Past

From cryptography to the proof of Fermat's Last Theorem, elliptic curves (those curves of the form y^2 = x^3 + ax+b) are ubiquitous in modern number theory.  In particular, much activity is focused on developing techniques to discover rational points on these curves. It turns out that finding a rational point on an elliptic curve is very much like finding the proverbial needle in the haystack -- in fact, there is currently no algorithm known to completely determine the group of rational points on an arbitrary elliptic curve.

 I'll introduce the ''real'' picture of elliptic curves and discuss why the ambient real points of these curves seem to tell us little about finding rational points. I'll summarize some of the story of elliptic curves over finite and p-adic fields and tell you about how I study integral points on (hyper)elliptic curves via p-adic integration, which relies on doing a bit of p-adic linear algebra.  Time permitting, I'll also give a short demo of some code we have to carry out these algorithms in the Sage Math Cloud.

In 2000 Eliashberg-Polterovich introduced the natural notion of orderability of contact manifolds; that is, the existence of a natural partial order on the group of contactomorphisms. I will explain how one can study orderability questions using the machinery of Rabinowitz Floer homology. We establish a link between orderable and hypertight contact manifolds, and show that the Weinstein Conjecture holds (i.e. there exists a closed Reeb orbit) whenever there exists a positive (not necessarily contractible) loop of contactomorphisms.

Joint work with Peter Albers and Urs Fuchs.

JH: Water filtration systems typically involve flow along a channel with permeable walls and suction applied across the wall. In this ``cross-flow'' arrangement, clean water leaves the channel while impurities remain within it. A limiting factor for the operation of cross-flow devices is the build-up of a high concentration of particles near the wall due to the induced flow. Termed concentration polarization (CP), this effect ultimately leads to the blocking of pores within the permeable wall and the deposition of a ``cake'' on the wall surface. Here we show that, through strategic choices in the spatial variations of the channel-wall permeability, we may reduce the effects of CP by allowing diffusion to smear out any build up of particles that may occur. We demonstrate that, for certain classes of variable permeability, there exist optimal choices that maximize the flux of clean water out of a device.

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IvG: TBC

We show that string theories admit chiral infinite tension analogues in which only the massless parts of the spectrum survive. Geometrically they describe holomorphic maps to spaces of complex null geodesics, known as ambitwistor spaces. They have the standard critical space–time dimensions of string theory (26 in the bosonic case and 10 for the superstring). Quantization leads to the formulae for tree– level scattering amplitudes of massless particles found recently by Cachazo, He and Yuan. These representations localize the vertex operators to solutions of the same equations found by Gross and Mende to govern the behaviour of strings in the limit of high energy, fixed angle scattering. Here, localization to the scattering equations emerges naturally as a consequence of working on ambitwistor space. The worldsheet theory suggests a way to extend these amplitudes to spinor fields and to loop level. We argue that this family of string theories is a natural extension of the existing twistor string theories. 

In the 1930's E. Artin conjectured that a form over a p-adic field of degree d has a non-trivial zero whenever n>d^2. In this talk we will discuss this relatively old conjecture, focusing on recent developments concerning quartic and quintic forms.

We show that the spatial norm in any critical homogeneous Besov

space in which local existence of strong solutions to the 3-d

Navier-Stokes equations is known must become unbounded near a singularity.

In particular, the regularity of these spaces can be arbitrarily close to

-1, which is the lowest regularity of any Navier-Stokes critical space.

This extends a well-known result of Escauriaza-Seregin-Sverak (2003)

concerning the Lebesgue space $L^3$, a critical space with regularity 0

which is continuously embedded into the spaces we consider. We follow the

``critical element'' reductio ad absurdum method of Kenig-Merle based on

profile decompositions, but due to the low regularity of the spaces

considered we rely on an iterative algorithm to improve low-regularity

bounds on solutions to bounds on a part of the solution in spaces with

positive regularity. This is joint work with I. Gallagher (Paris 7) and

F. Planchon (Nice).

Abstract : The Bayesian approach is a possible way to build estimators in statistical models. It consists in attributing a probability measure -the prior- to the unknown parameters of the model. The estimator is then the posterior distribution, which is a conditional distribution given the information contained in the data.

The Bernstein-von Mises theorem in parametric models states that under mild regularity conditions, the posterior distribution for the finite-dimensional model parameter is asymptotically Gaussian with `optimal' centering and variance.

In this talk I will discuss recent advances in the understanding of posterior distributions in nonparametric models, that is when the unknown parameter is infinite-dimensional, focusing on a concept of nonparametric Bernstein-von Mises theorem.

The HOMFLY polynomial is an invariant of knots in S^3 which can be

extended to an invariant of tangles in B^3. I'll give a geometrical

description of this invariant for rational tangles, and

explain how this description extends to a more general invariant

(the lambda^k colored HOMFLY polynomial of a rational tangle). I'll then

use this description to sketch a proof of a conjecture of Gukov and Stosic

about the colored HOMFLY homology of rational knots.

Parts of this are joint work with Paul Wedrich and Mihaljo Cevic.

This talk develops some aspects of stochastic calculus via regularization for processes with values in a general Banach space B.

A new concept of quadratic variation which depends on a particular subspace is introduced.

An Itô formula and stability results for processes admitting this kind of quadratic variation are presented.

Particular interest is devoted to the case when B is the space of real continuous functions defined on [-T,0], T>0 and the process is the window process X(•) associated with a continuous real process X which, at time t, it takes into account the past of the process.

If X is a finite quadratic variation process (for instance Dirichlet, weak Dirichlet), it is possible to represent a large class of path-dependent random variable h as a real number plus a real forward integral in a semiexplicite form.

This representation result of h makes use of a functional solving an infinite dimensional partial differential equation.

This decomposition generalizes, in some cases, the Clark-Ocone formula which is true when X is the standard Brownian motion W. Some stability results will be given explicitly.

This is a joint work with Francesco Russo (ENSTA ParisTech Paris)."

We will finish presenting Nyikos' counterexample to 
Bozyakova's conjecture: If e(Y) = L(Y) for every subspace Y of X, must X 
be hereditarily D?

The height of a rational number a/b (a,b integers which are coprime) is defined as max(|a|, |b|). A rational number with small (resp. big) height is a simple (resp. complicated) number. Though the notion height is so naive, height has played a fundamental role in number theory. There are important variants of this notion. In 1983, when Faltings proved the Mordell conjecture (a conjecture formulated in 1921), he first proved the Tate conjecture for abelian varieties (it was also a great conjecture) by defining heights of abelian varieties, and then deducing Mordell conjecture from this. The height of an abelian variety tells how complicated are the numbers we need to define the abelian variety. In this talk, after these initial explanations, I will explain that this height is generalized to heights of motives. (A motive is a kind of generalisation of abelian variety.) This generalisation of height is related to open problems in number theory. If we can prove finiteness of the number of motives of bounded height, we can prove important conjectures in number theory such as general Tate conjecture and Mordell-Weil type conjectures in many cases.

We study optimal portfolio strategies in a market

where the drift is driven by an unobserved Markov chain. Information on

the state of this chain is obtained from stock prices and from expert

opinions in the form of signals at random discrete time points. We use

stochastic filtering to transform the original problem into an

optimization problem under full information where the state variable is

the filter for the Markov chain. This problem is studied with dynamic

programming techniques and with regularization arguments. Finally we

discuss a number of numerical experiments

A projectile travelling supersonically in air creates a shock wave in the shape of a cone, with the projectile at the tip of the Mach cone. When the projectile travels over an array of microphones the shock wave is detected with different times of arrival at each microphone. Given measurements of the times of arrival, we are trying to calculate the azimuth direction of travel of the projectile. We have found a solution when the speed of the projectile is known. However the solution is ambiguous, and can take one of two possible values. Therefore we are seeking a new mathematical approach to resolve the ambiguity and thus find the azimuth direction of travel.

A major desideratum in transcendental number theory is a simple sufficient condition for a given real number to be irrational, or better yet transcendental. In this talk we consider various forms such a criterion might take, and prove the existence or non-existence of them in various settings.

By using the Weil-Gel'fand-Zak transform of Faddeev's quantum dilogarithm,

Andersen and Kasheav have proposed a new state-integral model for the

Andersen--Kashaev TQFT, where the circle valued state variables live on

the edges of oriented levelled shaped triangulations. I will look at a

couple of examples which give an idea of how the theories are coupled.

The problem of hydrodynamic turbulence is reformulated as a heat flow problem along a chain of mechanical systems which describe units of fluid of smaller and smaller spatial extent. These units are macroscopic but have few degrees of freedom, and can be studied by the methods of (microscopic) non-equilibrium statistical mechanics. The fluctuations predicted by statistical mechanics correspond to the intermittency observed in turbulent flows. Specically, we obtain the formula

$$ \zeta_p = \frac{p}{3} - \frac{1}{\ln \kappa} \ln \Gamma \left( \frac{p}{3} +1 \right) $$

for the exponents of the structure functions ($\left\langle \Delta_{r}v \rangle \sim r^{\zeta_p}$). The meaning of the adjustable parameter is that when an eddy of size $r$ has decayed to eddies of size $r/\kappa$ their energies have a thermal distribution. The above formula, with $(ln \kappa)^{-1} = .32 \pm .01$ is in good agreement with experimental data. This lends support to our physical picture of turbulence, a picture which can thus also be used in related problems.

In this talk I will introduce cluster categories and report on some new results on cluster categories of type E_6.

The context of data assimilation in oceanography will be described as well as the computational challenges associated with it. A class of numerical linear algebra methods is described whose purpose is to exploit the problem structure in order to reduce the computational burden and provide provable convergence results for what remains a (very large) nonlinear problem. This class belongs to the Krylov-space family of methods and the special structure used is the imbalance between the dimensions of the state space and the observation space. It is also shown how inexact matrix-vector products can be exploited. Finally, preconditioning issues and resulting adaptations of the trust-region methodology for nonlinear minimization will also be outlined.

By Serge Gratton, Selime Gurol, Philippe Toint, Jean Tshimanga and Anthony Weaver.

\textbf{Victor Fedyashov} \newline

\textbf{Title:} Ergodic BSDEs with jumps \newline

\textbf{Abstract:} We study ergodic backward stochastic differential equations (EBSDEs) with jumps, where the forward dynamics are given by a non-autonomous (time-periodic coefficients) Ornstein-Uhlenbeck process with Lévy noise on a separable Hilbert space. We use coupling arguments to establish existence of a solution. We also prove uniqueness of the Markovian solution under certain growth conditions using recurrence of the above mentioned forward SDE. We then give applications of this theory to problems of risk-averse ergodic optimal control.

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\textbf{Ruolong Chen} \newline

\textbf{Title:} tba \newline

\textbf{Abstract:}