Past

In 1890, G. H. Bryan demonstrated that when a ringing wine glass rotates, the shape of the vibration pattern precesses, and this effect is the basis for a family of high-precision gyroscopes. Mathematically, the precession can be described in terms of a symmetry-breaking perturbation due to gyroscopic effects of a geometrically degenerate pair of vibration modes.  Unfortunately, current attempts to miniaturize these gyroscope designs are subject to fabrication imperfections that also break the device symmetry. In this talk, we describe how these devices work and our approach to accurate and efficient simulations of both ideal device designs and designs subject to fabrication imperfections.

Quantum field theory (QFT) originated in physics in the context of

elementary particles. Although, over the years, surprising and profound

connections to very diverse branches of mathematics have been discovered,

QFT does not have, as yet, found a universally accepted "standard"

mathematical formulation. In this talk, I shall outline an approach to QFT

that emphasizes its underlying algebraic structure. Concretely, this is

represented by a concept called "Operator Product Expansion". I explain the

properties of such expansions, how they can be constructed in concrete QFT

models, and the emergent relationship between "perturbation theory" on the

physics side and

"Hochschild cohomology" on the physics side. This talk is based on joint

work

with Ch. Kopper and J. Holland from Ecole Polytechnique, Paris.

The conformal approach to scattering theory goes back to the 1960's

and 1980's, essentially with the works of Penrose, Lax-Phillips and

Friedlander. It is Friedlander who put together the ideas of Penrose

and Lax-Phillips and presented the first conformal scattering theory

in 1980. Later on, in the 1990's, Baez-Segal-Zhou explored Friedlander's

method and developed several conformal scattering theories. Their

constructions, just like Friedlander's, are on static spacetimes. The

idea of replacing spectral analysis by conformal geometry is however

the door open to the extension of scattering theories to general non

stationary situations, which are completely inaccessible to spectral

methods. A first work in collaboration with Lionel Mason explained

these ideas and applied them to non stationary spacetimes without

singularity. The first results for nonlinear equations on such

backgrounds was then obtained by Jeremie Joudioux. The purpose is now

to extend these theories to general black holes. A first crucial step,

recently completed, is a conformal scattering construction on

Schwarzschild's spacetime. This talk will present the history of the

ideas, the principle of the constructions and the main ingredients

that allow the extension of the results to black hole geometries.

In this presentation, we focus on the decay rate of the expected signature of a stopped Brownian motion; more specifically we consider two types of the stopping time: the first one is the Brownian motion up to the first exit time from a bounded domain $\Gamma$, denoted by $\tau_{\Gamma}$, and the other one is the Brownian motion up to $min(t, \tau_{\Gamma\})$. For the first case, we use the Sobolev theorem to show that its expected signature is geometrically bounded while for the second case we use the result in paper (Integrability and tail estimates for Gaussian rough differential equation by Thomas Cass, Christian Litterer and Terry Lyons) to show that each term of the expected signature has the decay rate like 1/ \sqrt((n/p)!) where p>2. The result for the second case can imply that its expected signature determines the law of the signature according to the paper (Unitary representations of geometric rough paths by Ilya Chevyrev)

In order to study the group of (outer) automorphisms of

any group G by geometric methods one needs a well-behaved "outer

space" with an interesting action of Out(G). If G is free abelian, the

classic symmetric space SL(n,R)/SO(n) serves this role, and if G is

free non-abelian an appropriate outer space was introduced in the

1980's. I will recall these constructions and then introduce joint

work with Ruth Charney on constructing an outer space for any

right-angled Artin group.

By solving the Homogeneous Monge-Ampere equation on the deformation to the normal cone of a complex submanifold of a Kahler manifold, we get a canonical tubular neighbourhood adapted to both the holomorphic and the symplectic structure. If time permits I will describe an application, namely an optimal regularity result for certain naturally defined plurisubharmonic envelopes.

Simple probabilistic models for sequential data (text, music...), e.g., hidden Markov models, cannot capture some structures such as
long-term dependencies or combinations of simultaneous patterns and probabilistic rules hidden in the data. On the other hand, models such as
recurrent neural networks can in principle handle any structure but are notoriously hard to learn given training data. By analyzing the structure of
neural networks from the viewpoint of Riemannian geometry and information theory, we build better learning algorithms, which perform well on difficult
toy examples at a small computational cost, and provide added robustness.

Dynamics in nature often proceed in the form of reactive events, aka activated processes. The system under study spends very long periods of time in various metastable states; only very rarely does it transition from one such state to another. Understanding the dynamics of such events requires us to study the ensemble of transition paths between the different metastable states. Transition path theory (TPT) is a general mathematical framework developed for this purpose. It is also the foundation for developing modern numerical algorithms such as the string method for finding the transition pathways or milestoning to calculate the reaction rate, and it can also be used in the context of Markov State Models (MSMs). In this talk, I will review the basic ingredients of the transition path theory and discuss connections with transition state theory (TST) as well as approaches to metastability based on potential theory and large deviation theory. I will also discuss how the string method arises in order to find approximate solutions in the framework of the transition path theory, the connections between milestoning and TPT, and the way the theory help building MSMs. The concepts and methods will be illustrated using examples from molecular dynamics, material science and atmosphere/ocean sciences.

We will describe a generalization of the algebra of differential operators, which gives a

geometric approach to quantization of cotangent field theories. This construction is compatible

with "integration" thus giving a local-to-global construction of volume forms on derived mapping

spaces using a version of non-abelian duality. These volume forms give interesting invariants of

varieties such as the Todd genus, the Witten genus and the B-model operations on Hodge

cohomology.

The observations of the first traces of cosmic structure in the

Cosmic Microwave Background are in excellent agreement with the

predictions of Inflation. However as we shall see, that account

is not fully satisfactory, as it does not address the transition

from an homogeneous and isotropic early stage to a latter one

lacking those symmetries. We will argue that new physics along the

lines of the dynamical quantum state reduction theories is needed

to account for such transition and, motivated by Penrose's ideas

suggest that quantum gravity might be the place from where

this new physics emerges. Moreover we will show that observations

can be used to constrain the various phenomenological proposals

made in this regard.

Microbial biofilms grow on most rock and stone surfaces and may play critical roles in weathering. With climate change and improving air quality in many cities in Europe biofilms are growing rapidly on many historic stone buildings and posing practical problems for heritage conservation. With many new field and lab techniques available it is now possible to identify the microbes present and start to clarify their roles. We now need help modelling microbial biofilm growth and impacts in order to provide better advice for conservators.

The question of how to derive useful bounds on

arbitrage-free prices of exotic options given only prices of liquidly

traded products like European call und put options has received much

interest in recent years. It also led to new insights about classic

problems in probability theory like the Skorokhod embedding problem. I

will take this as a starting point and show how this progress can be

used to give new results on general Monte-Carlo schemes.

This topic was the subject of an OCIAM workshop on 8th March 2013

given by Nick Hall Taylor . The presentation will start with a review

of the physical problem and experimental evidence. A mathematical

model leading to a hydrodynamic free boundary problem has been derived

and some mathematical and computational results will be described.

Finally we will assess the results so far and list a number of

interesting open problems.

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After the workshop and during coffee at 11:30, we will also give a preview of the

upcoming problems at the Malaysian Study Group (Mar. 17-21). Problem

descriptions can be found here:

www.maths.ox.ac.uk/~trinh/2014_studygroup_problems.pdf.

Several number theorists have stressed that the proofs of FLT focus on small concrete arithmetically defined groups rings and modules, so the steps can be checked by direct calculation in any given case. The talk looks at this in relation both to Hilbert's idea of contentual (inhaltlich) mathematics, and to formal provability in Peano arithmetic and other stronger and weaker axioms.

We study lower- and dual upper-bounds for Bermudan options in a MonteCarlo/MC setting and provide four contributions. 1) We introduce a local least-squares MC method, based on maximizing the Bermudan price and which provides a lower-bound, which "also" minimizes (not the dual upper-bound itself, but) the gap between these two bounds; where both bounds are specified recursively. 2) We confirm that this method is near optimal, for both lower- and upper-bounds, by pricing Bermudan max-call options subject to an up-and-out barrier; state-of-the-art methods including Longstaff-Schwartz produce a large gap of 100--200 basis points/bps (Desai et al. (2012)), which we reduce to just 5--15 bps (using the same linear basis of functions). 3) For dual upper-bounds based on continuation values (more biased but less time intensive), it works best to reestimate the continuation value in the continuation region only. And 4) the difference between the Bermudan option Delta and the intrinsic value slope at the exercise boundary gives the sensitivity to suboptimal exercise (up to a 2nd-order Taylor approximation). The up-and-out feature flattens the Bermudan price, lowering the Bermudan Delta well below one when the call-payoff slope is equal to one, which implies that optimal exercise "really" matters.

I plan to give a non technical introduction (i.e. no prerequisites required apart basic differential geometry) to some analytic aspects of the theory of harmonic maps between Riemannian manifolds, motivate it by briefly discussing some relations to other areas of geometry (like minimal submanifolds, string topology, symplectic geometry, stochastic geometry...), and finish by talking about the heat flow approach to the existence theory of harmonic maps with some open problems related to my research.

I will discuss a novel approach to estimating communication costs of an algorithm (also known as its I/O complexity), which is based on small-set expansion for computational graphs. Various applications and implications will be discussed as well, mostly having to do with linear algebra algorithms. This includes, in particular, first known (and tight) bounds on communication complexity of fast matrix multiplication.

Joint work with Grey Ballard, James Demmel, Benjamin Lipshitz and Oded Schwartz.