Colloquium

Approximation theory perspective on data processing

04/06/2016 - 4:15pm
04/06/2016 - 5:15pm
Speaker: 
Hrushikesh Mhaskar (CGU/Caltech)
Abstract: 

We argue that data based classification and regression problems can be viewed profitably as problems of function approximation. We describe in detail a case study in the theory of function approximation in case of approximation of continuous 2*pi–periodic functions by trigonometric polynomials. In particular, we will discuss the equivalence theorems, and some relatively new research regarding local approximation based on Fourier information. Several extensions to other settings are mentioned, and a few applications and potential applications are discussed.

Where: 
Argue Auditorium, Millikan, Pomona College

Fixed Point Free Permutations

02/17/2016 - 4:15pm
Speaker: 
Robert Guralnick (USC)
Abstract: 

A permutation g on a set X is said to be fixed point free (or a derangement) if g fixes no points.   If G is a finite group acting transitively on a set of cardinality greater than 1, then an elementary 1872 theorem of Jordan says that G contains derangements.  We will discuss generalizations of this result and applications to finite fields and number theory.

 

Where: 
Argue Auditorium, Millikan, Pomona College

Complicated triangulations and breast cancer research

02/10/2016 - 4:15pm
02/10/2016 - 5:15pm
Speaker: 
Mimi Tsuruga (UC Davis)
Abstract: 

Topology is considered to be one of the more pure (and perhaps? less useful) subjects in mathematics. However, the last two decades have seen a surge in research in applied topology, where topological tools have been employed to solve problems in image processing, dynamical systems, neuroscience, and virus evolution. In this talk, we will give a very basic introduction to a few topological tools and how some of them can be used to analyze genomic data of breast cancer.

Where: 
Argue Auditorium, Millikan, Pomona College

Dynamical Systems from a Number Theory Perspective

01/18/2016 - 3:45pm
01/18/2016 - 4:45pm
Speaker: 
Joe Silverman
Abstract: 

Dynamics is the study of iteration.  Classically one takes a rational map f(z)=F(z)/G(z) and attempts to describe the behavior of points under interation f^n(z) = f(f(f(...f(f(z))...))). The *f-orbit* of a point b is the set of images of b under the repeated iteration of f, Orbit of b = { b, f(b), f^2(b), f^3(b), ... }. The points with finite orbit are called *preperiodic points*. They play a particularly important role in the dynamics of f. For a number theorist, it is natural to take F(z) and G(z) to be polynomials with integer coefficients and to study the orbits of rational numbers b. In this talk I will survey some of the known results and some of the outstanding conjectures related to this number-theoretic view of dynamics.

Typical problems include: 

(1) How many preperiodic points can be rational numbers?

(2) For which rational maps f can the orbit of a rational number b contain infinitely many integers?

 

Where: 
Emmy Noether Room, Millikan 1021, Pomona College

Squaring the Circle and Other Impossible Dreams: Periods and Transcendence

04/13/2016 - 4:15pm
04/13/2016 - 5:15pm
Speaker: 
Paula Tretkoff (TAMU)
Abstract: 

The phrase “squaring the circle” is a recognized idiom of the English language that has come to mean “to solve an unusually hard problem”. This idiomatic meaning is very much reflected in the history of the ancient mathematical problem that gave rise to it: roughly speaking, the problem asks if it is possible to construct a square with the same area as the unit circle using compass and straightedge. The impossibility of squaring the circle is equivalent to the transcendence of the number π: that is, to showing there is no non-zero polynomial P(x) with rational coefficients such that P(π) = 0. Proving the transcendence of π was indeed unusually hard and was only achieved in 1882 by Lindemann, following a method due to Hermite, who proved the transcendence of e in 1873. The transcendence of π is equivalent to that of 2πi, a period of the function exp(z) in that exp(z + 2πi) = exp(z). Moreover e is the special value exp(1). The work of Hermite and Lindemann opened the way to a grand modern theory for proving the transcendence of numbers related to periods and values at algebraic points of other classical functions, like elliptic and, more generally, abelian functions, as well as hypergeometric functions. In this talk, we focus on periods in transcendental number theory, the evolution of results, for example, Hilbert’s seventh problem, the breakthroughs of Baker (Fields Medal 1970) and W¨ustholz, as well as some on some of my work, in part joint with M.D. Tretkoff. We also discuss challenges for the future. The talk presumes the intrigue of impossible dreams but does not assume any background in transcendence.

Where: 
Argue Auditorium, Millikan, Pomona College

Turan's Problem

03/23/2016 - 4:15pm
03/23/2016 - 5:15pm
Speaker: 
Annie Raymond (University of Washington)
Abstract: 

What is the maximum number of edges in a graph on n vertices without triangles? Mantel's answer in 1907 that at most half of the edges can be present started a new field: extremal combinatorics. More generally, what is the maximum number of edges in a n-vertex graph that does not contain any subgraph isomorphic to H? What about if you consider hypergraphs instead of graphs? We will explore different strategies to attack such problems, calling upon combinatorics, integer programming, semidefinite programming and flag algebras. We will conclude with some recent work where we embed the flag algebra techniques in more standard methods.

 

This is joint work with Mohit Singh and Rekha Thomas.

Where: 
Argue Auditorium, Millikan, Pomona College

Mathematical modeling of blood clot formation under flow and microorganism motility in heterogeneous fluids

03/09/2016 - 4:15pm
03/09/2016 - 5:15pm
Speaker: 
Karin Leiderman (UC Merced)
Abstract: 

Many biological fluid environments are inherently heterogeneous, comprised of macromolecules, polymers networks, and/or cells. These environments can be thought of, in some sense, as fluids interacting with biological porous materials. In this talk I will describe mathematical models of two such situations: dynamic formation of blood clots under flow and motility of microorganisms through the reproductive tract. The model of clot formation was developed to better understand the interplay between flow-mediated transport of clotting proteins and cells, blood cell deposition at an injury site, complex clotting biochemistry, and the overall effect of these on clot formation. A derivation of hindrance coefficients for the advection and diffusion of clotting proteins will be presented. Results show that the effects of such hindrance are profound and suggest a possible physical mechanism for limiting clot growth. Next I will describe regularized fundamental solutions to the governing equations of flow through porous material and the corresponding numerical method used to model a swimming microorganism. Results show that for certain ranges of porosity of the material through which they swim, a single swimmer's propulsion is faster and more efficient, and two swimmers are attracted toward one another.

Where: 
Argue Auditorium, Millikan, Pomona College

A Game Theoretic Approach to Pediatric Vaccine Pricing

03/02/2016 - 4:15pm
03/02/2016 - 5:15pm
Speaker: 
Banafsheh Behzad
Abstract: 

Pricing strategies in the United States pediatric vaccines market are studied using a Bertrand-Edgeworth-Chamberlin price game. The game analyzes the competition between asymmetric manufacturers with limited production capacities and linear demand, producing differentiated products. The model completely characterizes the unique pure strategy equilibrium in the Bertrand-Edgeworth-Chamberlin competition in an oligopoly setting. In addition, the complete characterization of mixed strategy equilibrium is provided for a duopoly setting. The results indicate that the pure strategy equilibrium exists if the production capacity of manufacturers is at their extreme. For the capacity regions where no pure strategy equilibrium exists, there exists a mixed strategy equilibrium. In a duopoly setting, the distribution functions of the mixed strategy equilibrium for both manufacturers are provided. The proposed game is applied to the United States pediatric vaccine market, in which a small number of asymmetric vaccine manufacturers produce differentiated vaccines. The sources of differentiation in the competing vaccines are the number of medical adverse events, the number of different antigens, and special advantages of those vaccines. The results indicate that the public sector prices of the vaccines are higher than the vaccine equilibrium prices. Furthermore, the situation when shortages of certain pediatric vaccines occur is studied. Market demand and degree of product differentiation are shown as two key factors in computing the equilibrium prices of the vaccines.

 

Where: 
Argue Auditorium, Millikan, Pomona College

Math clinic in the tropic: The USB Experience.

02/24/2016 - 4:15pm
02/24/2016 - 5:15pm
Speaker: 
Minaya Villasana de Armas
Abstract: 

The Claremont Colleges are well known for the Math Clinics, which has attracted the attention of scholars and students alike.  Perhaps unknowingly, the Claremont Colleges have exported that idea to other universities and even overseas!  In this talk I will address the strengths and the challenges of the Math Clinics in Venezuela and showcase one such industrial problem and the results obtained: Characterization of ideal defects.  The ultimate goal of this talk is to get feedback on how to improve the math clinic experience in the tropic from the experts!

 

Where: 
Argue Auditorium, Millikan, Pomona College

NP-Hardness: You keep using that term---I do not think it means what you think it means.

02/03/2016 - 4:15pm
02/03/2016 - 5:15pm
Speaker: 
David Morrison
Abstract: 

The computational notion of 'NP-hardness' is often used as an indication that a problem cannot be solved effectively in practice. However, this is in fact not true.  Many real-world instances of NP-hard problems such as the traveling salesman problem, graph coloring, and others, can be solved extremely quickly in practice. In this talk, we discuss reasons for this disconnect between theory and practice, and describe a number of recent results that improve the performance of algorithms for such problems even further. 

 

Where: 
Argue Auditorium, Millikan, Pomona College
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