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.

Kinematics of Circumgalactic Gas

Crystal Martin
University of California, Santa Barbara
November 27, 2018

Most of the baryons associated with galaxy halos reside in the
circumgalactic medium. A significant fraction of the mass is at
temperatures well below the halo virial temperature.  What holds
the cool CGM up?  I will present new measurements of the kinematics
of circumgalactic gas.  The data show that much of the low-ionization
CGM rotates in the same direction as the galactic disk, suggesting
substantial centrifugal support.  Disturbances from galactic winds

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.

The min-max width of unit volume three-spheres

Lucas Ambrozio
University of Warwick; Member, School of Mathematics
November 20, 2018
The (Simon-Smith) min-max width of a Riemannian three dimensional sphere is a geometric invariant that measures the tightest way, in terms of area, of sweeping out the three-sphere by two-spheres. In this talk, we will explore the properties of this geometric invariant as a functional on the space of unit volume

This is joint work with Rafael Montezuma.

Lyapunov exponents for small random perturbations of predominantly hyperbolic two dimensional volume-preserving diffeomorphisms, including the Standard Map

Alex Blumenthal
University of Maryland
November 19, 2018
An outstanding problem in smooth ergodic theory is the estimation from below of Lyapunov exponents for maps which exhibit hyperbolicity on a large but non- invariant subset of phase space. It is notoriously difficult to show that Lypaunov exponents actually reflect the predominant hyperbolicity in the system, due to cancellations caused by the“switching” of stable and unstable directions in those parts of phase space where hyperbolicity is violated.