I will give an overview of the interactions between chromatic homotopy theory and the algebraic K-theory of ring spectra, especially around the subject of Ausoni-Rognes's principle of "chromatic redshift," and some of the recent advances in this field.
In this talk, I would like to discuss Deligne’s version of Geometric Class Field theory, with special emphasis on the correspondence between rigidified 1-dimensional l-adic local systems on a curve and 1-dimensional l-adic local systems on Pic with certain compatibilities. We should like to give a sense of how this relates to the OG class field theory, and how Deligne demonstrates this correspondence via the geometry of the Abel-Jacobi Map. If time permits, we would also like to discuss the correspondence between continuous 1-dimensional l-adic representations of the etale fundamental group of a curve and local systems.
This talk will be about the stable boundary seen from different recent points of view.
Artin groups are a family of groups generalizing braid groups. The Tits conjecture, which was proved by Crisp-Paris, states that squares of the standard generators generate an obvious right-angled Artin subgroup. In a joint work with Kevin Schreve, we consider a larger collection of elements, and conjecture that their sufficiently large powers generate an obvious right-angled Artin subgroup. In the case of the braid group, regarded as a mapping class group of a punctured disc, these elements correspond to Dehn twist around the loops enclosing multiple consecutive punctures. This alleged right-angled Artin group is in some sense as large as possible; its nerve is homeomorphic to the nerve of the ambient Artin group. We verify this conjecture for some classes of Artin groups. We use our results to conclude that certain Artin groups contain hyperbolic surface subgroups, answering questions of Gordon, Long and Reid.
Given a curve , what is the surface that has smallest area among all surfaces spanning ? This classical problem and its generalizations are called Plateau's problem. In this talk we consider area minimizers among the class of integral currents, or roughly speaking, orientable manifolds. Since the 1960s a lot of work has been done by De Giorgi, Almgren, et al to study the interior regularity of these minimizers. Much less is known about the boundary regularity, in the case of codimension greater than 1. I will speak about some recent progress in this direction.
In some ways the theory of mapping class groups of 4-manifolds is in 2020 at the same place where the theory of mapping class groups of 2-manifolds was in 1973, before Thurston changed everything. In this talk I will describe some first steps in an ongoing joint project with Eduard Looijenga where we are trying to understand mapping class groups of certain algebraic surfaces (e.g. rational elliptic surfaces, and also K3 surfaces) from the Thurstonian point of view.
One can study periods of algebraic varieties by a process of "fibering out" in which the variety is fibred over a punctured curve $f:X->U$. I will explain this process and how it leads to the classical Picard Fuchs (or Gauss-Manin) differential equations. Periods are computed by integrating solutions of Picard Fuchs over suitable closed paths on $U$. One can also couple (i.e.tensor) the Picard Fuchs connection to given connections on $U$. For example, $t^s$ with $t$ a unit on $U$ and $s$ a parameter is a solution of the connection on $\mathscr{O}_U$ given by $\nabla(1) = sdt/t$. Our "periods" become integrals over suitable closed chains on $U$ of $f(t)t^sdt/t$. Golyshev called the resulting functions of $s$ "motivic Gamma functions".
Golyshev and Zagier studied certain special Picard Fuchs equations for their proof of the Gamma conjecture in mirror symmetry in the case of Picard rank 1. They write down a generating series, the Apéry series, the knowledge of the first few terms of which implied the gamma conjecture. We show their Apéry series is the Taylor series of a product of the motivic Gamma function times an elementary function of $s$. In particular, the coefficients of the Apéry series are periods up to inverting $2\pi i$. We relate these periods to periods of the limiting mixed Hodge structure at a point of maximal unipotent monodromy. This is joint work with M. Vlasenko.
We will discuss the similarity and difference between deterministic and stochastic NLS. Different notions (or possible formulations) of local solutions will also be discussed. We will also present a global well posedness result for stochastic mass critical NLS. Joint work with Weijun Xu (Oxford)
In 1943 Fisher, together with Corbet and Williams, published a study on the relation between the number of species and the number of individuals, which has since been recognized as one of the most influential papers in 20th century ecology. It was a combination of empirical work backed up by a simple theoretical argument, which describes a sort of universal law governing random partitions, such as the celebrated Ewens partition whose original derivation flows from the Fisher-Wright model. This talk will discuss several empirical studies of a similar sort, including Taylor's law and recent work related to Fairfield-Smith's work on the variance of spatial averages.