Talk

The proof of Fermat’s Last Theorem pioneered a new approach to resolving families of ternary Diophantine equations using modularity of residual Galois representations attached to Frey curves. In the case of differing exponents, Darmon gave a framework for resolving generalized Fermat equations in one varying exponent using Frey varieties. In this talk, I will survey the methods and techniques of recent progress on Darmon’s program as well as the challenges that remain in the study of generalized Fermat equations.

We study generalised polynomials, that is, functions that can be expressed using standard algebraic operations and the floor function. Generalised polynomials have long been studied (both explicitly and implicitly) in number theory and dynamics, often using dynamical and ergodic-theoretic methods (especially dynamics on nilmanifolds). These methods enable one to deduce precise information about the average behaviour of generalised polynomials, but allow for complicated behaviour on special sets of density zero. We will present some of the classical results and give examples of various interesting arithmetic and combinatorial behaviour occurring along sets of density zero as well as restrictions to what is possible. We will also state an analogue of Hadamard's quotient theorem for generalised polynomials (the classical version concerns linear recurrences). We will also formulate several open problems. The talk is based on joint work with Jakub Konieczny (Oxford).

Let C be a complex projective curve and let L and M be two line bundles on C.  One can associate with L and M  some natural maps, which are called Gaussian maps, between spaces of global sections of certain sheaves on C. These maps have been classically studied in connection with extendability questions of curves on surfaces. In this talk I will focus on the case of Enriques surfaces, presenting some natural questions that arise in this situation.

The study of fully faithful functors, including equivalences, between derived categories of smooth projective varieties (or, more generally, smooth proper triangulated categories) is, in many ways, analogous to the study of rational contractions in the minimal model program. For a Fano manifold, homological mirror symmetry predicts that its derived category admits canonical semi-orthogonal decompositions (related by the braid group action) with remarkable properties, such as compatibility with rational contractions. After discussing this motivation, I will survey potential constructions of canonical semi-orthogonal decompositions, focusing on the case where the Fano manifold is a moduli space of stable objects of some type on another manifold and where its birational geometry can be understood as a variation of the stability condition. As an application, we will construct the canonical semi-orthogonal decomposition of the derived category of the moduli space of stable vector bundles of rank 2 with a fixed determinant of odd degree on a smooth projective curve. When the degree is even, the moduli space is singular, and the construction provides canonical semi-orthogonal decompositions of its quasi-BPS categories; for example, they are compatible with the Hecke correspondence.
 

A log-canonical Poisson bracket is one of the form {x_i,x_j} = lambda_{ij} x_i x_j. Assuming there exists an action of an algebraic torus T that preserves the bracket and admits an open T-leaf (i.e. an open T-orbit of a symplectic leaf), I will describe all T-invariant Poisson deformations of { , }. The key result here is an unobstructedness phenomenon akin to the Bogomolov-Tian-Todorov theorem in the deformation theory of Calabi-Yau manifolds. Time permitting, I will discuss applications of this deformation-theoretic approach to the Poisson brackets on Bott-Samelson varieties and Poisson CGL extensions in the sense of Goodearl-Yakimov. This is joint work with Jiang-Hua Lu.

It is known that every term $U_n$ of a regular Lucas sequence has a primitive prime divisor if $n\ge 31$, i.e., a prime $p$ such that $p\mid U_n$ but $p\nmid U_k$, for all $1\leq k<n$. Can this be refined to specific sets of primes?
We ask a weaker question: what proportion of terms $U_n$ possesses a prime factor $p$ such that $Q$ is a quadratic non-residue modulo $p$? Here $Q$ is the second parameter of the Lucas sequence. We find a natural density of either $1/2$ or $1/4$ in many cases. As a consequence, infinitely many terms $U_n$ have a primitive prime divisor $p$ with $(Q/p)=-1$, with a positive lower density for certain values of $Q$. However, many cases remain in which our methods fail to apply. Exploring other ideas leads to other interesting density questions.