Past

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. 

Malliavin calculus is implemented in the context of [M. Hairer, A theory of regularity structures, Invent. Math. 2014]. This involves some constructions of independent interest, notably an extension of the structure which accommodates a robust and purely deterministic translation operator in L^2-directions between models. In the concrete context of the generalized parabolic Anderson model in 2D -one of the singular SPDEs discussed in the afore-mentioned article - we establish existence of a density at positive times.

Work of Schoen--Yau in the 70's/80's shows that area-minimizing (actually stable) two-sided surfaces in three-manifolds of non-negative scalar curvature are of a special topological type: a sphere, torus, plane or cylinder. The torus and cylinder cases are "borderline" for this estimate. It was shown by Cai--Galloway in the late 80's that the torus can only occur in a very special ambient three manifold. We complete the story by showing that a similar result holds for the cylinder. The talk should be accessible to those with a basic knowledge of curvature in Riemannian geometry.

In this talk, I will present recent results that give the necessary mathematical foundation for the study of rough path driven PDEs in the framework of weak solutions. The main tool is a new rough Gronwall Lemma argument whose application is rather wide: among others, it allows to derive the basic energy estimates leading to the proof of existence for e.g. parabolic RPDEs. The talk is based on a joint work with Aurelien Deya, Massimiliano Gubinelli and Samy Tindel.

It was discovered in the 1970s that black holes are thermodynamic objects carrying entropy, thus suggesting that they are really an ensemble of microscopic states. This idea has been realized in a remarkable manner in string theory, wherein one can describe these ensembles in a class of models. These ensembles are known, however, to contain configurations other than isolated black holes, and it remains an outstanding problem to precisely isolate a black hole in the microscopic ensemble. I will describe how this problem can be solved completely in N=4 string theory. The solution involves surprising relations to mock modular forms -- a class of functions first discovered by S. Ramanujan about 95 years ago. 

Who are you?  What motivates you?  What's important to you?  How do you react to challenges and adversities?  In this session we will explore the power of self-awareness (understanding our own characters, values and motivations) and introduce assertiveness skills in the context of building positive and productive relationships (with colleagues, collaborators, students and others).

Accurate methods for the first-order advection equation, used for example in tracking contaminants in fluids, usually exploit the theory of characteristics. Such methods are described and contrasted with methods that do not make use of characteristics.

Then the second-order wave equation, in the form of a first-order system, is considered. A review of the one-dimensional theory using solutions of various Riemann problems will be provided. In the special case that the medium has the ‘Goupillaud’ property, that waves take the same time to travel through each layer, one can derive exact solutions even when the medium is spatially heterogeneous. The extension of this method to two-dimensional problems will then be discussed. In two-dimensions it is not apparent that exact solutions can be found, however by exploiting a generalised Goupillaud property, it is possible to calculate approximate solutions of high accuracy, perhaps sufficient to be of benchmark quality. Some two-dimensional simulations, using exact one-dimensional solutions and operator splitting, will be described and a numerical evaluation of accuracy will be given.

The possibility of a landscape of metastable vacua raises the question of what fraction of vacua are truly long lived. Naively any would-be vacuum state has many nearby decay paths, and all possible decays must be suppressed. An interesting model of this phenomena consists of N scalars with a random potential of fourth order. We show that the scaling of the typical minimal bounce action with N is readily understood. We discuss the extension to more realistic landscape models as well as the effects of gravity. 

When estimated volatilities are not in perfect agreement with reality, delta hedged option portfolios will incur a non-zero profit-and-loss over time. There is, however, a surprisingly simple formula for the resulting hedge error, which has been known since the late 90s. We call this The Fundamental Theorem of Derivative Trading. This is a survey with twists of that result. We prove a more general version and discuss various extensions (including jumps) and applications (including deriving the Dupire-Gyo ̈ngy-Derman-Kani formula). We also consider its practical consequences both in simulation experiments and on empirical data thus demonstrating the benefits of hedging with implied volatility.