School of Mathematics

Mean action of periodic orbits of area-preserving annulus diffeomorphisms

Morgan Weiler
University of California, Berkeley
December 3, 2018
An area-preserving diffeomorphism of an annulus has an "action function" which measures how the diffeomorphism distorts curves. The average value of the action function over the annulus is known as the Calabi invariant of the diffeomorphism, while the average value of the action function over a periodic orbit of the diffeomorphism is the mean action of the orbit.

Recent Progress on Zimmer's Conjecture

David Fisher
Indiana University, Bloomington; Member, School of Mathematics
December 3, 2018
Lattices in higher rank simple Lie groups are known to be extremely rigid. Examples of this are Margulis' superrigidity theorem, which shows they have very few linear represenations, and Margulis' arithmeticity theorem, which shows they are all constructed via number theory. Motivated by these and other results, in 1983 Zimmer made a number of conjectures about actions of these groups on compact manifolds and in a recent breakthrough with Brown and Hurtado we have proven many of them.

Branched conformal structures and the Dyson superprocess

Govind Menon
Brown University; Member, School of Mathematics
November 30, 2018

In the early 1920s, Loewner introduced a constructive approach to the Riemann mapping theorem that realized a conformal mapping as the solution to a differential equation. Roughly, the “input” to Loewner’s differential equation is a driving measure and the “output” is a family of nested, conformally equivalent domains. This theory was revitalized in the late 1990s by Schramm. The Schramm-Loewner evolution (SLE) is a stochastic family of slit mappings driven by Loewner’s equation when the driving measure is an atom executing Brownian motion.

The Lucky Logarithmic Derivative

Will Sawin
Columbia University
November 29, 2018
We study the function field analogue of a classical problem in analytic number theory on the sums of the generalized divisor function in short intervals, in the limit as the degrees of the polynomials go to infinity. As a corollary, we calculate arbitrarily many moments of a certain family of L-functions, in the limit as the conductor goes to infinity. This is done by showing a cohomology vanishing result using a general bound due to Katz and some elementary calculations with polynomials.

Monotone Circuit Lower Bounds from Resolution

Mika Goos
Member, School of Mathematics
November 27, 2018
For any unsatisfiable CNF formula F that is hard to refute in the Resolution proof system, we show that a gadget-composed version of F is hard to refute in any proof system whose lines are computed by efficient communication protocols---or, equivalently, that a monotone function associated with F has large monotone circuit complexity. As an application, we show that a monotone variant of XOR-SAT has exponential monotone circuit complexity, which improves qualitatively on the monotone vs. non-monotone separation of Tardos (1988).

Homotopical effects of k-dilation

Larry Guth
Massachusetts Institute of Technology
November 27, 2018
Back in the 70s, Gromov started to study the relationship between the Lipschitz constant of a map (also called the dilation) and its topology. The Lipschitz constant describes the local geometric features of the map, and the problem is to understand how it relates to the global geometric features of the map -- a bit like trying to understand the relationship between the curvature of a Riemannian manifold and its topology.

Bubbling theory for minimal hypersurfaces

Ben Sharp
University of Warwick
November 27, 2018
We will discuss the bubbling and neck analysis for degenerating sequences of minimal hypersurfaces which, in particular, lead to qualitative relationships between the variational, topological and geometric properties of these objects. Our aim is to discuss the salient technical details appearing in both the closed and free-boundary setting, and to give an overview of the applications of such results. This will involve expositions of joint works with Lucas Ambrozio, Reto Buzano and Alessandro Carlotto.

Classical Verification of Quantum Computations

Urmila Mahadev
UC Berkeley
November 26, 2018
We present the first protocol allowing a classical computer to interactively verify the result of an efficient quantum computation. We achieve this by constructing a measurement protocol, which allows a classical string to serve as a commitment to a quantum state. The protocol forces the prover to behave as follows: the prover must construct an n qubit state of his choice, non-adaptively measure each qubit in the Hadamard or standard basis as directed by the verifier, and report the measurement results to the verifier.

Introduction to Query-to-Communication Lifting

Mika Goos
Member, School of Mathematics
November 20, 2018
I will survey new lower-bound methods in communication complexity that "lift" lower bounds from decision tree complexity. These methods have recently enabled progress on core questions in communication complexity (log-rank conjecture, classical--quantum separations) and related areas (circuit complexity, proof complexity, LP/SDP formulations). I will prove one concrete lifting theorem for non-deterministic (NP) query/communication models.

Almgren's isomorphism theorem and parametric isoperimetric inequalities

Yevgeny Liokumovich
Massachusetts Institute of Technology; Member, School of Mathematics
November 20, 2018
In the 60's Almgren initiated a program for developing Morse theory on the space of flat cycles. I will discuss some simplifications, generalizations and quantitative versions of Almgren's results about the topology of the space of flat cycles and their applications to minimal surfaces.

I will talk about joint works with F. C. Marques and A. Neves, and L. Guth.