Given a mod $p$ Galois representation, one often wonders whether it arises by reducing a $p$-adic one, and whether these lifts are suitably 'well-behaved'. In this talk, we discuss how ideas from homotopy theory aid the study of Galois deformations, reviewing work of Galatius-Venkatesh.
The arithmetic regularity lemma says that any dense set A in F_p^n can be cut along cosets of some small codimension subspace H <= F_p^n such that on almost all cosets of H, A is either random or structured (in a precise quantitative manner). A standard example shows that one cannot hope to improve "almost all" to "all", nor to have a good quantitative dependency between the constants involved. Adding a further combinatorial assumption on A to the arithmetic regularity lemma makes its conclusion so strong that one can essentially classify such sets A. In this talk, I will use use the analogous problem with F_p^n replaced with R^n as a way the motivate the funny title.
I will construct a fully-faithful functor from the category of co-admissible D-cap modules of Ardakov—Wadsley, to the category of quasi-coherent sheaves on the "analytic de Rham space”, at least in the case when the rigid variety is affinoid and étale over a polydisk.
Is it possible to tell the isomorphism type of an infinite group from its collection of finite quotients? This question, known as profinite rigidity, has deep roots in various areas of mathematics, ranging from arithmetic geometry to group theory. In this talk, I will introduce the question, its history and context. I will explain how profinite rigidity is studied using the machinery of profinite completions, including elementary proofs and counterexamples. Then I will outline some of the key results in the field, ranging from 1970 to the present day. Time permitting, I will elaborate on recent results of myself on the profinite rigidity of certain classes of solvable groups.
In the definition of the skein lasagna module of a $4$-manifold $X$, it is essential that the input TQFT be fully functorial for link cobordisms in $S^3 \times [0, 1]$. I will describe an approach to resolve existing sign ambiguities in Kronheimer and Mrowka's spectral sequence from Khovanov homology to singular instanton link homology. The goal is to obtain a theory that is fully functorial for link cobordisms in $S^3 \times [0,1]$, and where the $E_2$ page carries a canonical isomorphism to Khovanov-Rozansky $\mathfrak{gl}_2$ link homology. Possible applications include non-vanishing theorems for $4$-manifold Khovanov skein lasagna modules à la Ren-Willis.
In this talk, we give an accessible introduction to the theory of coarse cohomology of metric spaces in the sense of Margolis, which we present in direct analogy with group cohomology for discrete groups. We explain how this yields the robust notion of coarse cohomological dimension (due to Margolis), which is a genuine quasi-isometry invariant of metric spaces generalising the cohomological dimension of groups when the latter is finite. We then give applications to geometric properties of quasimorphisms and motivate how such considerations might be useful in the setting of non-positively curved groups. This is joint reading/work with Paula Heim.
The notion of a superconformal structure on a supermanifold goes back some forty years. I will discuss some recent work that shows how these structures and their deformations govern supersymmetric and superconformal field theories in geometric fashion. A superconformal structure equips a supermanifold with a sheaf of dg commutative algebras; the tangent sheaf of this dg ringed space reproduces the Weyl multiplet of conformal supergravity (equivalently, the superconformal stress tensor multiplet), in any dimension and with any amount of supersymmetry. This construction is uniform under twists, and thus provides a classification of relations between superconformal theories, chiral algebras, higher Virasoro algebras, and more exotic examples.
The mapping class group a solid handlebody of genus g sits between mapping class groups of surfaces and Out(F_n), in the sense there is an injective map to the mapping class group of the boundary and a surjective map to Out(F_g) via the action on the fundamental group. Similar behaviour happens with actions on associated spaces, such curve complexes and Teichmuller space. I’ll give an expository talk on this, partly in the context of our proof with Petersen that handlebody groups are virtual duality groups, and partly in the context of a problem list on handlebody groups written with Andrew, Hensel, and Hughes.
Given a smooth geometrically connected curve C over a perfect field k and a smooth commutative group scheme G defined over the function field K of C, one can consider isomorphism classes of G-torsors locally trivial at completions of K coming from closed points of C. They form a generalized Tate-Shafarevich group which specializes to the classical one in the case when k is finite. Recently, these groups have been studied over other base fields k as well, for instance p-adic or number fields. Surprisingly, finiteness can be proven in some cases but there are also quite a few open questions which I plan to discuss in my talk.