A polymer, or microscopic elastic filament, is often modelled as a linear chain of rigid bodies interacting both with themselves and a heat bath. Then the classic notions of persistence length are related to how certain correlations decay with separation along the chain. I will introduce these standard notions in mathematical terms suitable for non specialists, and describe the standard results that apply in the simplest cases of wormlike chain models that have a straight, minimum energy (or ground or intrinsic) shape. Then I will introduce an appropriate splitting of a matrix recursion in the group SE(3) which deconvolves the distinct effects of stiffness and intrinsic shape in the more complicated behaviours of correlations that arise when the polymer is not intrinsically straight. The new theory will be illustrated by fully implementing it within a simple sequence-dependent rigid base pair model of DNA. In that particular context, the persistence matrix factorisation generalises and justifies the prior scalar notions of static and dynamic persistence lengths.

# Past Forthcoming Seminars

In micro-fluidics, at small scales where inertial effects become negligible, surface to volume ratios are large and the interfacial processes are extremely important for the overall dynamics. Integral

equation based methods are attractive for the simulations of e.g. droplet-based microfluidics, with tiny water drops dispersed in oil, stabilized by surfactants. In boundary integral formulations for

Stokes flow, jumps in pressure and velocity gradients are naturally taken care of, viscosity ratios enter only in coefficients of the equations, and only the drop surfaces must be discretized and not the volume inside nor in between.

We present numerical methods for drops with insoluble surfactants, both in two and three dimensions. We discretize the integral equations using Nyström methods, and special care is taken in the evaluation of singular and also nearly singular integrals that is needed in the case of close drop interactions. A spectral method is used to solve the advection-diffusion equation on each drop surface that describes the evolution of surfactant concentration. The drop velocity and surfactant concentration couple together through an equation of state for the surface tension coefficient. An adaptive time-stepping strategy is developed for the coupled problem, with the constraint to minimize the number of Stokes solves, since this is the computationally most expensive part.

For high quality discretization of the drops throughout the simulations, a hybrid method is used in two dimensions, offering an arc-length parameterization of the interface. In three dimensions, a

reparameterization procedure is developed to optimize the spherical harmonics representation of the drop, while conserving the drop volume and amount of surfactant.

We present results from some validation tests and illustrate the ability of the numerical methods in different challenging problems.

The basic mathematical models that describe the behavior of fluid flows date back to the eighteenth century, and yet many phenomena observed in experiments are far from being well understood from a theoretical viewpoint. For instance, especially challenging is the study of fundamental stability mechanisms when weak dissipative forces (generated, for example, by molecular friction) interact with advection processes, such as mixing and stirring. The goal of this talk is to have an overview on recent results on a variety of aspects related to hydrodynamic stability, such as the stability of vortices and laminar flows, the enhancement of dissipative force via mixing, and the statistical description of turbulent flows.

We will discuss topological and algebraic aspects of splittings of free groups. In particular we will look at the core of two splittings in terms of CAT(0) cube complexes and systems of surfaces in a doubled handlebody.

Multivariate cryptography is one of a handful of proposals for post-quantum cryptographic schemes, i.e. cryptographic schemes that are secure also against attacks carried on with a quantum computer. Their security relies on the assumption that solving a system of multivariate (quadratic) equations over a finite field is computationally hard.

Groebner bases allow us to solve systems of polynomial equations. Therefore, one of the key questions in assessing the robustness of multivariate cryptosystems is estimating how long it takes to compute the Groebner basis of a given system of polynomial equations.

After introducing multivariate cryptography and Groebner bases, I will present a rigorous method to estimate the complexity of computing a Groebner basis. This approach is based on techniques from commutative algebra and is joint work with Alessio Caminata (University of Barcelona).

About fifteen years ago, Thomas Scanlon and I gave a description of sets that arise as the intersection of a subvariety with a finitely generated subgroup inside a semiabelian variety over a finite field. Inspired by later work of Derksen on the positive characteristic Skolem-Mahler-Lech theorem, which turns out to be a special case, Jason Bell and I have recently recast those results in terms of finite automata. I will report on this work, as well as on the work-in-progress it has engendered, also with Bell, on an effective version of the isotrivial Mordell-Lang theorem.

Starting from the seminal paper of Caporaso-Harris-Mazur, it has been proved that if Lang's Conjecture holds in arbitrary dimension, then it implies a uniform bound for the number of rational points in a curve of general type and analogue results in higher dimensions. In joint work with Kenny Ascher we prove analogue statements for integral points (or more specifically stably-integral points) on curves of log general type and we extend these to higher dimensions. The techniques rely on very recent developments in the theory of moduli spaces for stable pairs, a higher dimensional analogue of pointed stable curves.

If time permits we will discuss how very interesting problems arise in dimension 2 that are related to the geometry of the log-cotangent bundle.

Please note that this seminar will take place at the Physical and Theoretical Chemistry Laboratory within the

Department of Chemistry, room, PTCL lecture theatre.

We give an alternative, statistical physics based proof of the Ω(d2^{-d}) lower bound for the maximum sphere packing density in dimension d by showing that a random configuration from the hard sphere model has this density in expectation. While the leading constant we achieve is not the best known, we do obtain additional geometric information: we prove a lower bound on the entropy density of sphere packings at this density, a measure of how plentiful such packings are. This is joint work with Felix Joos and Will Perkins.

Within the wide field of sparse approximation, convolutional sparse coding (CSC) has gained considerable attention in the computer vision and machine learning communities. While several works have been devoted to the practical aspects of this model, a systematic theoretical understanding of CSC seems to have been left aside. In this talk, I will present a novel analysis of the CSC problem based on the observation that, while being global, this model can be characterized and analyzed locally. By imposing only local sparsity conditions, we show that uniqueness of solutions, stability to noise contamination and success of pursuit algorithms are globally guaranteed. I will then present a Multi-Layer extension of this model and show its close relation to Convolutional Neural Networks (CNNs). This connection brings a fresh view to CNNs, as one can attribute to this architecture theoretical claims under local sparse assumptions, which shed light on ways of improving the design and implementation of these networks. Last, but not least, we will derive a learning algorithm for this model and demonstrate its applicability in unsupervised settings.