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. 

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.