Mandatory for Math 298 Grad Seminar students who have been a TA for less than 2 quarters at UCI.

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. 
 

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.

The electromagnetic inverse scattering problem of determining the locations and polarization tensors of a collection of small scatterers is investigated. The locations of scatterers are determined by the multiple signal classification (MUSIC) method and the polarization tensors are retrieved by the two-step least squares method. Multiple scattering effect is taken into account and the inverse scattering problem is nonlinear. However, the proposed method does not involve iterative evaluations of the corresponding forward scattering problem. In addition, the method provides better imaging resolution than the standard MUSIC does, and applies to degenerate scatterers to which the standard MUSIC does not apply. The underlying mathematical principle and physical insight are discussed in detail.

Abstract:  In order to fight infection or heal injury, living cells have to be able to move in response to chemical cues around them. As a first step, an internal chemical "map" is rapidly induced in the cell, leading to its polarization, reorganization of structural proteins (cytoskeleton), shape change, and crawling motility. In my talk, I will summarize some of the work done in my group addressing these processes. I will explain the basic properties of the signalling proteins (small GTPases) and their role in cell polarization. I will motivate a sequence of mathematical models that we have studied to understand the underlying mechanisms. One theme in my talk will be the cross-fertilization of the complex biological problem and its simplified mathematical caricatures. Another theme is the effect of cell shape on internal chemical dynamics. I will conclude with recent developments on computations of cell shape dynamics in 2D simulations.