Student Seminar, Fall 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.

Fall term 2018

Our student seminars have concluded for the fall term. Please visit us back here in January when we begin our 2019 winter term.
 

Past Talks

 

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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