Past

 "We give a diophantine criterion for a polynomial with rational coefficients not to have any
rational zero, i.e. an existential formula in terms of the coefficients expressing this property. This can be seen as a kind of restricted
model-completeness for Q and answers a question of Koenigsmann."

One can ask whether the fundamental groups of 3-manifolds are distinguished by their sets of finite quotients. I will discuss the recent solution of this question for Seifert fibre spaces.

STAR-Vote is voting system that results from a collaboration between a number of
academics and the Travis County, Texas elections office, which currently uses a
DRE voting system and previously used an optical scan voting system. STAR-Vote
represents a rare opportunity for a variety of sophisticated technologies, such
as end-to-end cryptography and risk limiting audits, to be designed into a new
voting system, from scratch, with a variety of real world constraints, such as
election-day vote centers that must support thousands of ballot styles and run
all day in the event of a power failure.
We present and motivate the design of the STAR-Vote system, the benefits that we
expect from it, and its current status.

This is based on joint work with Josh Benaloh, Mike Byrne, Philip Kortum,
Neal McBurnett, Ron Rivest, Philip Stark, Dan Wallach
and the Office of the Travis County Clerk

A subshift is characterized by a set of allowable words on $d$ symbols. In a sense it encodes the allowable operations an automaton performs. In the late 1990's Matsumoto constructed a C*-algebra associated to a subshift, deriving initially his motivation from the work of Cuntz-Krieger. These C*-algebras were then studied in depth in a series of papers. In 2009 Shalit-Solel discovered a relation of the subshift algebras with their variants of operator algebras related to homogeneous ideals. In particular a subshift corresponds to a monomial ideal under this prism.

In a recent work with Shalit we take a closer look at these cases and study them in terms of classification programmes on nonselfadjoint operator algebras and Arveson's Programme on the C*-envelope. We investigate two nonselfadjoint operator algebras from one SFT and show that they completely classify the SFT: (a) up to the same allowable words, and (b) up to local conjugacy of the quantized dynamics. In addition we discover that the C*-algebra fitting Arveson's Programme is the quotient by the generalized compacts, rather than taking unconditionally all compacts as Matsumoto does. Actually there is a nice dichotomy that depends on the structure of the monomial ideal.

Nevertheless in the process we accomplish more in different directions. This happens as our case study is carried in the intersection of C*-correspondences, subproduct systems, dynamical systems and subshifts. In this talk we will give the basic steps of our results with some comments on their proofs.

Graded affine Hecke algebras were introduced by Lusztig for studying the representation theory of p-adic groups. In particular, some problems about extensions of representations of p-adic groups can be transferred to problems in the graded Hecke algebra setting. The study of extensions gives insight to the structure of various reducible modules. In this talk, I shall discuss some methods of computing Ext-groups for graded Hecke algebras.
The talk is based on arXiv:1410.1495, arXiv:1510.05410 and forthcoming work.

TBA

Recent results showed that the low energy expansion of closed superstring amplitudes can be expressed in terms of

single-valued multiple elliptic polylogarithms. I will explain how these functions may be defined as iterated integrals on the torus and

sketch how they arise from Feynman integrals.

A question by Poonen asks whether iterating the étale-Brauer set can give a finer obstruction set. We tackle the algebraic version of Poonen's question and give, in many cases, a negative answer.

I will explain some recent work using minimal surfaces to address problems in 3-manifold topology.  Given a Heegaard splitting, one can sweep out a three-manifold by surfaces isotopic to the splitting, and run the min-max procedure of Almgren-Pitts and Simon-Smith to construct a smooth embedded minimal surface.   If the original splitting were strongly irreducible (as introduced by Casson-Gordon), H. Rubinstein sketched an argument in the 80s showing that the limiting minimal surface should be isotopic to the original splitting.  I will explain some results in this direction and how jointly with T. Colding and D. Gabai we can use such min-max minimal surfaces to complete the classification problem for Heegaard splittings of non-Haken hyperbolic 3-manifolds.

First order asymptotic of scalar renewal sequences with infinite mean characterized by regular variation has been classified in the 60's (Garsia and Lamperti). In the recent years, the question of higher order asymptotic for renewal sequences with infinite mean was motivated by obtaining 'mixing rates' for dynamical systems with infinite measure. In this talk I will present the recent results we have obtained on higher order expansion for renewal sequences with infinite mean (not necessarily generated by independent processes) in the regime of slow regular variation (with small exponents).  I will also discuss some consequences of these results for error rates in certain limit theorems (such as arcsine law for null recurrent Markov processes).

 

The first reaction-diffusion equation developed and studied is the Fisher-KPP equation.  Introduced in 1937, it accounts for the spatial spreading and growth of a species.  Understanding this population-dynamics model is equivalent to understanding the distribution of the maximum particle in a branching Brownian motion.  Various generalizations of this model have been studied in the eighty years since its introduction, including a model with non-local reaction for the cane toads of Australia introduced by Benichou et. al.  I will begin the talk by giving an extended introduction on the Fisher-KPP equation and the typical behavior of its solutions.  Afterwards, I will describe the model for the cane toads equations and give new results regarding this model.  In particular, I will show how the model may be viewed as a perturbation of a local equation using a new Harnack-type inequality and I will discuss the super-linear in time propagation of the toads.  The talk is based on a joint work with Bouin and Ryzhik.

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Writing is a part of any career in science or mathematics. I will make some remarks about the role writing has played in my life and the role it might play in yours.

The InFoMM CDT Group Meetings will follow the format of the OCIAM group meetings. We hope they will facilitate good communication between the Academic and Student community so that the research activities remain closely connected, opportunities for additional interaction are easily identified, and cross-fertilisation of ideas can be catalysed. 

Tmoslav Plesa: Chemical Reaction Systems with a Homoclinic Bifurcation: An Inverse Problem, 25+5 min;

John Ockendon: Wave Homogenisation, 10 min + questions; 

Hilary Ockendon: Sloshing, 10 min + questions
 

Given a family $F$ of elliptic curves defined over $Q$, we are interested in the set $H(Y)$ of curves $E$ in $F$, of positive rank, and for which the minimum of the canonical heights of non-torsion rational points on $E$ is bounded by some parameter $Y$. When one can show that this set is finite, it is natural to investigate statistical properties of arithmetic objects attached to elliptic curves in the set $H(Y)$. We will describe some problems related to this, and will state some results in the case of families of quadratic twists of a fixed elliptic curve.

Motivated by the European sovereign debt crisis, we propose a hybrid sovereign default model which combines an accessible part which takes into account the movement of the sovereign solvency and the impact of critical political events, and a totally inaccessible part for the idiosyncratic credit risk. We obtain closed-form formulas for the probability that the default occurs at political critical dates in a Markovian CEV process setting. Moreover, we introduce a generalized density framework for the hybrid default times and deduce the compensator process of default. Finally we apply the hybrid model and the generalized density to the valuation of sovereign bond and explain the significant jumps in the long-term government bond yield during the sovereign crisis.

When assigned with the task of extracting information from given image data the first challenge one faces is the derivation of a truthful model for both the information and the data. Such a model can be determined by the a-priori knowledge about the image (information), the data and their relation to each other. The source of this knowledge is either our understanding of the type of images we want to reconstruct and of the physics behind the acquisition of the data or we can thrive to learn parametric models from the data itself. The common question arises: how can we customise our model choice to a particular application? Or better how can we make our model adaptive to the given data?

Starting from the first modelling strategy this talk will lead us from nonlinear diffusion equations and subdifferential inclusions of total variation type functionals as the most successful image modeltoday to non-smooth second- and third-order variational models, with data models for Gaussian and Poisson distributed data as well as impulse noise. These models exhibit solution-dependent adaptivities in form of nonlinearities or non-smooth terms in the PDE or the variational problem, respectively. Applications for image denoising, inpainting and surface reconstruction are given. After a critical discussion of these different image and data models we will turn towards the second modelling strategy and propose to combine it with the first one using a PDE constrained optimisation method that customises a parametrised form of the model by learning from examples. In particular, we will consider optimal parameter derivation for total variation denoising with multiple noise distributions and optimising total generalised variation regularisation for its application in photography.