Colloquium

Nonstandard Methods in Combinatorics

10/21/2015 - 4:15pm
10/21/2015 - 5:15pm
Speaker: 
Martino Lupini (Cal. Tech.)
Abstract: 

I will give an introduction to nonstandard analysis, and then present an overview of the application of nonstandard methods to problems in combinatorics of numbers.

Where: 
Argue Auditorium, Millikan, Pomona College

Linear Algebra as a Natural Language for Special Relativity and Its Paradoxes.

10/14/2015 - 4:15pm
Speaker: 
John de Pillis (UC Riverside)
Abstract: 

See Attachment

Where: 
Argue Auditorium, Millikan, Pomona College

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
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