School of Mathematics

For those who enjoy the beauty, clarity, and simplicity of Euclid's Elements, it might be strange to think that Greek geometry was more like a battlefield than a schoolroom; but such it was. Greek geometry arose in a climate of strong loyalties...

What is a random Hamiltonian diffeomorphism? In this talk, we propose an answer by constructing a family of probability measures on the group Ham(M) of Hamiltonian diffeomorphisms of a closed symplectic manifold M. We outline the construction, which...

A basic object in the modern study of symplectic topology is the space 𝑀𝑔(𝐴,𝐽)

of 𝐽-holomorphic curves of genus 𝑔 representing the homology class 𝐴

in a symplectic manifold (𝑀,𝑤). In a wide variety of situations, this space is a compact manifold (or...

Consider a 𝐶𝑘 function 𝐹:𝑀𝑚×𝑁𝑛⟶ℝ defined on the product of the manifold 𝑀𝑚

of dimension 𝑚 with the compact manifold 𝑁𝑛 of dimension 𝑛.

The function defined by inf𝑁𝑛𝐹:𝑀𝑚⟶ℝ is locally Lipschitz. Yomdin proved (in 1984) that if 𝑘𝑘(𝑚,𝑛), where the integer...

Recently, Zijian Rong and I defined a new differential graded algebra associated to a Legendrian knot Λ in a thickened convex surface Σ×ℝ. The algebra is defined combinatorially and in the same spirit as the Chekanov-Eliashberg DGA for Legendrians...

We study the online calibration of multidimensional forecasts over an arbitrary convex set. For forecasting over the hypercube, we give the first algorithm that achieves ε-calibration in a number of rounds polynomial in the dimension, exponentially...

In this talk, we will discuss finite-time blow-up of the three-dimensional incompressible Euler equations with smooth space-time forcing. We will describe a multiscale mechanism in which hydrodynamic instabilities amplify vorticity at successively...

I will discuss OpenAI's proposed solution of the unramified Langlands conjecture for function fields (problem #014 on the list). Recently, the geometric Langlands conjecture in characteristic 0 has been resolved, leading to substantial progress...

We introduce the notion of a Penner-type class of an exact symplectomorphism on a Liouville domain with an Ak-configuration of Lagrangian spheres for k⩾2, and prove that such a class has positive Floer-theoretic entropy. As a corollary, we construct...

From their origin sofic groups have had a strong connection to topological dynamics in that it was shown that sofic groups are surjunctive - a dynamical notion. Later on, Lewis Bowen developed an entropy theory for sofic groups. This led to further...