Past

This talk will discuss work-in-progress on the numerical approximation
of reflected diffusions arising from applications in engineering, finance
and network queueing models.  Standard numerical treatments with
uniform timesteps lead to 1/2 order strong convergence, and hence
sub-optimal behaviour when using multilevel Monte Carlo (MLMC).

In simple applications, the MLMC variance can be improved by through
a reflection "trick".  In more general multi-dimensional applications with
oblique reflections an alternative method uses adaptive timesteps, with
smaller timesteps when near the boundary.  In both cases, numerical
results indicate that we obtain the optimal MLMC complexity.

This is based on joint research with Eike Muller, Rob Scheichl and Tony
Shardlow (Bath) and Kavita Ramanan (Brown).

I will discuss some reaction-diffusion equations of bistable type motivated by biology and medicine. The aim is to understand the effect of the shape of the domain on propagation or on blocking of advancing waves. I will first describe the motivations of these questions and present a result about the existence of generalized “transition waves”. I will then discuss various geometric conditions that lead to either blocking, or partial propagation, or complete propagation. These questions involve new qualitative results for some non-linear elliptic and parabolic partial differential equations. I report here on joint work with Juliette Bouhours and Guillemette Chapuisat.

On the railway network, for example, there is a large base of installed equipment with a useful life of many years.  This equipment has condition monitoring that can flag a fault when a measured parameter goes outside the permitted range.  If we can use existing measurements to predict when this would occur, preventative maintenance could be targeted more effectively and faults reduced.  As an example, we will consider the current supplied to a points motor as a function of time in each operational cycle.

We say that a real closed field is an IPA-real closed field if it admits an integer part (IP) which is a model of Peano Arithmetic (PA). In [2] we prove that the value group of an IPA-real closed field must satisfy very restrictive conditions (i.e. must be an exponential group in the residue field, in the sense of [4]). Combined with the main result of [1] on recursively saturated real closed fields, we obtain a valuation theoretic characterization of countable IPA-real closed fields. Expanding on [3], we conclude the talk by considering recursively saturated o-minimal expansions of real closed fields and their IPs.

References:
[1] D'Aquino, P. - Kuhlmann, S. - Lange, K. : A valuation theoretic characterization ofrecursively saturated real closed fields ,
Journal of Symbolic Logic, Volume 80, Issue 01, 194-206 (2015)
[2] Carl, M. - D'Aquino, P. - Kuhlmann, S. : Value groups of real closed fields and
fragments of Peano Arithmetic, arXiv: 1205.2254, submitted
[3] D'Aquino, P. - Kuhlmann, S : Saturated o-minimal expansions of real closed fields, to appear in Algebra and Logic (2016)
[4] Kuhlmann, S. :Ordered Exponential Fields, The Fields Institute Monograph Series, vol 12. Amer. Math. Soc. (2000)
 

I will describe how the moduli of various congruences between Hecke eigenvalues of automorphic forms ought to show up in ratios of critical values of $\text{GSP}_2 \times \text{GL}_2$ L-functions. To test this experimentally requires the full force of Farmer and Ryan's technique for approximating L-values given few coefficients in the Dirichlet series.

We describe the possible influence of stochastic 
dependence on the evaluation of
the risk of joint portfolios and establish relevant risk bounds.Some 
basic tools for this purpose are  the distributional transform,the 
rearrangement method and extensions of the classical Hoeffding -Frechet 
bounds based on duality theory.On the other hand these tools find also 
essential applications to various problems of optimal investments,to the 
construction of cost-efficient payoffs as well as to various optimal 
hedging problems.We
discuss in detail the case of optimal payoffs in Levy market models as 
well as utility optimal payoffs and hedgings
with state dependent utilities.

We will review the basic building blocks of iterative solvers, i.e. sparse matrix-vector multiplication, in the context of GPU devices such 
as the cards by NVIDIA; we will then discuss some techniques in preconditioning by approximate inverses, and we will conclude with an 
application to an image processing problem from the biomedical field.

We will discuss families of Pell's equation in polynomials 
with one complex parameter. In particular the relation between 
the generic equation and its specializations. Our emphasis will
be on families with a triple zero. Then additive extensions enter 
the picture. 

I will present a basic overview of finiteness conditions, group cohomology, and related quasi-isometry invariance results. In particular, I will show that if a group satisfies certain finiteness conditions, group cohomology with group ring coefficients encodes some structure of the `homology at infinity' of a group. This is seen for hyperbolic groups in the work of Bestvina-Mess, which relates the group cohomology to the Čech cohomology of the boundary.

Trusted Platform Modules (TPMs) are currently used in large numbers of computers. In this talk, I will discuss the cryptographic algorithms supported by the current version of the Trusted Platform Modules (Version 1.2) and also those due to be included in the new version  (Version 2.0).  After briefly introducing the history of TPMs, and the difference between these two generations TPMs, I will focus on the challenges faced in developing Direct Anonymous Attestation (DAA) an algorithmic scheme designed to preserve privacy and included in TPMs.

The topological Fukaya category is a combinatorial model of the Fukaya category of exact symplectic manifolds which was first proposed by Kontsevich. In this talk I will explain work in progress (joint with J. Pascaleff and S. Scherotzke) on gluing techniques for the topological Fukaya category that are closely related to Viterbo functoriality. I will emphasize applications to homological mirror symmetry for three-dimensional CY LG models, and to Bondal's and Fang-Liu-Treumann-Zaslow's coherent constructible correspondence for toric varieties.  

In this seminar, I aim to go through the "main prequel" of the talk I gave during the first Advanced Class of this term, and provide a "simple" answer to Abraham Robinson's original question that he posed in 1973 regarding the (un)decidability of finitely generated extensions of undecidable fields. I will provide a quick introduction to, and some classical results from, the mathematical discipline of Field Arithmetic, and using these results show that one can construct undecidable (large) fields that have finitely generated extensions which are decidable. Of course, as I had mentioned in the advanced class, a counterexample to the "simple" question that I have been working on unfortunately does not seem to lie within this class of large fields. If time permits, I will provide a sneak peek into the possible "sequel" by briefly talking about what the main issue of solving the "simple" problem is, and how a "hide-and-seek" method might come in handy in tackling that problem.

Kerdock matrices are an attractive choice as deterministic measurement matrices for compressive sensing. I'll explain how Kerdock matrices are constructed, and then show how they can be adapted to one particular  strategy for quantizing measurements, in which measurements exceeding the desired dynamic range are rejected.

Ramsey classes may be viewed as the top of the line of Ramsey properties. Classical and not so classical examples of Ramsey classes of finite structures were recently extended by many new examples which make the characterisation of Ramsey classes  realistic (and in many cases known). Particularly I will cover recent  joint work with J. Hubicka.
 

The prime spectrum of a quantum algebra has a finite stratification in terms
of a set of distinguished primes called H-primes, and we can study these
strata by passing to certain nice localizations of the algebra.  H-primes
are now starting to show up in some surprising new areas, including
combinatorics (totally nonnegative matrices) and physics, and we can borrow
techniques from these areas to answer questions about quantum algebras and
their localizations.    In particular, we can use Grassmann necklaces -- a
purely combinatorial construction -- to study the topological structure of
the prime spectrum of quantum matrices.

Using the "trichotomy principle" by Boris Zilber I will give model theoretic proofs of appropriate versions of Torelli theorem and Borel-Tits theorem. The first one has interesting applications to anabelian geometry, I won't assume any prior knowledge in model theory.

While it is believed that many particle systems have periodic ground states, there are few rigorous crystallization results in two and more dimensions. In this talk I will show how results by the Hungarian geometer László Fejes Tóth can be used to prove that an idealised block copolymer energy is minimised by the triangular lattice. I will also discuss a numerical method for a broader class of optimal location problems and some conjectures about minimisers in three dimensions. This is joint work with Mark Peletier, Steven Roper and Florian Theil. 

Bordered Floer homology is a variant of Heegaard Floer homology adapted to manifolds with boundary. I will describe a class of three-manifolds with torus boundary for which these invariants may be recast in terms of immersed curves in a punctured torus. This makes it possible to recast the paring theorem in bordered Floer homology in terms of intersection between curves leading, in turn, to some new observations about Heegaard Floer homology. This is joint work with Jonathan Hanselman and Jake Rasmussen.