Let P: ... -> C_2 -> C_1 -> P^1 be a Z_p-cover of the projective line over a finite field of characteristic p which ramifies at exactly one rational point. In this talk, we study the p-adic Newton slopes of L-functions associated to characters of the Galois group of P. It turns out that for covers P such that the genus of C_n is a quadratic polynomial in p^n for n large, the Newton slopes are uniformly distributed in the interval [0,1]. Furthermore, for a large class of such covers P, these slopes behave in an even more regular way. This is joint work with Hui June Zhu.

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## Upcoming Seminars

### Tue Feb 21, 2017

We generalize the classical Bochner formula for the heat flow on a manifold M to martingales on path space PM, and develop a formalism to compute evolution equations for martingales on path space. We see that our Bochner formula on PM is related to two sided bounds on Ricci curvature in much the same manner as the classical Bochner formula on M is related to lower bounds on Ricci curvature. This establishes a new link between geometry and stochastic analysis, and provides a crucial new tool for the study of Einstein metrics and Ricci flow in the smooth and non-smooth setting. Joint work with Aaron Naber.

### Wed Feb 22, 2017

### Thu Feb 23, 2017

Abstract: We will consider recent numerous achievements in the area of understanding

the harmonic measure of general domains in\R^n.

These results become possible because of the recent breakthroughs in non-homogeneous

harmonic analysis.

The self-avoiding walk (SAW) is a model for polymers that assigns equal probability to all paths that do not return to places they have already been. The lattice version of this problem, while elementary to define, has proved to be notoriously difficult and is still open. It is initially more challenging to construct a continuous limit of the lattice model which is a random fractal. However, in two dimensions this has been done and the continuous model (Schram-Loewner evolution) can be analyzed rigorously and used to understand the nonrigorous predictions about SAWs. I will survey some results in this area and then discuss some recent work on this ``continuous SAW''.

### Fri Feb 24, 2017

I will introduce the main idea of scattering theory and asymptotic completeness. Then give a natural idea of proving the limiting absorption principle.

For a smooth curve, the natural paraemtrization

is parametrization by arc length. What is the analogue

for a random curve of fractal dimension d? Typically,

such curves have Hausdorff dmeasure 0. It turns out

that a different quantity, Minkowski content, is the

right thing.

I will discuss results of this type for the Schramm-Loewner

evolution --- both how to prove the content is well-defined

(work with M. Rezaei) and how it relates to the scaling

limit of the loop-erased random walk (work with F. Viklund

and C. Benes).

I will highlight some of the methods I've previously employed in undergraduate mathematics education. These methods were drawn from the research of others and some of my own inquires. I will give insight into how the methods are brought together so that they form a cohesive whole, while fitting the needs of the both the students and the department.

Central to the discussion is a particular enlightening experience I had with one of my students.

The experience has inspired my current methods. I will discuss this my vision for further developing curriculum and more so fostering a lively dynamic atmosphere around mathematics courses, both within and without the classroom walls.

Time permitting, I'll discuss some research that is underway with regards to these approaches.

Hindman’s theorem states that if one colors every natural number either red or blue, then there will be an infinite set X of natural numbers such that all finite sums of distinct elements from X have the same color. The original proof of Hindman’s theorem was a combinatorial mess and the slickest proof is via ultrafilters. In this talk, I will introduce the notion of an ultrafilter on a set, which is simply a division of the subsets of the set into two categories, “small" and “large", satisfying some natural axioms. We will then give the proof of Hindman’s theorem using ultrafilters that are idempotent with respect to a natural addition operation on the set of ultrafilters on the set of natural numbers. Finally, we will introduce an open conjecture of Erdos related to Hindman’s theorem, its reformulation in terms of ultrafilters, and some recent progress made on the problem by myself and my collaborators.

In this talk I will introduce the moduli space of curves and a class of vector bundles on it. I’ll discuss how these bundles, which have connections to algebraic geometry, representation theory, and mathematical physics, tell us about the moduli space of curves, and vice versa, focusing on just a few recent results and open problems.