On Langevin Dynamics in Machine Learning

Michael I. Jordan
University of California, Berkeley
June 11, 2020
Langevin diffusions are continuous-time stochastic processes that are based on the gradient of a potential function. As such they have many connections---some known and many still to be explored---to gradient-based machine learning. I'll discuss several recent results in this vein: (1) the use of Langevin-based algorithms in bandit problems; (2) the acceleration of Langevin diffusions; (3) how to use Langevin Monte Carlo without making smoothness assumptions.

New constraints on the Galois configurations of algebraic integers in the complex plane

Vesselin Dimitrov
University of Toronto
June 11, 2020
Fekete (1923) discovered the notion of transfinite diameter while studying the possible configurations of Galois orbits of algebraic integers in the complex plane. Based purely on the fact that the discriminants of monic integer irreducible polynomials P(X)∈ℤ[X] are at least 1 in magnitude (since they are non-zero integers), he found that the incidences (,P) between these polynomials P(X) and compacts ⊂ℂ of transfinite diameter d()

What Do Our Models Learn?

Aleksander Mądry
Massachusetts Institute of Technology
June 9, 2020
Large-scale vision benchmarks have driven---and often even defined---progress in machine learning. However, these benchmarks are merely proxies for the real-world tasks we actually care about. How well do our benchmarks capture such tasks?

Motives

Pierre Deligne
School of Mathematics
June 9, 2020
The IAS Number Theory group has started a new online seminar via Zoom to have more interactions among members during the Stay-at-Home Order. The seminar is modeled after the Basic Notions seminar at Harvard and is intended to be a one-hour-long Math Conversations on topics in Number Theory and related fields.

Infinite staircases of symplectic embeddings of ellipsoids into Hirzebruch surfaces

Morgan Weiler
Rice University
June 5, 2020
Gromov nonsqueezing tells us that symplectic embeddings are governed by more complex obstructions than volume. In particular, in 2012, McDuff-Schlenk computed the embedding capacity function of the ball, whose value at a is the size of the smallest four-dimensional ball into which the ellipsoid E(1,a) symplectically embeds. They found that it contains an “infinite staircase” of piecewise-linear sections accumulating from below to the golden ratio to the fourth power. However, infinite staircases seem to be rare for more general targets.

Real Lagrangian Tori in toric symplectic manifolds

Joé Brendel
University of Neuchâtel
June 5, 2020
In this talk we will be addressing the question whether a given Lagrangian torus in a toric monotone symplectic manifold can be realized as the fixed point set of an anti-symplectic involution (in which case it is called "real"). In the case of toric fibres, the answer depends on the geometry of the moment polytope of the ambient manifold. In the case of the Chekanov torus, the answer is always no. This can be proved using displacement energy and versal deformations.

Reeb orbits that force topological entropy

Abror Pirnapasov
Ruhr-Universität Bochum
June 5, 2020
A transverse link in a contact 3-manifold forces topological entropy if every Reeb flow possessing this link as a set of periodic orbits has positive topological entropy. We will explain how cylindrical contact homology on the complement of transverse links can be used to show that certain transverse links force topological entropy. As an application, we show that on every closed contact 3-manifold exists transverse knots that force topological entropy.

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.