15:45
Tail Estimates for Markovian Rough Paths
Abstract
We work in the context of Markovian rough paths associated to a class of uniformly subelliptic Dirichlet forms and prove an almost-Gaussian tail-estimate for the accumulated local p-variation functional, which has been introduced and studied by Cass, Litterer and Lyons. We comment on the significance of these estimates to a range of currently-studied problems, including the recent results of Ni Hao, and Chevyrev and Lyons.
Regular maps and simple groups
Abstract
A regular map is a highly symmetric embedding of a finite graph into a closed surface. I will describe a programme to study such embeddings for a rather large class of graphs: namely, the class of orbital graphs of finite simple groups.
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Abstract
Rational points on Kummer varieties can be studied through the variation of Selmer groups of quadratic twists of the underlying abelian variety, using an idea of Swinnerton-Dyer. We consider the case when the Galois action on 2-torsion has a large image. Under a mild additional assumption we prove the Hasse principle assuming the finiteness of relevant Shafarevich-Tate groups. This approach is inspired by the work of Mazur and Rubin.
Bifurcations in mathematical models of self-organization
Abstract
We consider self-organizing systems, i.e. systems consisting of a large number of interacting entities which spontaneously coordinate and achieve a collective dynamics. Sush systems are ubiquitous in nature (flocks of birds, herds of sheep, crowds, ...). Their mathematical modeling poses a number of fascinating questions such as finding the conditions for the emergence of collective motion. In this talk, we will consider a simplified model first proposed by Vicsek and co-authors and consisting of self-propelled particles interacting through local alignment.
We will rigorously study the multiplicity and stability of its equilibria through kinetic theory methods. We will illustrate our findings by numerical simulations.
15:45
The homological projective dual of Sym^2(P^n)
Abstract
In recent years, some powerful tools for computing semi-orthogonal decompositions of derived categories of algebraic varieties have been developed: Kuznetsov's theory of homological projective duality and the closely related technique of VGIT for LG models. In this talk I will explain how the latter works and how it can be used to understand the derived categories of complete intersections in Sym^2(P^n). As a consequence, we obtain a new proof of result of Hosono and Takagi, which says that a certain pair of non-birational Calabi-Yau 3-folds are derived equivalent.
The Crepant Transformation Conjecture and Fourier--Mukai Transforms
Abstract
A homotopy exact sequence and unipotent fundamental groups over function fields
Abstract
If X/F is a smooth and proper variety over a global function field of
characteristic p, then for all l different from p the co-ordinate ring of the l-adic
unipotent fundamental group is a Galois representation, which is unramified at all
places of good reduction. In this talk, I will ask the question of what the correct
p-adic analogue of this is, by spreading out over a smooth model for C and proving a
version of the homotopy exact sequence associated to a fibration. There is also a
version for path torsors, which enables me to define an function field analogue of
the global period map used by Minhyong Kim to study rational points.
Mirror symmetry without localisation
Abstract
Mirror Symmetry predicts a surprising relationship between the virtual numbers of degree-d rational curves in a target space X and variations of Hodge structure on a different space X’, called the mirror to X. Concretely, it predicts that one can compute genus-zero Gromov–Witten invariants (which are the virtual numbers of rational curves) in terms of hypergeometric functions (which are the solutions to a differential equation that controls the variation of Hodge structure). Existing proofs of this rely on beautiful but fearsomely complicated localization calculations in equivariant cohomology. I will describe a new proof of the Mirror Theorem, for a broad range of target spaces X, which is much simpler and more conceptual. This is joint work with Cristina Manolache.
Constrained rough paths
Abstract
I present some recent work with Bruce Driver and Christian Litterer on rough paths 'constrained’ to lie in a d - dimensional submanifold of a Euclidean space E. We will present a natural definition for this class of rough paths and then describe the (second) order geometric calculus which arises out of this definition. The talk will conclude with more advanced applications, including a rough version of Cartan’s development map.