Tue, 13 Oct 2015

15:45 - 16:45
L4

### D-modules from the b-function and Hamiltonian flow

Travis Schedler
(Imperial College London)
Abstract

Given a hypersurface, the Bernstein-Sato polynomial gives deep information about its singularities.  It is defined by a D-module (the algebraic formalism of differential equations) closely related to analytic continuation of the gamma function. On the other hand, given a hypersurface (in a Calabi-Yau variety) one can also consider the Hamiltonian flow by divergence-free vector fields, which also defines a D-module considered by Etingof and myself. I will explain how, in the case of quasihomogeneous hypersurfaces with isolated singularities, the two actually coincide. As a consequence I affirmatively answer a folklore question (to which M. Saito recently found a counterexample in the non-quasihomogeneous case): if c\$ is a root of the b-function, is the D-module D f^c / D f^{c+1} nonzero? We also compute this D-module, and for c=-1 its length is one more than the genus (conjecturally in the non-quasihomogenous case), matching an analogous D-module in characteristic p. This is joint work with Bitoun.

Thu, 26 Nov 2015

16:00 - 17:30
L4

### Nonlinear valuation under credit gap risk, collateral margins, funding costs and multiple curves

Damiano Brigo
(Imperial College London)
Abstract

Following a quick introduction to derivatives markets and the classic theory of valuation, we describe the changes triggered by post 2007 events. We re-discuss the valuation theory assumptions and introduce valuation under counterparty credit risk, collateral posting, initial and variation margins, and funding costs. A number of these aspects had been investigated well before 2007. We explain model dependence induced by credit effects, hybrid features, contagion, payout uncertainty, and nonlinear effects due to replacement closeout at default and possibly asymmetric borrowing and lending rates in the margin interest and in the funding strategy for the hedge of the relevant portfolio. Nonlinearity manifests itself in the valuation equations taking the form of semi-linear PDEs or Backward SDEs. We discuss existence and uniqueness of solutions for these equations. We present an invariance theorem showing that the final valuation equations do not depend on unobservable risk free rates, that become purely instrumental variables. Valuation is thus based only on real market rates and processes. We also present a high level analysis of the consequences of nonlinearities, both from the point of view of methodology and from an operational angle, including deal/entity/aggregation dependent valuation probability measures and the role of banks treasuries. Finally, we hint at how one may connect these developments to interest rate theory under multiple discount curves, thus building a consistent valuation framework encompassing most post-2007 effects.

Damiano Brigo, Joint work with Andrea Pallavicini, Daniele Perini, Marco Francischello.

Thu, 05 May 2016

16:00 - 17:00
L3

### Singular asymptotics of surface-plasmon resonance

Ory Schnitzer
(Imperial College London)
Abstract

Surface plasmons are collective electron-density oscillations at a metal-dielectric interface. In particular, highly localised surface-plasmon modes of nanometallic structures with narrow nonmetallic gaps, which enable a tuneable resonance frequency and a giant near-field enhancement, are at the heart of numerous nanophotonics applications. In this work, we elucidate the singular near-contact asymptotics of the plasmonic eigenvalue problem governing the resonant frequencies and modes of such structures. In the classical regime, valid for gap widths > 1nm, we find a generic scaling describing the redshift of the resonance frequency as the gap width is reduced, and in several prototypical dimer configurations derive explicit expressions for the plasmonic eigenvalues and eigenmodes using matched asymptotic expansions; we also derive expressions describing the resonant excitation of such modes by light based on a weak-dissipation limit. In the subnanometric nonlocal’’ regime, we show intuitively and by systematic analysis of the hydrodynamic Drude model that nonlocality manifests itself as a potential discontinuity, and in the near-contact limit equivalently as a widening of the gap. We thereby find the near-contact asymptotics as a renormalisation of the local asymptotics, and in particular a lower bound on plasmon frequency, scaling with the 1/4 power of the Fermi wavelength. Joint work with Vincenzo Giannini, Richard V. Craster and Stefan A. Maier.

Mon, 01 Jun 2015
14:15

### tba

Nikolas Kantas
(Imperial College London)
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

Alexei Skorobogatov
(Imperial College London)
Further Information
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
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