Past

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.

We investigate certain (hopefully new) arithmetic aspects of abelian varieties defined over function fields of curves over finitely generated fields. One of the key ingredients in our investigation is a new specialisation theorem a la N\'eron for the first Galois cohomology group with values in the Tate module, which generalises N\'eron specialisation theorem for rational points. Also, among other things, we introduce a discrete version of Selmer groups, which are finitely generated abelian groups. We also discuss an application of our investigation to anabelian geometry (joint work with Akio Tamagawa).

Familiar species; humans, mammals, fish, reptiles and plants represent only a razor’s edge of the Earth’s immense biodiversity. Most of the Earth’s multicellular species lie buried in soil, inside of plants, and in the undergrowth, and include millions of unknown species, almost half of which are thought to be fungi. Part of the amazing success of fungi may be the elegant solutions that they have evolved to the problems of dispersing, growing and adapting to changing environments. I will describe how we using both math modeling and experiments to discover some of these solutions. I will focus on (i) how cytoplasmic mixing enables some species to tolerate internal genetic diversity, making them better pathogens and more adaptable, and (ii) how self-organization of these flows into phases of transport and stasis enables cells to function both as transport conduits, and to perform other functions like growth and secretion.

Adaptive finite element methods (AFEMs) with Dörflers marking strategy are known to converge with

optimal asymptotical rates. Practical experiences show that AFEMs with a maximum marking strategy

produces optimal results thereby being less sensitive to choices of the marking parameter.

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In this talk, we prove that an AFEM with a modified maximum strategy is even instance optimal for the

total error, i.e., for the sum of the error and the oscillation. This is a non-asymptotical optimality result.

Our approach uses new techniques based on the minimisation of the Dirichlet energy and a newly

developed tree structure of the nodes of admissible triangulations.

I will discuss free-by-cyclic groups and cases where they can and cannot act on CAT(0) spaces. I will specifically go into a construction building CAT(0) 2-complexes on which free of rank 2-by-cyclic act. This is joint work with Martin Bridson and Martin Lustig.

I will discuss some very well studied cohomology groups that turn out to be captured by the machinery of critical CoHAs, for example the compactly supported cohomology of singular quiver varieties and untwisted character varieties. I will explain the usefulness of this extra CoHA structure on these groups, starting with a new proof of the Kac conjecture, and discuss a conjectural form for the CoHA associated to untwisted character varieties that provides a new way to think about the conjectures of Hausel and Rodriguez-Villegas. Finally I will discuss an approach to purity for the compactly supported cohomology of quiver varieties and a related approach to a conjecture of Shiffmann and Vasserot, analogous to Kirwan surjectivity for the stack of commuting matrices.