Many hard problems in Diophantine geometry have analogues over function fields which are less hard. I will give some examples.
*/ /*-->*/ Let G be a locally compact Hausdorff topological group. Examples are Lie groups, p-adic groups, adelic groups, and discrete groups. The BC (Baum-Connes) conjecture proposes an answer to the problem of calculating the K-theory of the convolution C* algebra of G. Validity of the conjecture has implications in several different areas of mathematics --- e.g. Novikov conjecture, Gromov-Lawson-Rosenberg conjecture, Dirac exhaustion of the discrete series, Kadison-Kaplansky conjecture. An expander is a sequence of finite graphs which is efficiently connected. Any discrete group which contains an expander as a sub-graph of its Cayley graph is a counter-example to the BC conjecture with coefficients. Such discrete groups have been constructed by Gromov-Arjantseva-Delzant and by Damian Osajda. This talk will indicate how to make a correction in BC with coefficients. There are no known counter-examples to the corrected conjecture, and all previously known confirming examples remain confirming examples.
We show how to get coarse bounds on the number of (non-simple) closed geodesics on a surface, given upper bounds on both length and self-intersection number. Recent work by Mirzakhani and by Rivin has produced asymptotics for the growth of the number of simple closed curves and curves with one self-intersection (respectively) with respect to length. However, no asymptotics, or even bounds, were previously known for other bounds on self-intersection number. Time permitting, we will discuss some applications of this result
It has been conjectured that the fundamental theory of strings and branes has an $E_{11}$ symmetry. I will explain how this conjecture leads to a generalised space-time, which is automatically equipped with its own geometry, as well as equations of motion for the fields that live on this generalised space-time.
Shear Thickening fluids such as cornstarch and water show remarkable response under impact, which allows, for example, a person to run on the surface of the suspension. We perform constant velocity impact experiments along with imaging and particle tracking in a shear thickening fluid at velocities lower than 500 mm/s and suspension heights of a few cm. In this regime, where inertial effects are insignificant, we find that a solid-like dynamically jammed region with a propagating front is generated under impact. The suspension is able to support large stresses like a solid only when the front reaches the opposite boundary. These impact-activated fronts are generated only above a critical velocity. We construct a model by taking into account that sufficiently large stresses are generated when this solid like region spans to the opposite boundary and the work necessary to deform this solid like material dissipates the kinetic energy of the impacting object. The model shows quantitative agreement of the measured penetration depth using high speed video of a person running on cornstarch and water suspensions.
Data in many areas of science and sociology is now routinely represented in the form of networks. A fundamental task often required is to compare two datasets (networks) to assess the level of similarity between them. In the context of biological sciences, networks often represent either direct or indirect molecular interactions and an active research area is to assess the level of conservation of interaction patterns across species.
Currently biological network comparison software largely relies on the concept of alignment where close matches between the nodes of two or more networks are sought. These node matches are based on sequence similarity and/or interaction patterns. However, because of the incomplete and error-prone datasets currently available, such methods have had limited success. Moreover, the results of network alignment are in general not amenable for distance-based evolutionary analysis of sets of networks. In this talk I will describe Netdis, a topology-based distance measure between networks, which offers the possibility of network phylogeny reconstruction.
In a series of recent papers David Masser and Umberto Zannier proved the relative Manin-Mumford conjecture for abelian surfaces, at least when everything is defined over the algebraic numbers. In a further paper with Daniel Bertrand and Anand Pillay they have explained what happens in the semiabelian situation, under the same restriction as above.
At present it is not clear that these results are effective. I'll discuss joint work with Philipp Habegger and Masser and with Harry Schimdt in which we show that certain very special cases can be made effective. For instance, we can effectively compute a bound on the order of a root of unity t such that the point with abscissa 2 is torsion on the Legendre curve with parameter t.
**Note change of room**
We present a dynamic theory for time-inconsistent stopping problems. The theory is developed under the paradigm of expected discounted
payoff, where the process to stop is continuous and Markovian. We introduce equilibrium stopping policies, which are imple-mentable
stopping rules that take into account the change of preferences over time. When the discount function induces decreasing impatience, we
establish a constructive method to find equilibrium policies. A new class of stopping problems, involving equilibrium policies, is
introduced, as opposed to classical optimal stopping. By studying the stopping of a one-dimensional Bessel process under hyperbolic discounting, we illustrate our theory in an explicit manner.
I describe joint work with Alastair Irving in which we improve a result of
D.H.J. Polymath on the length of intervals in $[N,2N]$ that can be shown to
contain $m$ primes. Here $m$ should be thought of as large but fixed, while $N$
tends to infinity.
The Harman sieve is the key to the improvement. The preprint will be
available on the Math ArXiv before the date of the talk.
Can we extend the FEM to general polytopic, i.e. polygonal and polyhedral, meshes while retaining
the ease of implementation and computational cost comparable to that of standard FEM? Within this talk, I present two approaches that achieve just that (and much more): the Virtual Element Method (VEM) and an hp-version discontinuous Galerkin (dG) method.
The Virtual Element spaces are like the usual (polynomial) finite element spaces with the addition of suitable non-polynomial functions. This is far from being a novel idea. The novelty of the VEM approach is that it avoids expensive evaluations of the non-polynomial "virtual" functions by basing all
computations solely on the method's carefully chosen degrees of freedom. This way we can easily deal
with complicated element geometries and/or higher continuity requirements (like C1, C2, etc.), while
maintaining the computational complexity comparable to that of standard finite element computations.
As you might expect, the choice and number of the degrees of freedom depends on such continuity
requirements. If mesh flexibility is the goal, while one is ready to give up on global regularity, other approaches can be considered. For instance, dG approaches are naturally suited to deal with polytopic meshes. Here I present an extension of the classical Interior Penalty dG method which achieves optimal rates of convergence on polytopic meshes even under elemental edge/face degeneration.
The next step is to exploit mesh flexibility in the efficient resolution of problems characterised by
complicated geometries and solution features, for instance within the framework of automatic FEM
adaptivity. I shall finally introduce ongoing work in this direction.
I will present some recent results concerning the higher gradient integrability of
σ-harmonic functions u with discontinuous coefficients σ, i.e. weak solutions of
div(σ∇u) = 0. When σ is assumed to be symmetric, then the optimal integrability
exponent of the gradient field is known thanks to the work of Astala and Leonetti
& Nesi. I will discuss the case when only the ellipticity is fixed and σ is otherwise
unconstrained and show that the optimal exponent is attained on the class of
two-phase conductivities σ: Ω⊂R27→ {σ1,σ2} ⊂M2×2. The optimal exponent
is established, in the strongest possible way of the existence of so-called
exact solutions, via the exhibition of optimal microgeometries.
(Joint work with V. Nesi and M. Ponsiglione.)
I will report on some joint work with Jacob Tsimerman
concerning multiplicative relations among singular moduli.
Our results rely on the ``Ax-Schanuel'' theorem for the j-function
recently proved by us, which I will describe.
I will discuss various types of filling functions on topological spaces, stating some results in the area. I will then go onto prove that a finitely presented subgroup of a hyperbolic group of cohomological dimension 2 is hyperbolic. On the way I will prove a stronger result about filling functions of subgroups of hyperbolic groups of cohomological dimension $n$.
We review a class of systems of non-linear PDEs, derived from the Cahn--Hilliard and Ohta--Kawasaki functionals, that describe the energy evolution of diblock copolymers. These are long chain molecules that can self assemble into repeating patterns as they cool. We are particularly interested in finite element numerical methods that approximate these PDEs in the two-phase (in which we model the polymer only) and three-phase (in which we imagine the polymer surrounded by, and interacting with, a void) cases.
We present a brief derivation of the underlying models, review a class of numerical methods to approximate them, and showcase some early results from our codes.
The talk will present ongoing work on medical image reconstruction from x-ray scanners. A suitable method for reconstruction of these undersampled systems is compressed sensing. The presentation will show respective reconstruction methods and their analysis. Furthermore, work in progress about extensions of the standard approach will be shown.
[based on joint work with Li Guo and Bin Zhang]
We apply to the study of exponential sums on lattice points in
convex rational polyhedral cones, the generalised algebraic approach of
Connes and Kreimer to perturbative quantum field theory. For this purpose
we equip the space of cones with a connected coalgebra structure.
The algebraic Birkhoff factorisation of Connes and Kreimer adapted and
generalised to this context then gives rise to a convolution factorisation
of exponential sums on lattice points in cones. We show that this
factorisation coincides with the classical Euler-Maclaurin formula
generalised to convex rational polyhedral cones by Berline and Vergne by
means of an interpolating holomorphic function.
We define renormalised conical zeta values at non-positive integers as the
Taylor coefficients at zero of the interpolating holomorphic function. When
restricted to Chen cones, this yields yet another way to renormalise
multiple zeta values at non-positive integers.
We consider the system of nonlinear differential equations
\begin{equation}
(1) \qquad
\begin{cases}
\dot u_n(t) + \lambda^{2n} u_n(t)
- \lambda^{\beta n} u_{n-1}(t)^2 + \lambda^{\beta(n+1)} u_n(t) u_{n+1}(t) = 0,\\
u_n(0) = a_n, n \in \mathbb{N}, \quad \lambda > 1, \beta > 0.
\end{cases}
\end{equation}
In this talk we explain why this system is a model for the Navier-Stokes equations of hydrodynamics. The natural question is to find a such functional space, where one could prove the existence and the uniqueness of solution. In 2008, A. Cheskidov proved that the system (1) has a unique "strong" solution if $\beta \le 2$, whereas the "strong" solution does not exist if $\beta > 3$. (Note, that the 3D-Navier-Stokes equations correspond to the value $\beta = 5/2$.) We show that for sufficiently "good" initial data the system (1) has a unique Leray-Hopf solution for all $\beta > 0$.
We will give a sketch overview of Scholze's theory of perfectoid spaces and the tilting equivalence, starting from Huber's geometric approach to valuation theory. Applications to weight-monodromy and p-adic Hodge theory we will only hint at, preferring instead to focus on examples which illustrate the philosophy of tilting equivalence.
Representations of free loop groups possess an operation, akin to
tensor product, under which they form a braided tensor category. I
will discuss a similar operation, which is present on the category of
representations of the based loop groups, and which equips it with the
structure of a monoidal cateogory. Finally, I will present a recent
result, according to which the Drinfel'd centre of the category of
representations of a based loop group is equivalent to the category of
representations of the corresponding free loop group.
: Estimating volatility from recent high frequency data, we revisit the question of the smoothness of the volatility process. Our main result is that log-volatility behaves essentially as a fractional Brownian motion with Hurst exponent H of order 0.1, at any reasonable time scale.
This leads us to adopt the fractional stochastic volatility (FSV) model of Comte and Renault.
We call our model Rough FSV (RFSV) to underline that, in contrast to FSV, H<1/2.
We demonstrate that our RFSV model is remarkably consistent with financial time series data; one application is that it enables us to obtain improved forecasts of realized volatility.
Furthermore, we find that although volatility is not long memory in the RFSV model, classical statistical procedures aiming at detecting volatility persistence tend to conclude the presence of long memory in data generated from it.
This sheds light on why long memory of volatility has been widely accepted as a stylized fact.
Finally, we provide a quantitative market microstructure-based foundation for our findings, relating the roughness of volatility to high frequency trading and order splitting.
This is joint work with Jim Gatheral and Thibault Jaisson.