cities without going through B’s

Can cities be allocated in a way that leaves neither country jealous of the other’s share?

No – not in general. But with certain road networks one can always get within one city of this ideal. Which networks are these? With more than 2 countries, the question gets harder . . . in interesting ways.

In its “classical” setting Fair Division concerns sharing a single, continuously divisible resource. Some solutions can be adapted to this new setting, but a lot needs to change.

Forms of Remigration – Émigré Jewish Mathematicians and Germany in the Immediate Post-War Period

Volker Remmert, , of the Bergische Universitaet Wuppertal

Tuesday, Oct 9th, 4:45pm, Bailey 207, with reception at 4:15pm in Bailey 204

Over the last twenty years or so there has been a steady flow of historical studies on remigration into Germany in the immediate post-war period. These studies have described three main forms of academic remigration to Germany after World War II:

1) returning to universities in Germany on a permanent basis as university professors;

2) returning as visiting professors, assessing Germany without any obligation to stay;

3) returning for guest lectures and academic visits.

In this context my interest is in Jewish émigré mathematicians and their stance to Germany in the immediate post-war period.

 

Math, Music, and Health Science

Danielle Gregg ’19 and Robert Righi ’19, Union College Undergraduates

Thursday, Oct 4th, 1:00pm, Bailey 207

Much recent research has focused on discerning topological and geometric features of data. For example, by observing the “birth” and “death” of holes via an algebraic method known as persistent homology, we can distinguish noise from significant features in data. In analyzing the “shape” of data our research diverges into two separate fields: music and health science. How can one use geometric and topological methods to classify a variety of degenerative diseases of the eye or compare songs within an artist’s discography? Come learn about what two Union students researched over the past summer as well as the often non-linear research process.

Action Graphs and Catalan Numbers

Julie Bergner, University of Virginia and Cornell University

Thursday, Sept 27th, 1:00pm, Bailey 207

The Catalan numbers are given by a recursively-defined sequence and arise from over 200 different kinds of combinatorial objects. In 2013, two of my undergraduate research students, Gerardo Alvarez and Ruben Lopez, showed that a family of directed graphs called action graphs gives a new way to obtain this sequence. Since these graphs are defined inductively, one might ask what sequences we can get by using a different initial graph but the same induction process. Last year, three more students, Cedric Harper, Ryan Keller, and Mathilde Rosi-Marshall, looked into this question. They found new families, called $k$-initial action graphs, which produce self-convolutions of the Catalan sequence. In this talk we’ll introduce the sequences and graphs involved and talk about how these comparisons were made.

 

Cutting Up Space: Hilbert’s Third Problem and the Dehn Invariant

Jonathan Campbell, Vanderbilt University

Friday, Sept 21st, 1:00pm, Bailey 207

Give two polyhedra of equal volume, can you cut up one into a finite number of pieces, and reassemble it into the other? This was a problem posed by Hilbert in a famous address. I’ll go through the two dimensional analogue of this problem, and present Dehn’s beautiful solution to Hilbert’s question. Time permitting, I’ll give some hint of how this easily stated problem shows up in my own research.

 

Counting sudokus

Professor Brenda Johnson, Union College

Thursday, Sept 13th, 1:00pm, Bailey 207

Sudoku is a popular puzzle involving a 9×9 grid in which one has to arrange the numbers 1 through 9 so that each row, column, and block contains all nine numbers.  There are many interesting mathematical questions involving sudoku puzzles.  In this talk, we’ll focus on a couple of  questions related to counting sudokus.  After discussing how many possible solutions there are for 9×9 and 4×4 sudokus, we’ll look at ways in which one can generate new sudokus from old ones, and whether or not  these techniques can be used to generate all sudokus of a given size.

Please view a list of seminars from previous years here.

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