Talk

W. Lowen and M. Van den Bergh defined a deformation theory for Abelian categories, such that deformations of the category of (quasi)coherent sheaves on an affine variety Spec A coincide with deformations of the algebra A, which are controlled by the Hochschild cochain complex equipped with the Gerstenhaber bracket. For categories of coherent sheaves on a (non-affine) quasi-projective variety X, one should replace the single commutative algebra A by a "diagram" of commutative algebras, obtained as the restriction of the structure sheaf of the variety to an affine open cover which is closed under intersections.

Deformations of QCoh(X) are then controlled by the so-called Gerstenhaber-Schack complex of this diagram of algebras, whose cohomology is isomorphic to the Hochschild cohomology of X. First-order deformations of QCoh(X) are parametrized by HH²(X), which for smooth complex varieties includes deformations of the complex structure as well as (complex) deformation quantizations.

I will explain how to obtain an explicit L-infinity algebra structure on this complex, controlling the higher deformation theory of QCoh(X), in case X can be covered by two affine open sets and explain how this point of view may be used to study the effect of deformations of QCoh(X) on moduli spaces of vector bundles or instanton moduli. This latter application can be viewed as an analogue of work by Nekrasov-Schwarz on non-commutative instantons on R⁴.

This talk is based on joint work with Yaël Frégier and work in progress joint with Zhengfang Wang.

We propose a general framework to inductively prove new results for counting number fields. By using this method, we prove the precise asymptotic distribution of G -extensions for a family of Galois groups G  that could be constructed via taking towers of extensions. The key ingredient is a uniform estimate on the number of relative extensions with dependency on the base field. This is a joint work with Robert J.Lemke Oliver and Melanie Matchett Wood.

Choose some positive $k$ and a rational elliptic curve $E$, and choose $k$ pairs of primes $(p_i, p'_i)$. Take $d_0$ to be $p_1 p_2 \dots p_k$, and consider the family of $d$ given by replacing $p_i$ with $p'_i$ for some set of $i$. Under special circumstances, we show that $2^k$-Selmer elements of a twist $E^{d_0}$ can be constructed from $2^k$-Selmer elements of the remaining twists $E^d$. By elaborating on this strategy we show that, in a grid of twists of $E$, some information about the distribution of $2^k$-Selmer groups over this grid can be found from symbols whose values are subject to analytic control.

Take $E$ to be an elliptic curve over a number field whose four torsion obeys certain technical conditions. In this talk, we will outline a proof that $100\%$ of the quadratic twists of $E$ have rank at most one. To do this, we will find the distribution of $2^k$-Selmer ranks in this family for every $k > 1$. Using this framework, we will also find the distribution of the $2^k$-class ranks of the imaginary quadratic fields for all $k > 1$.

The Q_l-pro-unipotent non-abelian Kummer map associated to a curve X is a certain function controlling the existence of Galois-invariant paths between points of X, and plays an important role in the non-abelian Chabauty method for finding rational points. In this talk, I will report on a project with Netan Dogra, in which we compute these maps explicitly when the base field is p-adic, obtaining a description of them in terms of harmonic analysis on the reduction graph of X. As a result, we are able to prove injectivity results for these maps.

We consider the family of normal octic fields with Galois group $D_4$, ordered by their discriminant. In joint work with Arul Shankar, we verify the strong Malle conjecture for this family of number fields, obtaining the order of growth as well as the constant of proportionality. In this talk, we will discuss and review the combination of techniques from analytic number theory and geometry-of-numbers methods used to prove
these results.