Past

Oxford Nanopore Technologies aim to enable the analysis of any living thing, by any person, in any environment. The world's first and only nanopore DNA
sequencer, the MinION is a portable, real time, long-read, low cost device that has been designed to bring easy biological analyses to anyone, whether in
scientific research, education or a range of real world applications such as disease/pathogen surveillance, environmental monitoring, food chain
surveillance, self-quantification or even microgravity biology. Gordon will talk the about the technology, applications and future direction.
Stuart will talk about the nanopore signal, computational methods and informatics challenges associated with reading DNA directly.

Speaker: Yixuan Wang
Titile: Minimum resting time with market orders
Abstract:  Regulators have been discussing possible rules to control high frequency trading and decrease market speed, and minimum resting time is one of them. We develop a simple mathematical model, and derive an asymptotic expression of the expected PnL, which is also the performance criteria that a market maker would like to maximize by choosing the optimal depth at which she posts the limit order. We investigate the comparative statistics of the optimal depth with each parameters, an in particular the comparative statistics show that the minimum resting time will decrease the market liquidity, forcing the market makers to post limit orders of volume 1.


Speaker: Marco Pangallo
Title: Does learning converge in generic games?
Abstract: In game theory, learning has often been proposed as a convincing method to achieve coordination on an equilibrium. But does learning converge, and to what? We start investigating the drivers of instability in the simplest possible non-trivial setting, that is generic 2-person, 2-strategy normal form games. In payoff matrices with a unique mixed strategy equilibrium the players may follow the best-reply cycle and fail to converge to the Nash Equilibrium (NE): we rather observe limit cycles or low-dimensional chaos. We then characterize the cyclic structure of games with many moves as a combinatorial problem: we quantify exactly how many best-reply configurations give rise to cycles or to NE, and show that acyclic (e.g. coordination, potential, supermodular) games become more and more rare as the number of moves increases (a fortiori if the payoffs are negatively correlated and with more than two players).  In most games the learning dynamics ends up in limit cycles or high-dimensional chaotic attractors, preventing the players to coordinate. Strategic interactions would then be governed by learning in an ever-changing environment, rather than by rational and fully-informed equilibrium thinking.
Collaborators: J. D. Farmer, T. Galla, T. Heinrich, J. Sanders

to come

Using some new results in probabilistic recognition of black box groups
(Joint work with Sukru Yalcinkaya), I will discuss some emerging
structures that beg for a model-theoretic analysis.

In this talk on joint work with REBECCA BELLOVIN we discuss the “local ε-isomorphism” conjecture of Fukaya and Kato for (crystalline) families of G_{Q_p}-representations. This can be regarded as a local analogue of the global Iwasawa main conjecture for families, extending earlier work of Kato for rank one modules, of Benois and Berger for crystalline representations with respect to the cyclotomic extension as well as of Loeffler, Venjakob and Zerbes for crystalline representations with respect to abelian p-adic Lie extensions of Q_p. Nakamura has shown Kato’s - conjecture for (ϕ,\Gamma)-modules over the Robba ring, which means in particular only after inverting p, for rank one and trianguline families. The main ingredient of (the integrality part of) the proof consists of the construction of families of Wach modules generalizing work of Wach and Berger and following Kisin’s approach via a corresponding moduli space.
 

We consider impatient decision makers when their assets' prices are in undesirable low regions for a significant amount of time, and they are risk averse to negative price jumps. We wish to study the unusual reactions of investors under such adverse market conditions. In mathematical terms, we study the optimal exercising of an American call option in a random time-horizon under spectrally negative Lévy models. The random time-horizon is modeled by an alarm of the so-called Omega default clock in insurance, which goes off when the cumulative amount of time spent by the asset price in an undesirable low region exceeds an independent exponential random time. We show that the optimal exercise strategies vary both quantitatively and qualitatively with the levels of impatience and nervousness of the investors, and we give a complete characterization of all optimal exercising thresholds. 

In variational imaging and other inverse problem modeling, regularisation plays a major role. In recent years, high order regularizers such as the total generalised variation, the mean curvature and the Gaussian curvature are increasingly studied and applied, and many improved results over the widely-used total variation model are reported.
Here we first introduce the fractional order derivatives and the total fractional-order variation which provides an alternative  regularizer and is not yet formally analysed. We demonstrate that existence and uniqueness properties of the new model can be analysed in a fractional BV space, and, equally, the new model performs as well as the high order regularizers (which do not yet have much theory). 
In the usual framework, the algorithms of a fractional order model are not fast due to dense matrices involved. Moreover, written in a Bregman framework, the resulting Sylvester equation with Toeplitz coefficients can be solved efficiently by a preconditioned solver. Further ideas based on adaptive integration can also improve the computational efficiency in a dramatic way.
 Numerical experiments will be given to illustrate the advantages of the new regulariser for both restoration and registration problems.
 

Using results by Eisenbud, Schoutens and Zilber I will propose a model theoretic structure that aims to capture the algebra (or geometry) of a non reduced scheme over an algebraically closed field. 

In this seminar, we present a fast fully homomorphic encryption (FHE) based on GSW and its ring variants. The cryptosystem relies on the hardness of lattice problems in the unique domain (e.g. the LWE family). After a brief presentation of these lattice problems, with a few notes on their asymptotic and practical average case hardness, we will present our homomorphic cryptosystem TFHE, based on a ring variant of GSW. TFHE can operate in two modes: The first one is a leveled homomorphic mode, which has the ability to evaluate deterministic automata (or branching programs) at a rate of 1 transition every 50microseconds. For the second mode, we also show that this scheme can evaluate its own decryption in only 20milliseconds, improving on the the construction by Ducas-Micciancio, and of Brakerski-Perlman. This makes the scheme fully homomorphic by Gentry's bootstrapping principle, and for instance, suitable for representing fully dynamic encrypted databases in the cloud.

An expander is a family of finite graphs of uniformly bounded degree, increasing number of vertices and Cheeger constant bounded away from zero. They occur throughout mathematics and computer science; the most famous constructions of expanders rely on powerful results in geometric group theory and number theory, while expanders are used in everything from error-correcting codes, through disproving the strongest version of the Baum-Connes conjecture, to affine sieve theory and the twin prime, Mersenne prime and Hardy-Littlewood conjectures.

However, very little was known about how different the geometry of two expanders could be. This question was raised by Ostrovskii in 2013, and a year later Mendel and Naor gave the first example of two 'distinct' expanders.

In this talk I will construct a continuum of expanders which are, in a certain sense, geometrically incomparable. Once the existence of a single expander is accepted, the remainder of the proof is a heady mix of counting, addition, multiplication, and just for the experts, a little bit of division. Two very different - and very interesting - continuums of 'distinct' expanders have since been constructed by Khukhro-Valette and Das.

We consider two types of actions on moduli spaces of quiver representations over a field k and we decompose their fixed loci using group cohomology. First, for a perfect field k, we study the action of the absolute Galois group of k on the points of this quiver moduli space valued in an algebraic closure of k; the fixed locus is the set of k-rational points and we obtain a decomposition of this fixed locus indexed by the Brauer group of k. Second, we study algebraic actions of finite groups of quiver automorphisms on these moduli spaces; the fixed locus is decomposed using group cohomology and each component has a modular interpretation. If time permits, we will describe the symplectic and holomorphic geometry of these fixed loci in hyperkaehler quiver varieties. This is joint work with Florent Schaffhauser.

We prove tight upper and lower bounds on an observable of the antiferromagnetic Potts model. From this we deduce the case d=3 of a conjecture of Galvin and Tetali on maximising the number of proper colourings in d-regular graphs.

I will discuss recent results in finite simple groups. These include growth, generation (with a number theoretic flavour), and conjectures of Gowers and Viola on mixing and complexity whose proof requires representation theory as a main tool.
 

Joint work with Wolfgang Hackbusch and Daniel Kressner

I will talk about connections between the compressible and incompressible Navier-Stokes systems. In case of the compressible model, as the bulk (volume) viscosity is very high, the divergence of the velocity becomes small, in the limit it is zero and we arrive at the case of incompressible system. An important role here is played by the inhomogeneous version of the classical Navier-Stokes equations. I plan to discuss analytical obstacle appearing within the analysis. The considerations are done in the framework of regular solutions in Besov and Sobolev spaces. The results which will be discussed are joint with Raphael Danchin from Paris.

Motivated by a problem in quasiconformal mapping, we introduce a new type of problem in complex analysis, with its roots in the mathematical physics of the Bose-Einstein condensates in superconductivity.The problem will be referred to as \emph{geometric zero packing}, and is somewhat analogous to studying Fekete point configurations.The associated quantity is a density, denoted  $\rho_\C$ in the planar case, and $\rho_{\mathbb{H}}$ in the case of the hyperbolic plane.We refer to these densities as \emph{discrepancy densities for planar and hyperbolic zero packing}, respectively, as they measure the impossibility of atomizing the uniform planar and hyperbolic area measures.The universal asymptoticvariance $\Sigma^2$ associated with the boundary behavior of conformal mappings with quasiconformal extensions of small dilatation is related to one of these discrepancy densities: $\Sigma^2= 1-\rho_{\mathbb{H}}$.We obtain the estimates$2.3\times 10^{-8}<\rho_{\mathbb{H}}\le0.12087$, where the upper estimate is derived from the estimate from below on $\Sigma^2$ obtained by Astala, Ivrii, Per\"al\"a,  and Prause, and the estimate from below is much more delicate.In particular, it follows that $\Sigma^2<1$, which in combination with the work of Ivrii shows that the maximal fractal dimension of quasicircles conjectured by Astala cannot be reached.Moreover, along the way, since the universal quasiconformal integral means spectrum has the asymptotics$\mathrm{B}(k,t)\sim\frac14\Sigma^2 k^2|t|^2$ for small $t$ and $k$, the conjectured formula $\mathrm{B}(k,t)=\frac14k^2|t|^2$ is not true.As for the actual numerical values of the discrepancy density $\rho_\C$, we obtain the estimate from above $\rho_\C\le0.061203\ldots$ by using the equilateral triangular planar zero packing, where the assertion that equality should hold can be attributed to Abrikosov. The values of $\rho_{\mathbb{H}}$ is expected to be somewhat close to the value of $\rho_\C$.

Elliptic cohomology is a family of generalised cohomology theories
$Ell_E^*$ parametrised by an elliptic curve $E$ (over some ring $R$).
Just like many other cohomology theories, elliptic cohomology admits
equivariant versions. In this talk, I will recall an old conjectural
description of elliptic cohomology, due to G. Segal, S. Stolz and P.
Teichner. I will explain how that conjectural description led me to
suspect that there should exist a generalisation of equivariant
elliptic cohomology, where the group of equivariance gets replaced by
a fusion category. Finally, I will construct $C$-equivariant elliptic
cohomology when $C$ is a fusion category, and $R$ is a ring of
characteristc zero.

One of the challenges of 21st-century science is to model the evolution of complex systems.  One example of practical importance is urban structure, for which the dynamics may be described by a series of non-linear first-order ordinary differential equations.  Whilst this approach provides a reasonable model of urban retail structure, it is somewhat restrictive owing to uncertainties arising in the modelling process.

We address these shortcomings by developing a statistical model of urban retail structure, based on a system of stochastic differential equations.   Our model is ergodic and the invariant distribution encodes our prior knowledge of spatio-temporal interactions.  We proceed by performing inference and prediction in a Bayesian setting, and explore the resulting probability distributions with a position-specific metrolpolis-adjusted Langevin algorithm.