Colloquium

Biquandle Brackets

10/07/2015 - 4:15pm
10/07/2015 - 5:15pm
Speaker: 
Sam Nelson (Claremont Mckenna College)
Abstract: 

Biquandles are algebraic structures with axioms inspired by knot theory. Given a finite biquandle X and a commutative ring with identity R, we define an algebraic structure known as a biquandle bracket. Biquandle brackets can be used to define a family of knot and link invariants

known as quantum enhancements which include biquandle cocycle invariants and skein polynomials such as the Alexander, Jones and HOMFLYpt polynomials as special cases. As an application we will see a new skein invariant which is not determined by the knot group, the knot quandle or the HOMFLYpt polynomial.

Where: 
Argue Auditorium, Millikan, Pomona College

Energy driven pattern formation: The good, the bad and the beautiful

09/30/2015 - 4:15pm
09/30/2015 - 5:15pm
Speaker: 
Andy Bernoff (Harvey Mudd College)
Abstract: 

A wide variety of physical and biological systems can be described as continuum limits of interacting particles. Their dynamics can often be described in terms of a monotonically decreasing interaction energy. We show how to exploit these energies numerically, analytically and asymptotically to characterize the observed behavior. Examples are drawn from the dynamics of thin fluid layers including ferrofluids.

Where: 
Argue Auditorium, Millikan, Pomona College

The Intermediate Value Theorem and Generalizations

09/23/2015 - 4:15pm
09/23/2015 - 5:15pm
Speaker: 
Alfonso Castro (HMC)
Abstract: 

In mathematics, the Art of Solving Equations, the intermediate value theorem plays a central role. I will be discussing its impact in finding fixed points, analyzing functions of complex variables, and solving equations in ordered Banach spaces.

Where: 
Argue Auditorium, Millikan, Pomona College

Poster Session

09/16/2015 - 4:15pm
09/16/2015 - 5:15pm
Speaker: 
Students
Abstract: 

See attached. 

Where: 
Millikan Courtyard

COMPARING THE DEGREES OF UNCONSTRAINED AND CONSTRAINED APPROXIMATION BY POLYNOMIALS

09/17/2015 - 4:15pm
09/17/2015 - 5:15pm
Speaker: 
Dany Leviatan
Abstract: 

See attachment.

Where: 
Argue Auditorium, Millikan, Pomona College

For linear algebra lovers: Carathéodory's theorem and its relatives

09/09/2015 - 4:15pm
09/09/2015 - 5:15pm
Speaker: 
Jesus De Loera (UC Davis)
Abstract: 

Are you someone that found linear algebra the most awesome and beautiful subject in the universe?  Do linear equations and vectors make you smile? Then I have the theorem for you; Carath\'eodory’s theorem! It states that any vector in the convex hull of a subset $X$ of $R^d$ can be expressed as a linear convex combination of at most $d+1$ vectors of the set $X$.  It is a variation on the basic fact that vectors in $R^d$ can be expressed as linear combination of a basis (with $d$ vectors). This talk will consider the lovely Carath\'eodory’s theorem, and its many relatives and variations. I will show the many applications (e.g., in economics, logistics, and signal processing) and how this theorem touches in the inner depths of the mathematician's soul. I will offer many open questions for people young solve, so come prepared to fall in love with $Ax=b$.

 

Where: 
Argue Auditorium, Millikan, Pomona College

Towards a mathematical understanding of geometric clustering problems

04/29/2015 - 4:15pm
04/29/2015 - 5:15pm
Speaker: 
Rachel Ward, University of Texas, Austin
Abstract: 

It is often said that  "clustering is difficult only when it does not matter".  We aim to make this statement more mathematically precise in the setting of clustering points in Euclidean space.  We will focus on the k-means optimization problem, arguably the most popular algorithm for unsupervised clustering.  While the k-means objective is highly nonconvex and hard to optimize in the worst case, heuristic algorithms are known to empirically produce good clusters "when it matters".   To explain these observations, we introduce geometric conditions on a set of points under which the k-means objective can be optimized efficiently.  For points drawn from separated balls, the important quantities are the distances between the centers of the balls compared to the relative density of points within the balls.  We will also discuss the case of outliers and overlapping point clouds, relationships and implications for other clustering algorithms such as spectral clustering, and conclude by discussing open questions related to this work.  This is joint work with P. Awasthi, A. Bandeira, M. Charikar, R. Krishnaswamy, and S. Villar.

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

Bringing Mathematical Creativity to the Forefront

04/22/2015 - 4:15pm
04/22/2015 - 5:15pm
Speaker: 
Gail Tang, University of La Verne
Abstract: 

Creativity is an important aspect of a mathematicians’ work (Sriraman, 2009), whether it is an enlightenment that is somewhat unexpected (Poincar, 1946) or a product that is aesthetically pleasing (Borwein, Liljedahl, & Zhai, 2014). There is a significant amount of mathematical creativity literature at the K-12 level (e.g., Silver, 1997; Lev-Zamir & Leikin, 2013). However, there is less at the undergraduate level, where students are at the early stages of becoming mathematicians. Our research group begins to address this deficit by examining how mathematicians perceive creativity in their work and in their teaching. In this talk, I will share participating mathematicians’ views and how those views contributed to the development of the Creativity-in-Progress Rubric (CPR) on proving. In addition, I will discuss implementation of the rubric on student data. Finally, conjectures on how instructors can foster the discussion of creativity in the classroom will be presented. This work is done in collaboration with M. Savic, G. Karakok, H. El Turkey, and E. Naccarato.

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

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
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