Mathematical quasicrystals and their spectral properties

Speaker: 

Anton Gorodetski

Institution: 

UC Irvine

Time: 

Friday, November 30, 2012 - 4:00pm

Location: 

MSTB 120

Penrose tilings and substitution sequences, spectral properties of operators in Hilbert space and dynamical systems, fractals and convolutions of singular measures - we will see how all these topics meet in the study of mathematical quasicrystals. 

Curve shortening: an introduction to geometric evolutions

Speaker: 

Jeff Streets

Institution: 

UC Irvine

Time: 

Friday, October 12, 2012 - 4:00pm to 5:00pm

Location: 

MSTB 120

What happens when we decrease the length of a closed curve in
the plane as fast as possible? This seemingly simple question has a very
nice answer which involves a beautiful combination of partial differential
equations and planar geometry. Come and get a glimpse of the amazing
subject of geometric flows!

Rational points on Artin-Schreier curves

Speaker: 

Alexander Mueller

Institution: 

University of Michigan

Time: 

Tuesday, March 5, 2013 - 2:00pm to 3:00pm

Host: 

Daqing Wan

Location: 

RH306

An Artin-Schreier curve X (associated with an equation of the form y^p- y = f(x)) must satisfy the familiar Weil bound
 
||X(F_{p^n} )| - (p^n + 1)| < (degf - 1)(p - 1)p^{n/2}
 
but in many cases stronger bounds hold. In particular, Rojas-Leon and Wan proved such a curve must satisfy a bound of the form

||X(F_{p^n}) - (p^n +1)|<C_{d,n} p^{(n+1)/2}
 
where C{d,n} is a constant that depends on d := degf and n but not p. I will talk about how to use the representation theory of the symmetric group to prove both similar bounds and related statements about the zeros of the zeta function of X. Specifically, I will define a class of auxiliary varieties Y_n, each with an action of S_n, and explain how the S_n representation H^{n-1}(Y_n) contains useful arithmetic information about X. To provide an example, I will use these techniques to show that if d is “small” relative to p, then (a/b)^p = 1 for “most” pairs of X zeta zeroes a and b. 
 

 

Positive Equilibrium Solutions in Structured Population Dynamics

Speaker: 

Christoph Walker

Institution: 

Leibniz Universität Hannover

Time: 

Thursday, October 18, 2012 - 3:00pm

Location: 

RH 306

The talk focuses on positive equilibrium (i.e. time-independent) solutions to mathematical models for the dynamics of populations structured by age and spatial position. This leads to the study of quasilinear parabolic equations with nonlocal and possibly nonlinear initial conditions. We shall see in an abstract functional analytic framework how bifurcation techniques may be combined with optimal parabolic regularity theory to establish the existence of positive solutions. As an application of these results we give a description of the geometry of coexistence states in a two-parameter predator-prey model.

Common boundary values of holomorphic functions for two-sided complex structures

Speaker: 

Xianghong Gong

Institution: 

University of Wisconsin--Madison

Time: 

Tuesday, October 23, 2012 - 3:00pm to 4:00pm

Host: 

Song-Ying Li

Location: 

RH 306

Let M be a smooth hypersurface in R^{2n}. Suppose that on each side of M, there is a complex structure which is smooth up to M. Assume that the difference of the two complex structures is sufficiently small on M. We show that if a continuous function is holomorphic with respect to both complex structures, the function must be smooth from both sides of M. The regularity proof makes essential use of J-holomorphic curves and Fourier transform on families of curves.

This is joint work with Florian Bertrand and Jean-Pierre Rosay.

Experimental and computational analysis of motility initiation and turning of motile cells

Speaker: 

Alex Mogilner

Institution: 

UC Davis

Time: 

Monday, February 25, 2013 - 4:00pm to 5:00pm

Host: 

German A. Enciso Ruiz

Location: 

RH 306

Animal cells crawl on flat surfaces using lamellipodium – dynamic network of actin polymers and myosin motors enveloped by the cell membrane. Experimental analysis of the lamellipodial geometry, cell speed and actin dynamics in fish keratocyte cells combined with computational modeling suggested that steady crawling of the motile cells is based on a force balance between actin growth and myosin contraction. However, explanation of unsteady movements, especially of motility initiation and turning, remains elusive. I will present simulations of a 2D model of viscous contractile actin-myosin network with free boundary that, coupled with experimental data, suggests that stick-slip nonlinear adhesion is the key to understanding polarization and turning of the motile cells.

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