We will discuss some geometric methods to study Diophantine equations. We focus on the case of elliptic curves and their natural generalisations: Abelian varieties, Calabi-Yau manifolds and hyperelliptic curves.
(joint work with Aurélien Alfonsi and Arturo Kohatsu-Higa)
We are interested in the Wasserstein distance on the space of continuous sample-paths equipped with the supremum norm between the laws of a uniformly elliptic one-dimensional diffusion process and its continuous-time Euler scheme with N steps. This distance controls the discretization biais for a large class of path-dependent payoffs.
Its convergence rate to 0 is clearly intermediate between -the rate -1/2 of the strong error estimation obtained when coupling the stochastic differential equation and its Euler scheme with the same Brownian motion -and the rate -1 of the weak error estimation obtained when comparing the expectations of the same function of the diffusion and its Euler scheme at the terminal time.
For uniformly elliptic one-dimensional stochastic differential equations, we prove that this rate is not worse than -2/3.
Abstract: In 2010, Erschler, Tóth and Werner introduced the so-called Stuck Walks, which are a class of self-interacting random walks on Z for which there is competition between repulsion at small scale and attraction at large scale. They proved that, for any positive integer L, if the relevant parameter belongs to a certain interval, then such random walks localize on L + 2 sites with positive probability. They also conjectured that it is the almost sure behaviour. We settle this conjecture partially, proving that the walk localizes on L + 2 or L + 3 sites almost surely, under the same assumptions.
According to Witten a gauge theory with a mass gap contains a possibly trivial Topological Field Theory (TFT) in the infrared. We show that in SU(N) YM it there exists a trivial TFT defined by twistor Wilson loops whose v.e.v. is 1 in the large-N limit for any shape of the loops supported on certain Lagrangian submanifolds of space-time that lift to Lagrangian submanifolds of twistor space.
We derive a new version of the Makeenko-Migdal loop equation for the topological twistor Wilson loops, the holomorphic loop equation, that involves the change of variables in the YM functional integral from the connection to the anti-selfdual part of the curvature and the choice of a holomorphic gauge.
Employing the holomorphic loop equation and viewing Floer homology the other way around,
we associate to arcs asymptotic in both directions to the cusps of the Lagrangian submanifolds the critical points of an effective action implied by the holomorphic loop equation. The critical points of the effective action, being associated to the homology of the punctured Lagrangian submanifolds, consist of surface operators of the YM theory, supported on the punctures. The correlators of surface operators in the TFT satisfy for large momentum the constraint that follows by the renormalization group and by the asymptotic freedom and they are saturated by an infinite sum of pure poles of scalar and pseudoscalar glueballs, whose joint spectrum is exactly linear in the mass squared.
For several physical purposes we outline a related construction of a twistorial Topological String Theory dual to the TFT, that involves the Chern-Simons action on Lagrangian submanifolds of
twistor space.
Relative entropy weighted optimization is convex optimization problem over the space of probability measures. Many convex optimization problems can be rephrased as such a problem. This is particularly useful since this problem type admits a quasi-explicit solution (i.e. as the expectation over a random variable), which immediately provides a Monte-Carlo method for numerically computing the solution of the optimization problem.
In this talk we discuss the background and application of this approach to stochastic optimal control problems, which may be considered as relative entropy weighted problems with Wiener space as probability space, and its connection with the theory of large deviations for Brownian functionals. As a particular application we discuss the minimization of the local time in a given point of Brownian motion with drift.
Recent years have seen rapid expansion of the capabilities
to recreate in vivo conditions using in vitro microfluidic assays.
A wide range of single cell and cell population behaviors can now
be replicated, controlled and imaged for detailed studies to gain
new insights. Such experiments also provide useful fodder for
computational models, both in terms of estimating model parameters
and for testing model-generated hypotheses. Our experiments have
focused in several different areas.
1) Single cell migration experiments in 3D collagen gels have
revealed that interstitial flow can lead to biased cell migration
in the upstream direction, with important implications to cancer
invasion. We show this phenomenon to be a consequence of
integrin-mediated mechanotransduction.
2) Endothelial cells seeded in fibrin gels form perfusable
vascular networks within 2-3 days through a process termed
“vasculogenesis”. The process by which cells sense their
neighbours, extend projections and form anastomoses, and
generate interconnected lumens can be observed through time-lapse
microscopy.
3) These vascular networks, once formed, can be perfused with
medium containing cancer cells that become lodged in the
smaller vessels and proceed to transmigrate across the endothelial
barrier and invade into the surrounding matrix. High resolution
imaging of this process reveals a fascinating sequence of events
involving interactions between a tumour cell, endothelial cells,
and underlying matrix. These three examples will be presented
with a view toward gaining new insights through computational
modelling of the associated phenomena.
We will present in this lecture the global existence of weak solutions and the local existence and uniqueness of strong-in-time solutions for the fully-coupled Navier-Stokes/Q-tensor system on a bounded domain $\O\subset\mathbb{R}^d$ ($d=2,3$) with inhomogenerous Dirichlet and Neumann or mixed boundary conditions. Our result is valid for any physical parameter $\xi$ and we consider the Navier-Stokes equations with a general (but smooth) viscosity coefficient.
The good use of condiments is one of the secrets of a tasty quiche. If you want to delight your guests, add a pinch of ground pepper or cinnamon to the yellow liquid formed by the mix of the eggs and the crème fraiche. Here, is a surprise : even if the liquid is at rest, the pinch of milled pepper spreads by itself at the surface of the mixture. It expands in a circular way, and within a few seconds, it covers an area equal to several times its initial one. Why does it spread like that ? What factors influence this dispersion ? I will present some experiments and mathematical models of this process.
***** PLEASE NOTE THIS SEMINAR TAKES PLACE ON TUESDAY *****
Reduced ODE models describing coarsening dynamics of droplets in nanometric polymer film interacting on solid substrate in the presence of large slippage at the liquid/solid interface are derived from one-dimensional lubrication equations. In the limiting case of the infinite slip length corresponding to the free suspended films a collision/absorption model then arises and is solved explicitly. The exact collision law is derived. Existence of a threshold at which the collision rates switch from algebraic to exponential ones is shown.
***** PLEASE NOTE THIS SEMINAR TAKES PLACE ON TUESDAY *****