In this talk, I consider the problem of

hedging European and Bermudan option with a given probability. This

question is

more generally linked to portfolio optimisation problems under weak

stochastic target constraints.

I will recall, in a Markovian framework, the characterisation of the

solution by

non-linear PDEs. I will then discuss various numerical algorithms

to compute in practice the quantile hedging price.

This presentation is based on joint works with B. Bouchard (Université

Paris Dauphine), G. Bouveret (University of Oxford) and ongoing work

with C. Benezet (Université Paris Diderot).

# Past Forthcoming Seminars

In a recent preprint, Basterra, Bobkova, Ponto, Tillmann and Yeakel

defined operads with homological stability (OHS) and showed that after

group-completion, algebras over an OHS group-complete to infinite loop

spaces. This can in particular be used to put a new infinite loop space

structure on stable moduli spaces of high-dimensional manifolds in the

sense of Galatius and Randal-Williams, which are known to be infinite

loop spaces by a different method.

To complicate matters further, I shall introduce a mild strengthening of

the OHS condition and construct yet another infinite loop space

structure on these stable moduli spaces. This structure turns out to be

equivalent to that constructed by Basterra et al. It is believed that

the infinite loop space structure due to Galatius--Randal-Williams is

also equivalent to these two structures.

Spline curves represent a simple and efficient tool for data interpolation in Euclidean space. During the past decades, however, more and more applications have emerged that require interpolation in (often high-dimensional) nonlinear spaces such as Riemannian manifolds. An example is the generation of motion sequences in computer graphics, where the animated figure represents a curve in a Riemannian space of shapes. Two particularly useful spline interpolation methods derive from a variational principle: linear splines minimize the average squared velocity and cubic splines minimize the average squared acceleration among all interpolating curves. Those variational principles and their discrete analogues can be used to define continuous and discretized spline curves on (possibly infinite-dimensional) Riemannian manifolds. However, it turns out that well-posedness of cubic splines is much more intricate on nonlinear and high-dimensional spaces and requires quite strong conditions on the underlying manifold. We will analyse and discuss linear and cubic splines as well as their discrete counterparts on Riemannian manifolds and show a few applications.

In the study of variational models for non-linear elasticity in the context of proving regularity we are led to the challenging so-called Ball-Evan's problem of approximating a Sobolev homeomorphism with diffeomorphisms in its Sobolev space. In some cases however we are not able to guarantee that the limit of a minimizing sequence is a homeomorphism and so the closure of Sobolev homeomorphisms comes into the game. For $p\geq 2$ they are exactly Sobolev monotone maps and for $1\leq p<2$ the monotone maps are intricately related to these limits. In our paper we prove that monotone maps can be approximated by diffeomorphisms in their Sobolev (or Orlicz-Sobolev) space including the case $p=1$ not proven by Iwaniec and Onninen.

Warped cones are infinite metric spaces that are associated with actions by homeomorphisms on metric spaces. In this talk I will try to explain why the coarse geometry of warped cones can be seen as an invariant of the action and what it can tell us about the acting group.

Broken cryptographic algorithms and hardness assumptions are a constant

threat to real-world protocols. Prominent examples are

hash functions for which collisions become known, or number-theoretic

assumptions which are threatened by advances in quantum computing.

Especially when it comes to key exchange protocols, the switch to

quantum-resistant primitives has begun and aims to protect today’s

secrets against future developments, moving from common Diffie–Hellman

based solutions to Learning-With-Errors-based approaches. Remarkably,

the authentication step in such protocols is usually still carried out

with quantum-vulnerable signature schemes. The intuition here is that

the adversary would need to break this protocol primitive today, without

having quantum power yet. The question we address here is if this

intuition is justified, and if so, if we can show this rigorously. We

particularly consider the authenticated variant of the recently

introduced post-quantum secure key exchange protocol NewHope (Alkim et

al., USENIX Security 2016), as well as by TLS 1.3, which is currently

being developed by the Internet Engineering Task Force.

I was speak on the way Newton carries out his calculus in the Principia in the framework of classical geometry rather than with fluxions, his deficiencies, and the relation of this work to inverse-square laws.

(Joint with Gareth Boxall) In this talk I will introduce some properties of distal theories. I will remark that distality is preserved neither under reducts nor expansions of the language. I will then go on to discuss a recent result that the Shelah expansion of a theory is distal if and only if the theory itself is distal.

Understanding the distribution of subgraph counts has long been a central question in the study of random graphs. In this talk, we consider the distribution of Sn, the number of K4 subgraphs, in the Erdös Rényi random graph G(n, p). When the edge probability p \in (0, 1) is constant, a classical central limit theorem for Sn states that (Sn−µn)/σn converges in distribution. We establish a stronger form of convergence, namely the corresponding local limit theorem, which is joint work with O. Riordan.

In mirror symmetry, the prepotential on the Kahler side has an expansion, the constant term of which is a rational multiple of zeta(3)/(2 pi i)^3 after an integral symplectic transformation. In this talk I will explain the connection between this constant term and the period of a mixed Hodge-Tate structure constructed from the limit MHS at large complex structure limit on the complex side. From Ayoub’s works on nearby cycle functor, there exists an object of Voevodsky’s category of mixed motives such that the mixed Hodge-Tate structure is expected to be a direct summand of the third cohomology of its Hodge realisation. I will present the connections between this constant term and conjecture about how mixed Tate motives sit inside Voevodsky’s category, which will also provide a motivic interpretation to the occurrence of zeta(3) in prepotential.