Events

In 1945, Siegel showed that the expected value of the lattice-sums of a function over all the lattices of unit covolume in an n-dimensional real vector space is equal to the integral of the function. In 2012, Venkatesh restricted the lattice-sum function to a collection of lattices that had a cyclic group of symmetries and proved a similar mean value theorem. Using this approach, new lower bounds on the most optimal sphere packing density in n dimensions were established for infinitely many n. In the talk, we will outline some analogues of Siegel's mean value theorem over lattices. This approach has modestly improved some of the best known lattice packing bounds in many dimensions. We will speak of some variations and related ideas. (Joint work with V. Serban, M. Viazovska)

抖抖与势的基本对象geometric representation theory and important also in Donaldson-Thomas theory of Calabi-Yau 3-categories. In this talk, we will introduce quantum corrections to such objects by counting quasimaps from curves to the critical locus of the potential. Our construction is based on the theory of gauged linear sigma model (GLSM) and uses recent development of DT theory of CY 4-folds. Joint work with Gufang Zhao.

This talk is about joint work with Yann Guggisberg. The main result is that the set of generalized symplectic capacities is a complete invariant for every symplectic category whose objects are of the form $(M,\omega)$, such that $M$ is compact and 1-connected, $\omega$ is exact, and there exists a boundary component of $M$ with negative helicity. This answers a question of Cieliebak, Hofer, Latschev, and Schlenk. It appears to be the first result concerning this question, except for results for manifolds of dimension 2, ellipsoids, and polydiscs in $\mathbb{R}^4$. If time permits, then I will also present some answers to the following question and problem of Cieliebak, Hofer, Latschev, and Schlenk: Question: Which symplectic capacities are connectedly target-representable? Problem: Find a minimal generating set of symplectic capacities.

Lagrangian cobordisms induce exact triangles in the Fukaya category. But how many exact triangles can be recovered by Lagrangian cobordisms? One way to measure this is by comparing the Lagrangian cobordism group to the Grothendieck group of the Fukaya category. In this talk, we discuss the setting of exact conical Lagrangian submanifolds in Liouville manifolds and compute Lagrangian cobordism groups of Weinstein manifolds. As an application, we get a geometric interpretation for Viterbo restriction for Lagrangian cobordism groups.

Given a smooth algebraic variety (or orbifold) X with a normal crossings divisor D, the space X decomposes into strata describing the loci where various components of D (self-)intersect. In this series of four lectures, we will study the intersection theory of X, as well as certain birational modifications of X obtained by iteratively blowing up some of the strata. In recent years, the logarithmic Chow ring of the pair (X,D) was defined to encode the intersection theory of all such iterated blow-ups simultaneously. This allows to perform calculations with strict transforms of cycles on X without specifying a concrete birational model of X in which they live, and has been applied successfully in studying certain geometric cycles on moduli spaces. In the first lecture, we give an overview of the relevant definitions, a sketch of some of their applications, and a roadmap for the following talks. The second lecture talks about the case where X is a toric variety (and D its toric boundary), and explains that here the entire intersection theory is described in terms of convex geometric data (the fan Sigma of X and piecewise polynomial functions on Sigma). Then we show how the language of cone stacks and Artin fans can be used to generalize from the toric situation to arbitrary pairs (X,D). Finally, we talk about ongoing work on the logarithmic Chow ring of the moduli space of curves.

Time-harmonic Maxwell's equations model the propagation of electromagnetic waves, and their numerical discretization by finite elements is instrumental in a large array of applications. In the simpler setting of acoustic waves, it is known that (i) the Galerkin Lagrange finite element approximation to a Helmholtz problem becomes asymptotically optimal as the mesh is refined. Similarly, (ii) asymptotically constant-free a posteriori error estimates are available for Helmholtz problems. In this talk, considering Nédélec finite element discretizations of time-harmonic Maxwell's equations, I will show that (i) still holds true and propose an a posteriori error estimator providing (ii). Both results appear to be novel contributions to the existing literature.

In this talk we quantify the asymptotic behaviour of multidimensional drifltess diffusions in domains unbounded in a single direction, with asymptotically normal reflections from the boundary. We identify the critical growth/contraction rates of the domain that separate stability, null recurrence and transience. In the stable case we prove existence and uniqueness of the invariant distribution and establish the polynomial rate of decay of its tail. We also establish matching polynomial upper and lower bounds on the rate of convergence to stationarity in total variation. All exponents are explicit in the model parameters that determine the asymptotics of the growth rate of the domain, the interior covariance, and the reflection vector field. Proofs are probabilistic, and use upper and lower tail bounds for additive functionals up to return times to compact sets, for which we develop novel sub/supermartingale criteria, applicable to general continuous semimartingales. Time permitting, I will discuss the main ideas behind the proofs in the talk. This is joint work with Miha Bresar (Warwick) and Andrew Wade (Durham).

Let X be a smooth projective variety. Define a stable map f : C → X to be eventually smoothable if there is an embedding X → PN such that (C,f) occurs as the limit of a 1-parameter family of stable maps to PN with smooth domain curves. Via an explicit deformation-theoretic construction, we produce a large class of stable maps (called stable maps with model ghosts), and show that they are eventually smoothable. This is joint work with Mohan Swaminathan.