We study the small Deborah number limit of the Doi-Onsager equation for the dynamics of nematic liquid crystals. This is a Smoluchowski-type equation that characterizes the evolution of a number density function, depending upon both particle position and its orientation vector, which lies on the unit sphere. We prove that, in the low temperature regime, when the Deborah number tends to zero, the family of solutions with rough initial data near local equilibria will converge to a local equilibrium distribution prescribed by a weak solution of the harmonic map heat flow into the sphere. This flow is a special case of the gradient flow to the Oseen-Frank energy functional for nematic liquid crystals and the existence of its global weak solution was first obtained by Y.M Chen, using Ginzburg-Landau approximation.  The key ingredient of our result is to show the strong compactness of the family of number density functions and the proof relies on the strong compactness of the corresponding second moment (or the Q-tensor), a spectral decomposition of the linearized operator near the limiting local equilibrium distribution, as well as the energy dissipation estimates.  This is a joint work with Wei Wang in Zhejiang university.
 

Simple mathematical models have had remarkable successes in biology, framing how we understand a host of mechanisms and processes. However, with the advent of a host of new experimental technologies, the last ten years has seen an explosion in the amount and types of data now being generated. Increasingly larger and more complicated processes are now being explored, including large signalling or gene regulatory networks, and the development, dynamics and disease of entire cells and tissues. As such, the mechanistic, mathematical models developed to interrogate these processes are also necessarily growing in size and complexity. These detailed models have the potential to provide vital insights where data alone cannot, but to achieve this goal requires meeting significant mathematical challenges. In this talk, I will outline some of these challenges, and recent steps we have taken in addressing them.

It is well known that a rough path is uniquely determined by its signature (the collection of global iterated path integrals) up to tree-like pieces. However, the proof the uniqueness theorem is non-constructive and does not give us information about how quantitative properties of the path can be explicitly recovered from its signature. In this talk, we examine the quantitative relationship between the local p-variation of a rough path and the tail asymptotics of its signature for the simplest type of rough paths ("line segments"). What lies at the core of the work a novel technique based on the representation theory of complex semisimple Lie algebras. 

This talk is based on joint work with Horatio Boedihardjo and Nikolaos Souris

We discuss some results on integration of ``rough differential forms'', which are generalizations of classical (smooth) differential forms to similar objects involving Hölder continuous functions that may be nowhere differentiable. Motivations arise mainly from geometric problems related to irregular surfaces, and the techniques are naturally related to those of Rough Paths theory. We show in particular that such a geometric integration can be constructed substituting appropriately differentials with more general asymptotic expansions (of Stratonovich or Ito type) and by summing over a refining sequence of partitions, leading to a two-dimensional extension of the classical Young integral, that coincides with the integral introduced recently by R. Züst. We further show that Stratonovich sums gives an advantage allowing to weaken the requirements on Hölder exponents, and discuss some work in progress in the stochastic case. Based on joint works with E. Stepanov, G. Alberti and I. Ballieul.

Gibbs measures of nonlinear Schrödinger equations are a fundamental object used to study low-regularity solutions with random initial data. In the dispersive PDE community, this point of view was pioneered by Bourgain in the 1990s. We study the problem of the derivation of Gibbs measures as high-temperature limits of thermal states in many-body quantum mechanics.

In our work, we apply a perturbative expansion in the interaction. This expansion is then analysed by means of Borel resummation techniques. In two and three dimensions, we need to apply a Wick-ordering renormalisation procedure. Moreover, in one dimension, our methods allow us to obtain a microscopic derivation of the time-dependent correlation functions for the cubic nonlinear Schrödinger equation. This is based partly on joint work with Jürg Fröhlich, Antti Knowles, and Benjamin Schlein.

Using ideas from paracontrolled calculus, we prove local well-posedness of a renormalized version of the three-dimensional stochastic nonlinear wave equation with quadratic nonlinearity forced by an additive space-time white noise on a periodic domain. There are two new ingredients as compared to the parabolic setting. (i) In constructing stochastic objects, we have to carefully exploit dispersion at a multilinear level. (ii) We introduce novel random operators and leverage their regularity to overcome the lack of smoothing of usual paradifferential commutators

In this talk I will discuss some recent results that allow to approximate a real singularity given by polynomial equations of degree d (e.g. the zero set of a polynomial, or the number of its critical points of a given Morse index) with a singularity which is diffeomorphic to the original one, but it is given by polynomials of degree O(d^(1/2)log d).
The approximation procedure is constructive (in the sense that one can read the approximating polynomial from a linear projection of the given one) and quantitative (in the sense that the approximating procedure will hold for a subset of the space of polynomials with measure increasing very quickly to full measure as the degree goes toinfinity).

The talk is based on joint works with P. Breiding, D. N. Diatta and H. Keneshlou