Yau's conjecture states that the volume of the nodal set of
Laplace eigenfunctions on a compact Riemannian manifold is comparable to
the square root of the corresponding eigenvalue. Donnelly and Fefferrman
proved Yau's conjecture for real analytic metrics but the conjecture stays
widely open for smooth metrics specially in dimensions n>2. Recently
Sogge-Zelditch and Colding-Minicozzi have established new lower bounds for
the volume of the nodal sets. In this talk we give a new proof of
Colding-Minicozzi's result using a different method. This is a joint work
with Christopher Sogge and Zuoqin Wang.

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.