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.
Fri, 08 Nov 2013

16:00 - 17:00
L4

Optimal Collateralization with Bilateral Default Risk

Enrico Biffis
(Imperial College London)
Abstract
We consider over-the-counter (OTC) transactions with bilateral default risk, and study the optimal design of the Credit Support Annex (CSA). In a setting where agents have access to a trading technology, default penalties and collateral costs arise endogenously as a result of foregone investment opportunities. We show how the optimal CSA trades off the costs of the collateralization procedure against the reduction in exposure to counterparty risk and expected default losses. The results are used to provide insights on the drivers of different collateral rules, including hedging motives, re-hypothecation of collateral, and close-out conventions. We show that standardized collateral rules can have a detrimental impact on risk sharing, which should be taken into account when assessing the merits of standardized vs. bespoke CSAs in non-centrally cleared OTC instruments. This is joint work with D. Bauer and L.R. Sotomayor (GSU).
Thu, 02 May 2013

14:00 - 15:00
L2

Sheafy matrix factorizations and bundles of quadrics

Ed Segal
(Imperial College London)
Abstract

A Landau-Ginzburg B-model is a smooth scheme $X$, equipped with a global function $W$. From $(X,W)$ we can construct a category $D(X,W)$, which is called by various names, including ‘the category of B-branes’. In the case $W=0$ it is exactly the derived category $D(X)$, and in the case that $X$ is affine it is the category of matrix factorizations of $W$. There has been a lot of foundational work on this category in recent years, I’ll describe the most modern and flexible approach to its construction. I’ll then interpret Nick Addington’s thesis in this language. We’ll consider the case that $W$ is a quadratic form on a vector bundle, and the corresponding global version of Knorrer periodicity. We’ll see that interesting gerbe structures arise, related to the bundle of isotropic Grassmannians.

Thu, 02 May 2013

14:00 - 15:00
L2

Sheafy matrix factorizations and bundles of quadrics

Ed Segal
(Imperial College London)
Abstract
A Landau-Ginzburg B-model is a smooth scheme $X$, equipped with a global function $W$. From $(X,W)$ we can construct a category $D(X,W)$, which is called by various names, including ‘the category of B-branes’. In the case $W=0$ it is exactly the derived category $D(X)$, and in the case that $X$ is affine it is the category of matrix factorizations of $W$. There has been a lot of foundational work on this category in recent years, I’ll describe the most modern and flexible approach to its construction.

I’ll then interpret Nick Addington’s thesis in this language. We’ll consider the case that $W$ is a quadratic form on a vector bundle, and the corresponding global version of Knorrer periodicity. We’ll see that interesting gerbe structures arise, related to the bundle of isotropic Grassmannians.

Thu, 07 Feb 2013

16:00 - 17:00
L3

C-groups

Kevin Buzzard
(Imperial College London)
Abstract

Toby Gee and I have proposed the definition of a "C-group", an extension of Langlands' notion of an L-group, and argue that for an arithmetic version of Langlands' philosophy such a notion is useful for controlling twists properly. I will give an introduction to this business, and some motivation. I'll start at the beginning by explaining what an L-group is a la Langlands, but if anyone is interested in doing some background preparation for the talk, they might want to find out for themselves what an L-group (a Langlands dual group) is e.g. by looking it up on Wikipedia!

Thu, 31 Jan 2013

16:00 - 17:00
L3

Classicality for overconvergent eigenforms on some Shimura varieties.

Christian Johansson
(Imperial College London)
Abstract
A well known theorem of Coleman states that an overconvergent modular eigenform of weight k>1 and slope less than k-1 is classical. This theorem was later reproved and generalized using a geometric method very different from Coleman's cohomological approach. In this talk I will explain how one might go about generalizing the cohomological method to some higher-dimensional Shimura varieties.
Thu, 24 Jan 2013

16:00 - 17:00
L3

p-adic functoriality for inner forms of unitary groups.

Judith Ludwig
(Imperial College London)
Abstract

In this talk I will explain a notion of p-adic functoriality for inner forms of definite unitary groups. Roughly speaking, this is a morphism between so-called eigenvarieties,  which are certain rigid analytic spaces parameterizing p-adic families  of automorphic forms. We will then study certain properties of classical Langlands functoriality that allow us to prove p-adic functoriality in some "stable" cases.

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