Colloquium

Generalizations of Euler’s Theorem to Matrices

04/15/2015 - 4:15pm
04/15/2015 - 5:15pm
Speaker: 
Bogdan Petrenko, Eastern Illinois University
Abstract: 

This talk will be based on my work with Marcin Mazur (Binghamton University). V.I. Arnold’s investigations of discrete dynamical systems led him to many intriguing number-theoretical ques- tions. In particular, he was motivated by dynamical considerations when he asked whether it would be possible to extend Euler’s totient theorem to square matrices with integer entries. This theorem states that if a and n ≥ 2 are relatively prime positive integers, then n divides aφ(n) − 1, where φ(n) is the number of integers from 1,2,...,n−1 that are relatively prime to n. When n is a power of a prime number, such an extension was obtained by W. J ̈anichen in 1921, and it was independently rediscovered by I. Schur in 1937, and by V.I. Arnold in 2001. In this talk, I will present such an extension for all 2-by-2 integer matrices and all n ≥ 2; this extension implies Euler’s totient theorem. We will also see that such an extension does not exist for all r-by-r integer matrices, where r ≥ 3 is fixed, and all n ≥ 2. It is very likely, however, that for each such r, there exist interesting two-term congruences that remain to be discovered.

Where: 
Shanahan Center for Teaching and Learning (SCTL), at Harvey Mudd, Basement, B460

Clots or not? Mathematics making the invisible visible

04/08/2015 - 4:15pm
04/08/2015 - 5:15pm
Speaker: 
Ami Radunskaya, Pomona College
Abstract: 

Mathematical modeling is the art of representing real-world processes in mathematical terms.  These abstractions can be used to understand the behavior of devices, to predict outcomes, or to test hypotheses.  In this talk I will describe current work whose goal is to understand how traditional in vitro coagulation tests compare to what is actually going on in vivo. Since these in vitro tests are used to prescribe anti-coagulants, it is crucial to know whether they are indeed a measure of how quickly a clot will form.  I will show how differential equations  can be used to ``see” what is happening inside the blood vessel.   I will discuss the mathematical challenges inherent in this type of research, as well as the potential for discovery.

No expertise in mathematical modeling OR anti-coagulants is assumed.  Please come join the discussion!

This is joint work with the WhAM! clotbusters research group.

Where: 
Shanahan Center for Teaching and Learning (SCTL), at Harvey Mudd, Basement, B460

Counting faces on simplicial complexes: V-E+F and beyond

04/01/2015 - 4:15pm
04/01/2015 - 5:15pm
Speaker: 
Steven Klee, Seattle University
Abstract: 

A graph is a combinatorial object that is built out of vertices and edges.  More generally, a simplicial complex is a combinatorial object that is built out of vertices, edges, triangles, tetrahedra, and their higher-dimensional cousins.  The most natural combinatorial statistics to collect on a simplicial complex are its face numbers, which count the number of vertices, edges, and higher-dimensional faces in the complex.  

This talk will give a survey on face numbers of simplicial complexes, beginning with planar graphs and moving on to graphs on other surfaces, such as tori or projective planes.   From there, we will study spheres and manifolds of higher dimensions.  We will undertake two main questions in this talk: First, what is the relationship between the face numbers of a simplicial complex and its underlying geometric structure? Second, how can we infer extra combinatorial information from properties of the underlying graph of a simplicial complex, such as graph connectivity or graph colorability?

Where: 
Shanahan Center for Teaching and Learning (SCTL), at Harvey Mudd, Basement, B460

Universal Spaces

03/25/2015 - 4:15pm
03/25/2015 - 5:15pm
Speaker: 
Asuman Aksoy, Claremont McKenna College
Abstract: 

In a paper published posthumously in 1927 (Bull. Sci. Math 51(1927) 43-64 and 74–96), P.S. Urysohn constructed a complete, separable metric space that contains an isometric copy of every complete separable metric space. This result is now referred to as the Urysohn universal space. In this talk, I examine Urysohn’s original construction and various convexity properties of this “special” universal space and show that it has a finite ball intersection property even though the Urysohn universal space is not hyperconvex. This is joint work with Z. Ibragimov, California State University, Fullerton.

Where: 
Shanahan Center for Teaching and Learning (SCTL), at Harvey Mudd, Basement, B460

Through the luring graph and what analysts found there

03/11/2015 - 4:15pm
03/11/2015 - 5:15pm
Speaker: 
Yves Van Gennip, University of Nottingham
Abstract: 

Lured in by applications in image processing and data analysis, in recent years a number of analysts have turned their attention to graph based problems. Studies of some of these problems, which can be interpreted as analogues to classical continuum partial differential equation models, not only are very useful in practice, but also show interesting connections between the continuum results and the graph problems.


In this talk we will explore some of these graph based PDE type problems and their applications to image and data science.

Where: 
Shanahan Center for Teaching and Learning (SCTL), at Harvey Mudd, Basement, B460

No Colloquium--Spring Break

03/18/2015 - 12:00am
Speaker: 
N/A
Abstract: 

N/A

Where: 
N/A

Action graphs and Catalan numbers

03/04/2015 - 4:15pm
03/04/2015 - 5:15pm
Speaker: 
Julie Bergner, University of California, Riverside
Abstract: 

Action graphs are labeled directed graphs that arose in the study of group actions on other algebraic objects. The 0th action graph consists of a vertex and no edges, and new vertices and edges are added at each stage by an inductive process. We will prove that the number of new vertices (and edges) given at the nth step is given by the nth Catalan number. We will then give a direct comparison between these action graphs and planar rooted trees, which give another known method for producing Catalan numbers. Lastly, we will look at the motivation for defining action graphs and some of their generalizations. This work was done in collaboration with P. Hackney, G. Alvarez, and R. Lopez.

Where: 
Shanahan Center for Teaching and Learning (SCTL), at Harvey Mudd, Basement, B460

Mathematics and Precision Medicine

02/25/2015 - 4:15pm
02/25/2015 - 5:15pm
Speaker: 
Bob Palais, University of Utah
Abstract: 

Mathematics will play a key role in the recently announced Precision Medicine Initiative. We will describe mathematical methods being developed to analyze and interpret DNA and other molecules that impact our health, and used to enhance diagnosis and therapy. These include rapid economical tests to identify and quantify genetic variations without sequencing, used in tests for transplant compatibility, simultaneous detection of a variety of pathogens including Ebola, newborn screening, and a cancer therapy. We will see how some surprising and interesting mathematical connections can appear in the process.

Where: 
Shanahan Center for Teaching and Learning (SCTL), at Harvey Mudd, Basement, B460

Sandpiles and Dominos

02/18/2015 - 4:15pm
02/18/2015 - 5:15pm
Speaker: 
David Perkinson, Reed College
Abstract: 

The Abelian Sandpile Model (ASM) is a mathematical model devel- oped by physicists around 1990 to elucidate self-organized criticality, a phe- nomenon claimed to be ubiquitous in nature. Roughly, self-organized criti- cality describes a system that naturally evolves into a state at the border of stability, with instabilities over time characterized by scale invariance. The Gutenberg-Richter law in geophysics and Zipf’s law in linguistics are often cited as real-world examples. The ASM has been shown to have connections to algebraic geometry and commutative algebra, combinatorics, potential theory, and number theory.
In this talk, I will present work done with undergraduate students con- necting the sandpile model with domino tilings. We will be interested in tiling an m × n checkerboard (m rows and n columns) with dominos. A domino covers exactly two squares of the checkerboard, and a tiling consists of covering the checkerboard with non-overlapping dominos.
As warm-up for the talk you may want to answer the following two ques- tions: (i) How many ways are there of tiling a 4 × 4 checkerboard with dominos? (ii) Take a flexible 4 × 4 checkboard and glue one of its edges to the opposite edge with a twist to get a Mo ̈bius band. How many ways are there of tiling this twisted checkerboard?

Where: 
Shanahan Center for Teaching and Learning (SCTL), at Harvey Mudd, Basement, B460

Can Zombies Do Math? OR Humanism as a Philosophy of Mathematics

02/11/2015 - 4:15pm
02/11/2015 - 5:15pm
Speaker: 
Gizem Karaali, Pomona College
Abstract: 

Skimming through recent book and movie titles, one might imagine that we are headed for a zombie apocalypse. Many have written about what this would entail for our civilization, for our culture, and even for our consumerist tendencies. In this talk we will look at yet another facet of this phenomenon: What would happen to our mathematics? Guided by the history and the philosophy of mathematics, we will pose and search for answers to fundamental questions about the nature of mathematics and how it relates to our humanity. It is this speaker's main goal that by the end of the talk, the audience will be able to answer the question on the title, along with a few other, possibly more respectable, philosophical questions, such as "What is 3?"

Where: 
Shanahan Center for Teaching and Learning (SCTL), at Harvey Mudd, Basement, B460
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