Math Department Events Listing

In part II of this introduction to the representations of the symmetric group, we will finish the proof that the Young symmetrizers generate all of the ordinary irreducible representations of S_n. After briefly discussing the intertwining number approach, we will then begin a discussion of Specht modules.

Many problems in pure and applied mathematics may be reduced to that of determining the spectrum of a linear operator. This is the case for the linear stability analysis of equilibrium solutions of finite or infinite-dimensional evolution systems, and for the forward scattering problem associated with any integrable system. In this talk, I will show how a method which goes back to Hill (1886) may be used to compute spectra of linear operators with periodic coefficients. It may be extended to problems on an infinite domain. The method is algorithmic in nature and as such its only competitor are finite-difference methods. Hill's method converges exponentially, due to its spectral origins. It also incorporates Floquet theory, allowing for the determination of the entire spectrum, as opposed to isolated elements of it. I will illustrate the method using a variety of examples.

In 1992 Milne made a conjecture on the existence of integral canonical models of Shimura varieties. These integral models are supposed to generalize Mumford's moduli schemes of principally polarized abelian schemes. Classical works of Zink, Rapoport, and Langlands proved the existence of such integral models for Shimura varieties of PEL type using the deformation theories of Serre–Tate and Grothendieck–Messing. In the last 12 years we developed five new methods in proving the existence of integral canonical models of Shimura varieties of Hodge type. In this talk we report on the fourth method which works even for the small primes 2 and 3.

A soccer ball is essentially a polyhedron built out of pentagons and hexagons. If you've ever tried to count the number of pentagons and hexagons on a soccer ball (as I did when I taught Math 302B), you have probably wished for an easier way to keep track of all these shapes. In this talk, we will discuss a nice way to do this. Time permitting, we will also discuss some connections between this topic and a few other ideas in math and science.

In 1694 Jacob Bernoulli published an article in Acta Eruditorum on an algebraic plane curve which he called the "lemniscus." Bernoulli's orginal title is the latin word for a ribbon, but today the curve is known as Bernoulli's lemniscate. It has many fascinating properties which can be explored using only calculus and elementary algebra and is thus a gem of an example to have in your teaching bag. This will be a survey talk exhibiting many of these properties and connections with plane differential geometry. We will see use of circles and hyperbolas, trig and hyperbolic functions, completing the square, polar coordinates, slope fields, compass and ruler constructions, the torus of revolution, stereographic projection for the sphere, and envelopes of families enter in our study.

Groundbreaking works of Evans and Searles in 1994 and of Cohen and Gallavotti in 1995 started a new period in the development of statistical mechanics, giving rise to numerous new results, centered around fluctuation theorems-- quantitative statements about entropy production rate in nonequilibrium systems. We will review some of this emerging field, starting from a paper by C. Maes, which interprets fluctuation theorems using equilibrium statistical mechanics in space-time. This is a natural continuation (but it does not assume familiarity with the content) of the recent talks by Bill Faris and by Daniel Ueltschi in the Mathematical Physics Seminar.

Sign up for a time with Christa King. Please bring your Consent Form & Insurance Card

Erdos was one of the 20th Century's most prolific, and most unusual, mathematicians. He lived a peripatetic life, traveling from university to institute to university, living out of two suitcases- one filled with mathematical papers. In his wake he left hundreds of jointly-authored papers and colleagues inspired to investigate new questions. He is credited with initiating many new areas of mathematical investigation.

We are currently developing an efficient and accurate computational technique that can capture the dynamics of the ocean and the erodible bottom topography even when the shore profile and basin morphology change significantly in time. This fairly robust technique will find practical use in beach erosion studies, tidal flooding, tsunami wave runup, basin drainage, and valley flooding.

Currently the computational technique is employed in one-dimension for non-eroding bottom topography. The shallow water wave equations (PDEs) are integrated with a conservative finite volume method. Front tracking using an ODE integrator is employed to integrate the basin boundaries. An adaptive mesh stitches the edge of the computational mesh where the front tracking occurs to the remaining finite volume cells.

I will present our current technique for our model as well as some results from our actual code. I will also discuss possible approaches to overcome difficulties encountered with stability in our code. Finally, I will present other future work that we hope to accomplish including generalization of the code to two-dimensions and incorporation of changing basin topography into our model.

Determinantal Random Point Processes appear naturally in Random Matrix Theory, Random Growth Models, and related areas of Probability, Discrete Mathematics and Mathematical Physics. The first part of the talk will be devoted to examples and some general results about Determinantal Random Point Processes. In the second part of the talk I will discuss Central Limit Theorem type results for linear statistics in Determinantal Random Point Processes. The talk should be accessible to non-experts.