Lecture 1: L-functions and Number Theory, November 14, 2016, room and time: TBA

Lecture 2: The Trace Formula and Automorphic Forms, November 15, 2016, room and time: TBA

Lecture 3: Beyond Endoscopy and Functoriality, November 16, 2016, room and time: TBA

Abstract: Number theory is founded on the basic properties of integers and prime numbers. But its study these days is increasingly leading us to the far reaches of some of the most diverse and powerful areas of mathematics. Nowhere is this more apparent than in the Langlands program, which represents a profound unifying force for mathematics.

We shall try to introduce the Langlands program through the theory of L-functions. These are infinite series that look like the famous Riemann zeta function, except that they have nontrivial coefficients. The information that goes into the coefficients is in fact very interesting, and gives an elegant way of organizing fundamental data from number theory, representation theory and algebraic geometry. The Langlands program postulates deep relationships among different L-functions, and hence also the data in their coefficients.

We shall discuss these matters, and explain how they are part of the theory of automorphic forms. We shall then describe the trace formula, which has led to important results in the classification of automorphic representations. If time permits, we shall also say something about Beyond Endoscopy, a proposal by Langlands for attacking the central conjecture of the subject known as the Principle of Functoriality.

We are happy to announce that the department of mathematics' Outstanding Graduate Teaching Assistant award for 2015-16 has been awarded to Xiang Wan. As winner of the departmental award, Xiang was nominated to compete for a University-wide award, and was selected as one of the top 15 of a very accomplished group of teachers. He was presented with a Jefferson Cup in honor of his achievement.

The department also recognizes two graduate teaching assistants with Honorable Mentions for our teaching award: they are Peter Bonventre and Jonathan Simone. All three of these outstanding GTAs will receive recognition and a cash prize at this year’s graduation ceremony.

Congratulations to Xiang, Peter, and Jon!

In 2016, the Final Exercises ceremony for the College and Graduate School of Arts & Sciences will be on **Saturday, May 21** with the department graduation ceremonies following that afternoon. The fair-, inclement-, and severe-weather ceremony sites for **Mathematics** will be:

**Fair Weather:** *Finals on the Lawn, fair-weather sites for department graduation ceremonies***Pavilion I Lower Garden
Ceremony Start Time: 12:15 p.m.**

**Inclement Weather:** *Finals on the Lawn, all department ceremonies inside***Gilmer Hall Room 130
Ceremony Start Time: 12:45 p.m.**

**Severe Weather:** *Finals in John Paul Jones Arena, all department ceremonies inside***Gilmer Hall Room 130
Ceremony Start Time: 12:45 p.m.**

**Please note that Gilmer Hall Room 130 is a Remote Viewing Location for Finals on the Lawn. We anticipate that the remote viewing will conclude prior to 12 p.m.*

Under the severe-weather plan, ALL degree candidates (undergraduate and graduate) will participate in the ceremony at John Paul Jones Arena. Each graduating student will receive **six** guest seating tickets for the Lawn. Since we cannot accommodate as many people in the arena compared to the Lawn, graduates will be restricted to **three** guests if the ceremony is moved inside. Tickets are required for all guests either on the Lawn or at the arena. Guests without tickets can watch a live broadcast of the ceremony at one of the remote viewing locations across Grounds.

Complimentary parking will be available at Scott Stadium and at John Paul Jones Arena with shuttle bus service to Central Grounds. Buses will load at the entrance of the Student Activities Building (located in the Scott Stadium west parking lot) and near the west entrance of the arena. Return shuttles to these lots will run continuously from Central Grounds throughout Saturday afternoon. Parking is also available in the Emmet/Ivy parking garage ($5 per vehicle), which is about a 10 minute walk to the Lawn (no shuttle service).

For additional information on Finals Weekend 2016, including the remote viewing locations, please visit our web site at: www.virginia.edu/finals .

**Karen Smith (University of Michigan) **

*Algebra, Geometry and Analysis over Finite Fields*

**Lecture 1:** Monday, February 29th, 2016

Time: 5:00PM-6:00PM

Location: Clark 108

Abstract: In this lecture, we review how Noether's introduction of the concept of an abstract ring changed the course of mathematics in the twentieth century by enabling us to apply the methods of "reduction modulo p" to solve problems in algebraic geometry. Specifically, I'll discuss how understanding solutions to polynomials over finite fields can help understand the geometry of geometric objects (called varieties) defined by real or complex polynomials. Miraculously, rings of characteristic p have some very special properties that can be powerful tools in analyzing them, often replacing tools like integration for real manifolds.

**Lecture 2: **Tuesday, March 1st, 2016

Time: 5:00PM-6:00PM

Location: Clark 108

Abstract: In the second lecture, we review Hironaka's famous theorem on the resolution of singularities of a complex algebraic variety. We show this theorem can help us understand and measure the singularities of complex varieties. Amazingly, it turns out that the only algebraic characterization of a geometric condition called "rational singularities" involves reduction to characteristic p. Specifically, we will see how algebraic tools such as Frobenius splitting impact different areas of math, including the minimal model program for complex algebraic varieties and cluster algebras in combinatorics/representation theory.

**Lecture 3**: Wednesday, March 2nd, 2016

Time: 3:30-4:30

Location: Physics 203

Abstract: In the final lecture, we discuss a numerical invariant of singularities called the analytic index of singularities, which is defined by the convergence of a certain integral. Amazingly, this invariant turns out to have a prime characteristic description as well, as the limit, over all primes p, of another invariant called the F-pure threshold. The study of these F-pure thresholds leads to some very interesting and mysterious fractal like behavior.

We are proud to report that Jiahua Liu has been selected to receive the International Studies Office Award for international undergraduate student academic excellence. He will be honored at the awards ceremony for graduating international students and their families this May. Congratulations, Jiahua Liu!

The 2016 Putnam Award Winners include Sifan Ye, Juan Velasco, and Arun Kannan. Congratulations to all three on this impressive accomplishment!

The award announcements at the Gordon E. Keller Mathematics Majors Dinner revealed this year's recipient of the E.J. McShane Prize in Mathematics to be Alexander Grieser. Congratualtions to Alexander on this accomplishment!

Congratulations to Ben Webster! He is one of two 2016 recipients of the prestigious Cory Family Teaching Awards, and joins David Sherman (2013) among the award recipients from the Mathematics department. The Cory Family Teaching Awards are "designed to reward and incentivize excellence in teaching among junior faculty." Since 2013 and continuing through 2017, two junior faculty members are chosen as recipients and honored at Fall Convocation. Each recipient is awarded $25,000 thanks to the generosity of Mr. and Mrs. Charles R. Cory. Well done, Ben!

One of the central problems in solid state physics consists of finding

effective models which describe the dynamics of electrons in periodic

potentials. Exponentially localized Wannier functions, if they exist,

enable us to replace the periodic and unbounded Schroedinger operator

with a discrete Jacobi-type infinite matrix.

We shall consider a real analytic and time reversal symmetric family of

Bloch projections of rank N and construct an orthonormal basis for its

range, which is both real analytic and periodic with respect to its

d-dimensional quasi-momenta when $1\leq d\leq 3$ and $N\geq 1$. We will

also show what can go wrong in dimensions higher than three, and make

the connection with topological degree theory.

This lecture is intended to anyone who has a basic knowledge of

functional analysis and a minimal interest in the rigorous mathematical

description of solid state physics. Following the first lecture would

help a lot.