Mon, 11 May 2015
15:45

Tail Estimates for Markovian Rough Paths

Marcel Ogrodnik
(Imperial College London)
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.

Tue, 27 Jan 2015

17:00 - 18:00
C2

Regular maps and simple groups

Martin Liebeck
(Imperial College London)
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.

Thu, 04 Dec 2014

16:00 - 17:00
L5

Twitter Video Download

Alexei Skorobogatov
(Imperial College London)
Further Information

Twitter Video Download: https://indireyim.com/

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.

Mon, 24 Nov 2014

15:30 - 16:30
L2

Bifurcations in mathematical models of self-organization

Pierre Degond
(Imperial College London)
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.

Tue, 02 Dec 2014
15:45
L4

The homological projective dual of Sym^2(P^n)

Jorgen Rennemo
(Imperial College London)
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.

Tue, 13 May 2014

14:00 - 15:00
L4

The Crepant Transformation Conjecture and Fourier--Mukai Transforms

Tom Coates
(Imperial College London)
Abstract

Suppose that X and Y are Kahler manifolds or orbifolds which are related by a crepant resolution or flop F.  It is expected that the Gromov--Witten potentials of X and Y should be related by analytic continuation in Kahler parameters combined with a linear symplectomorphism between Givental's symplectic spaces for X and Y.  This linear symplectomorphism is expected to coincide, in a precise sense which I will explain, with the Fourier--Mukai transform on K-theory induced by F.  In this talk I will prove these conjectures, as well as their torus-equivariant generalizations, in the case where X and Y are toric.  
This is joint work with Hiroshi Iritani and Yunfeng Jian
Thu, 12 Jun 2014

16:00 - 17:00
L5

A homotopy exact sequence and unipotent fundamental groups over function fields

Christopher Lazda
(Imperial College London)
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.

Tue, 13 May 2014

15:30 - 16:30
L4

Mirror symmetry without localisation

Tom Coates
(Imperial College London)
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.

Mon, 24 Feb 2014

15:45 - 16:45
Eagle House

Constrained rough paths

THOMAS CASS
(Imperial College London)
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.

Mon, 21 Oct 2013

14:15 - 15:15
Oxford-Man Institute

Asymptotic independence of three statistics of the maximal increments of random walks and Levy processes

Aleksandar Mijatovic
(Imperial College London)
Abstract
Abstract: Let $H(x) = \inf\{n:\, \exists\, k x\}$ be the first epoch that an increment of the size larger than $x>0$ of a random walk $S$ occurs and consider the path functionals: $$ R_n = \max_{m\in\{0, \ldots, n\}}\{S_{n} - S_m\}, R_n^* = \max_{m,k\in\{0, \ldots, n\}, m\leq k} \{S_{k}-S_m\} \text{and} O_x=R_{H(x)}-x.$$ The main result states that, under Cram\'{e}r's condition on the step-size distribution of $S$, the statistics $R_n$, $R_n^* -y$ and $O_{x+y}$ are asymptotically independent as $\min\{n,y,x\}\uparrow\infty$. Furthermore, we establish a novel Spitzer-type identity characterising the limit law $O_\infty$ in terms of the one-dimensional marginals of $S$. If $y=\gamma^{-1}\log n$, where $\gamma$ is the Cram\'er coefficient, our results together with the classical theorem of Iglehart (1972) imply the existence of a joint weak limit of the three statistics and identify its law. As corollary we obtain a new factorization of the exponential distribution as a convolution of the asymptotic overshoot $O_\infty$ and the stationary distribution of the reflected random walk $R$. We prove analogous results for the corresponding statistics of a L\'{e}vy process. This is joint work with M. Pistorius.
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