Independence of ℓ for Frobenius conjugacy classes attached to abelian varieties

Rong Zhou
Imperial College London
June 18, 2020
Let A be an abelian variety over a number field E⊂ℂ and let v be a place of good reduction lying over a prime p. For a prime ℓ≠p, a result of Deligne implies that upon replacing E by a finite extension, the Galois representation on the ℓ-adic Tate module of A factors as ρℓ:Gal(E⎯⎯⎯⎯/E)→GA, where GA is the Mumford--Tate group of Aℂ. For p>2, we prove that the conjugacy class of ρℓ(Frobv) is defined over ℚ and independent of ℓ. This is joint work with Mark Kisin.

The challenges of model-based reinforcement learning and how to overcome them

Csaba Szepesvári
University of Alberta
June 18, 2020
Some believe that truly effective and efficient reinforcement learning algorithms must explicitly construct and explicitly reason with models that capture the causal structure of the world. In short, model-based reinforcement learning is not optional. As this is not a new belief, it may be surprising that empirically, at least as far as the current state of art is concerned, the majority of the top performing algorithms are model-free.

On learning in the presence of biased data and strategic behavior

Avrim Blum
Toyota Technological Institute at Chicago
June 16, 2020
In this talk I will discuss two lines of work involving learning in the presence of biased data and strategic behavior. In the first, we ask whether fairness constraints on learning algorithms can actually improve the accuracy of the classifier produced, when training data is unrepresentative or corrupted due to bias. Typically, fairness constraints are analyzed as a tradeoff with classical objectives such as accuracy. Our results here show there are natural scenarios where they can be a win-win, helping to improve overall accuracy.

Floer Cohomology and Arc Spaces

Mark McLean
Stony Brook University
June 12, 2020
Let f be a polynomial over the complex numbers with an isolated singular point at the origin and let d be a positive integer. To such a polynomial we can assign a variety called the dth contact locus of f. Morally, this corresponds to the space of d-jets of holomorphic disks in complex affine space whose boundary `wraps' around the singularity d times. We show that Floer cohomology of the dth power of the Milnor monodromy map is isomorphic to compactly supported cohomology of the dth contact locus.

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.