School of Mathematics

Dynamical generalizations of the Prime Number Theorem and disjointness of additive and multiplicative actions

Florian Richter
Northwestern University
June 4, 2020
One of the fundamental challenges in number theory is to understand the intricate way in which the additive and multiplicative structures in the integers intertwine. We will explore a dynamical approach to this topic. After introducing a new dynamical framework for treating questions in multiplicative number theory, we will present an ergodic theorem which contains various classical number-theoretic results, such as the Prime Number Theorem, as special cases.

Winding for Wave Maps

Max Engelstein
University of Minnesota
June 1, 2020
Wave maps are harmonic maps from a Lorentzian domain to a Riemannian target. Like solutions to many energy critical PDE, wave maps can develop singularities where the energy concentrates on arbitrary small scales but the norm stays bounded. Zooming in on these singularities yields a harmonic map (called a soliton or bubble) in the weak limit. One fundamental question is whether this weak limit is unique, that is to say, whether different bubbles may appear as the limit of different sequences of rescalings.

Joint equidistribution of adelic torus orbits and families of twisted L-functions

Farrell Brumley
Université Sorbonne Paris Nord
May 28, 2020
The classical Linnik problems are concerned with the equidistribution of adelic torus orbits on the homogeneous spaces attached to inner forms of GL2, as the discriminant of the torus gets large. When specialized, these problems admit beautiful classical interpretations, such as the equidistribution of integer points on spheres, of Heegner points or packets of closed geodesics on the modular surface, or of supersingular reductions of CM elliptic curves.

Mirrors of curves and their Fukaya categories

Denis Auroux
Harvard University
May 22, 2020
Homological mirror symmetry predicts that the derived category of coherent sheaves on a curve has a symplectic counterpart as the Fukaya category of a mirror space. However, with the exception of elliptic curves, this mirror is usually a symplectic Landau-Ginzburg model, i.e. a non-compact manifold equipped with the extra data of a "stop" in its boundary at infinity.

Iwasawa theory and Bloch-Kato conjecture for unitary groups

Xin Wan
Morningside Center of Mathematics, Chinese Academy of Sciences
May 21, 2020
We describe a new method to study Eisenstein family and Iwasawa theory on unitary groups over totally real fields of general signatures. As a consequence we prove that if the central L-value of a cuspidal eigenform on the unitary group twisted by a CM character is 0, then the corresponding Selmer group has positive rank. The method also has a byproduct the p-adic functional equations for p-adic L-functions and p-adic families of Eisenstein series on unitary groups.

Forecasting Epidemics and Pandemics

Roni Rosenfeld
Carnegie Mellon University
May 21, 2020
Epidemiological forecasting is critically needed for decision making by national and local governments, public health officials, healthcare institutions and the general public. The Delphi group at Carnegie Mellon University was founded in 2012 to advance the theory and technological capability of epidemiological forecasting, and to promote its role in decision making, both public and private. Our long term vision is to make epidemiological forecasting as useful and universally accepted as weather forecasting is today.

Neural SDEs: Deep Generative Models in the Diffusion Limit

Maxim Raginsky
University of Illinois Urbana-Champaign
May 19, 2020
In deep generative models, the latent variable is generated by a time-inhomogeneous Markov chain, where at each time step we pass the current state through a parametric nonlinear map, such as a feedforward neural net, and add a small independent Gaussian perturbation. In this talk, based on joint work with Belinda Tzen, I will discuss the diffusion limit of such models, where we increase the number of layers while sending the step size and the noise variance to zero.

The Non-Stochastic Control Problem

Elad Hazan
Princeton University
May 18, 2020
Linear dynamical systems are a continuous subclass of reinforcement learning models that are widely used in robotics, finance, engineering, and meteorology. Classical control, since the work of Kalman, has focused on dynamics with Gaussian i.i.d. noise, quadratic loss functions and, in terms of provably efficient algorithms, known systems and observed state. We'll discuss how to apply new machine learning methods which relax all of the above: efficient control with adversarial noise, general loss functions, unknown systems, and partial observation.