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

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

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

03/18/2015 - 12:00am

Speaker:

N/A

Abstract:

N/A

Where:

N/A

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

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

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

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

01/28/2015 - 4:15pm

01/28/2015 - 5:15pm

Speaker:

Sinai Robins, Nanyang Technological University, Singapore and Brown University

Abstract:

We study the problem of covering Euclidean space R^d by possibly overlapping translates of a convex body P, such that almost every point is covered exactly k times, for a fixed integer k. Such a covering of Euclidean space by translations of P is called a k-tiling. Classical tilings by translations (which we call 1-tilings in this context) began with the work of the famous crystallographer Fedorov and with the work of Minkowski, who founded the Geometry of Numbers. Some 50 years later Venkov and McMullen gave a complete characterization of all convex objects that 1-tile Euclidean space.

Today we know that k-tilings can be tackled by methods from Fourier analysis, though some of their aspects can also be studied using purely combinatorial means. For many of our results, there is both a combinatorial proof and a harmonic analysis proof. For k larger than 1, the collection of convex objects that k-tile is much wider than the collection of objects that 1-tile. So it's a more diverse subject with plenty (infinite families) of examples in R^2 as well. There is currently no complete knowledge of the polytopes that k-tile in dimension 3 or larger, and even in 2 dimensions it is quite challenging. We will cover both ``ancient'', as well as very recent, results concerning 1-tilings and other k-tilings. This is based on some joint work with Nick Gravin, Dmitry Shiryaev, and Mihalis Kolountzakis.

Where:

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

02/04/2015 - 4:15pm

02/04/2015 - 5:15pm

Speaker:

Cynthia Flores, California State University, Channel Islands

Abstract:

A classical problem in the theory of Partial Differential Equations (PDEs) is knowing when a solution to a given Initial Value Problem (IVP) exists, is unique and in what space does the solution persist. In this talk, we will motivate the definition of weighted Sobolev spaces and their role in describing solutions to the Initial Value Problem (IVP) for the Benjamin-Ono equation. Specifically, we will discuss decay properties of solutions corresponding to the IVP in weighted Sobolev spaces. This talk will be aimed at a broad audience.

Where:

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