Events

In this talk we will discuss two results on the multiplicity problem for prime closed orbits of dynamically convex Reeb flows on the boundary of a star-shaped domain in R²ⁿ. The first result asserts that such a flow has at least n prime closed Reeb orbits, improving the previously known lower bound by a factor of two. The second main theorem is that when, in addition, the domain is centrally symmetric and the Reeb flow is non-degenerate, the flow has either exactly n or infinitely many prime closed orbits. This is a joint work with Basak Gurel and Viktor Ginzburg.

A long-standing topic of interest is to understand the desingularization problem in vortex dynamics for the incompressible 2D Euler equations: solutions of the system that approximate point vortices in the sense that the vorticity of the solution stays highly concentrated around a finite number of points on some interval of time. There are a large class of steady states that satisfy this behaviour, and also solutions that exhibit this behaviour dynamically on finite time intervals. We exhibit a solution of 2D Euler that is genuinely dynamic, and also retains this concentration of vorticity around points for all time: a configuration approximating two vortex pairs separating at a linear rate.

The affine self-diffeomorphism group of a translation surface can be rather large. For example, the affine diffeos of a 2-torus contain a lattice.  In the case of a torus $T$, Ghosh, Gorodnik, and Nevo proved that for any $\eta > 0$, for almost every $y \in T$ and for every $x \in T$, there exist infinitely many $\gamma \in SL_2(Z)$ so that $\| \gamma x- y \|\leq \|\gamma\|^{-1-\eta}$. We show that a similar result holds for translation surfaces with the lattice property. This is joint work with Josh Southerland.

Piecewise polynomials and Chow rings

As is well-known, Koiter's model is often used in numerical simulations, because it is a two-dimensional model that captures well the "membrane-dominated" and "flexural-dominated" effects that arise in a nonlinearly elastic shell subjected to applied forces and specific boundary conditions. Finding a satisfactory existence theory for this nonlinear shell model has stood as an open problem for a very long time. The present work, which is a joint work with Cristinel Mardare, provides a two-dimensional model that preserves all the virtues of Koiter's model, while being in addition amenable to a satisfactory existence theory. More precisely, our new two-dimensional mathematical model for a nonlinearly elastic shell takes the form of a minimization problem with a stored energy function that is polyconvex and orientation-preserving, and more generally satisfies all the other assumptions of John Ball's existence theorem. In addition, the most noteworthy feature of this model is that it is "of Koiter's type", in the sense that for a specific class of deformations that are "to within the first order" identical to those introduced by W.T. Koiter for defining his model, the "lowest order part" of its stored energy function coincides with the stored energy function of Koiter's model.

In 1990 Le Gall showed an asymptotic expansion of the epsilon-neighborhood of a planar Brownian trajectory (Wiener sausage) into powers of 1/|log eps|, that involves the renormalized self-intersection local times. In my talk I will present an analogue of this in the case of the 2D GFF. In the latter case, there is an asymptotic expansion of the epsilon-neighborhood of a sign cluster of the 2D GFF into half-integer powers of 1/|log eps|, with the coefficients of the expansion being related to the renormalized (Wick) powers of the GFF.

A core problem in statistics and machine learning is to approximate difficult-to-compute probability distributions. This problem is especially important in Bayesian statistics, which frames all inference about unknown quantities as a calculation about a conditional distribution. In this talk I review and discuss innovations in variational inference (VI), a method that approximates probability distributions through optimization. VI has been used in myriad applications in machine learning and Bayesian statistics. After quickly reviewing the basics, I will discuss two lines of research in VI. I first describe stochastic variational inference, an approximate inference algorithm for handling massive datasets, and demonstrate its application to probabilistic topic models of millions of articles. Then I discuss black box variational inference, a more generic algorithm for approximating the posterior. Black box inference applies to many models but requires minimal mathematical work to implement. I will demonstrate black box inference on deep exponential families---a method for Bayesian deep learning---and describe how it enables powerful tools for probabilistic programming. Finally, I will highlight some more recent results in variational inference, including statistical theory, score-based objective functions, and interpolating between mean-field and fully dependent variational families.

A core problem in statistics and machine learning is to approximate difficult-to-compute probability distributions. This problem is especially important in Bayesian statistics, which frames all inference about unknown quantities as a calculation about a conditional distribution. In this talk I review and discuss innovations in variational inference (VI), a method that approximates probability distributions through optimization. VI has been used in myriad applications in machine learning and Bayesian statistics. After quickly reviewing the basics, I will discuss two lines of research in VI. I first describe stochastic variational inference, an approximate inference algorithm for handling massive datasets, and demonstrate its application to probabilistic topic models of millions of articles. Then I discuss black box variational inference, a more generic algorithm for approximating the posterior. Black box inference applies to many models but requires minimal mathematical work to implement. I will demonstrate black box inference on deep exponential families---a method for Bayesian deep learning---and describe how it enables powerful tools for probabilistic programming. Finally, I will highlight some more recent results in variational inference, including statistical theory, score-based objective functions, and interpolating between mean-field and fully dependent variational families.

In the theory of p-adic L-functions a p-adic Gross–Zagier formula gives interpretation to special values of p-adic L-functions outside the region of interpolation using p-adic integration. Seen as a first step towards "explicit reciprocity laws" they have important applications towards proving various instances of the Bloch-Kato conjecture. We construct a p-adic twisted triple product L-function associated to finite slope families of Hilbert modular forms, assuming p unramified in the totally real fields. In joint work with Ting-Han Huang, we prove a p-adic Gross–Zagier formula for this L-function for a pair of an elliptic modular form and a quadratic Hilbert modular form. This generalises work of Blanco-Chacon and Fornea for the case of Hida families, and we overcome a technical assumption in their work of p being split in the quadratic field.

I will present a result connecting higher genus descendant logarithmic Gromov-Witten invariants of toric surfaces to refined counts of tropical curves. I will then discuss some applications. First, how it can be used to relate higher genus descendant invariants of log Calabi-Yau surfaces with quantum scattering diagrams, generalising an expectation coming from mirror symmetry. I will then briefly mention how these tropical correspondence theorems may help with computing the quantum cohomology ring of the Hilbert scheme of points on an elliptic surface. This is based on joint work Patrick Kennedy-Hunt and Ajith Urundolil Kumaran, as well as conversations with Georg Oberdieck.