Thu, 09 Feb 2017

16:00 - 17:00
L6

A logarithmic interpretation of Edixhoven's jumps for Jacobians

Johannes Nicaise
(Imperial College London)
Abstract

Let A be an abelian variety over a strictly henselian discretely valued field K. In his 1992 paper "Néron models and tame ramification", Edixhoven has constructed a filtration on the special fiber of the Néron model of A that measures the behaviour of the Néron model with respect to tamely ramified extensions of K. The filtration is indexed by rational numbers in [0,1], and if A is wildly ramified, it is an open problem whether the places where it jumps are always rational. I will explain how an interpretation of the filtration in terms of logarithmic geometry leads to explicit formulas for the jumps in the case where A is a Jacobian, which confirms in particular that they are rational. This is joint work with Dennis Eriksson and Lars Halvard Halle.

Thu, 24 Nov 2016

16:00 - 17:30
L4

The Randomised Heston model

Jack Jacquier
(Imperial College London)
Abstract

We propose a randomised version of the Heston model--a widely used stochastic volatility model in mathematical finance--assuming that the starting point of the variance process is a random variable. In such a system, we study the small- and large-time behaviours of the implied volatility, and show that the proposed randomisation generates a short-maturity smile much steeper (`with explosion') than in the standard Heston model, thereby palliating the deficiency of classical stochastic volatility models in short time. We precisely quantify the speed of explosion of the smile for short maturities in terms of the right tail of the initial distribution, and in particular show that an explosion rate of $t^\gamma$ (gamma in [0,1/2]) for the squared implied volatility--as observed on market data--can be obtained by a suitable choice of randomisation. The proofs are based on large deviations techniques and the theory of regular variations. Joint work with Fangwei Shi (Imperial College London)

Mon, 09 May 2016

15:45 - 16:45
C6

Global quantizations with and without symmetries

MICHAEL RUZHANSKY
(Imperial College London)
Abstract

In this talk we will give an overview of the recent research on global quantizations on spaces of different types: compact and nilpotent Lie groups, general locally compact groups, compact manifolds with boundary.

Fri, 27 May 2016

13:00 - 14:30
L6

Deep Learning for Modeling Financial Data

Justin Sirignano, postdoc at Imperial College.
(Imperial College London)
Abstract
Deep learning has emerged as one of the forefront areas in machine learning, achieving major success in imaging, speech recognition, and natural language processing. We apply deep learning to two areas in finance: (1) mortgage delinquency and prepayment and (2) limit order books. Using datasets unprecedented in size, we show that deep neural networks outperform several status quo approaches. Due to the heavy computational cost from both the size of the models and the data, we use GPU clusters to train the models.
Tue, 09 Feb 2016

14:00 - 15:00
L4

Virtual signed Euler characteristics and the Vafa-Witten equations

Richard Thomas
(Imperial College London)
Abstract

I will describe 5 definitions of Euler characteristic for a space with perfect obstruction theory (i.e. a well-behaved moduli space), and their inter-relations. This is joint work with Yunfeng Jiang. Then I will describe work of Yuuji Tanaka on how to this can be used to give two possible definitions of Vafa-Witten invariants of projective surfaces in the stable=semistable case.

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)
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