# Mathematics Colloquium 2010-11 Abstracts

## SUMMER TERM

Abstract:

Networks are common within biological systems and have been characterised in a range of different contexts that include metabolism, protein-protein interaction, neuronal circuits and ecological food webs. However, one area that has received relatively little attention is analysis of organisms whose entire growth form is as a network. Unlike vascular transport systems in plants and animals, these networks are not constrained to a predictable structure, but continuously grow and have to adapt to exploit patchy and ephemeral resources, and also maintain their integrity in the face of predation or random damage. We use live-cell imaging techniques, network analysis and mathematical modelling to analyse how these networks resolve these conflicting demands. The networks are constructed by local iterative developmental processes rather than centralized control, with growth involving over-production of links and nodes, followed by selective pruning of some links and reinforcement of others. Such a process mimics the process of Darwinian evolution in which natural selection removes less fit offspring. This “Darwinian network model” may represent a generalized model for growth of physical biological networks. We believe that the generic ingredients include a non-linear positive reinforcement term related to the local flux and a linear decay term. However, experimental tests of these ideas using radiotracers and scintillation imaging have revealed additional complexity, most notably strong oscillatory components superimposed on the underlying nutrient flux, which become self-organised into well demarcated domains differing in phase. Furthermore, fusion between compatible individuals leads to rapid nutrient re-distribution and formation of a fully synchronised oscillating super-colony.

Essentially all known results on (weakly) self-avoiding walks assume the symmetry the distribution of the walk under lattice isometries. If one drops this assumption, some new phenomena appear. We adapt a method originally developed with Christine

Ritzmann to this situation. (Joint work with Felix Rubin)

Abstract:

In a famous paper Timothy Gowers introduced a sequence of norms U(k) defined for functions on abelian groups. He used these norms to give quantitative bounds for Szemeredi's theorem on arithmetic progressions.

The behavior of the U(2) norm is closely tied to Fourier analysis. In this talk we present a generalization of Fourier analysis (called k-th order Fourier analysis) that is related in a similar way to the U(k+1) norm.

Ordinary Fourier analysis deals with homomorphisms of abelian groups into the circle group.

We view k-th order Fourier analysis as a theory which deals with morphisms of abelian groups into algebraic structures that we call "k-step nilspaces". These structeres are variants of structures introduced by Host and Kra (called parallelepiped structures) and they are close relatives of nil-manifolds. Our approach has two components. One is an uderlying algebraic theory of nilspaces and the other is a variant of ergodic theory on ultra product groups.

Using this theory, we obtain inverse theorems for the U(k) norms on arbitrary abelian groups that generalize results by Green, Tao and Ziegler. As a byproduct we also obtain an interesting limit theory for functions on abelian groups in the spirit of the recently developed graph limit theory.

*Sharp estimate of singular integrals with and without stochastic optimal control*by**Alexander Volberg**

Abstract:

The talk is devoted to sharp estimates of weighted Calderon--Zygmund operators. We explain a) why the need for such estimates appear in PDE, b) why this is a difficult problem. Recently it has been solved by Tuomas Hyt\"onen, and we will explain how the original proof worked, and how one can simplify it considerably by using the methods of Stochastic Optimal Control. The methods include random geometric constructions of independent interest, which first appeared in works on non-homogeneous Harmonic Analysis by Nazarov--Treil--Volberg and subsequently brought the solution of Painlev\'e problem by Xavier vTolsa. We will touch upon this direction too.

Abstract**:
**Consider a large cloud of particles which are moving around in space due to a random transport process such as diffusion. If these particles are "sticky" and clump together irreversibly upon contact then the resulting distribution of cluster sizes evolves in time as smaller clusters stick to each other to produce larger ones. The statistical dynamics of such sticky particles has applications in surface physics, colloids, granular materials, bio-physics and atmospheric science. It also provides a rich variety of non-equilibrium phenomena for theoretical analysis. One of the most striking of these phenomena is the so-called gelation transition which may be

interpreted as corresponding to the generation of clusters of infinite size in a finite time. In this talk, I will discuss the scaling theory of cluster aggregation at the level of mean field theory and explain the meaning of the gelation transition. At the end I will discuss some recent results on the somewhat mysterious phenomenon of "instantaneous" gelation and its relation to some problems in cloud physics.

Abstract:

There is an extensive applied mathematics literature developed for problems in the biological and physical sciences. Our understanding of social science problems from a mathematical standpoint is less developed, but also presents some very interesting problems, especially for young researchers. This lecture uses crime as a case study for using applied mathematical techniques in a social science application and covers a variety of mathematical methods that are applicable to such problems. We will review recent work on agent based models, methods in linear and nonlinear partial differential equations, variational methods for inverse problems and statistical point process models. From an application standpoint we will look at problems in residential burglaries and gang crimes. Examples will consider both ``bottom up'' and ``top down'' approaches to understanding the mathematics of crime, and how the two approaches could converge to a unifying theory.

## SPRING TERM

Abstract:

Tropical geometry is geometry over the tropical semiring, where multiplication is replaced by addition and addition is replaced by minimum. This "tropicalization" procedure turns algebraic varieties (solutions to polynomial equations) into polyhedral complexes, which are combinatorial objects. A surprisingly large amount of information about the variety is still present in tropical "combinatorial shadow". In this talk I will introduce tropical varieties, and indicate some of their applications, both inside and outside algebraic geometry.

Abstract:

The talk introduces the concept of covering spaces for the study of (polyhedral) surfaces and differential forms in shape modeling. A covering surface covers its base surface with multiple sheets, and connections between the sheets determine the topology of the covering. Covering surfaces naturally appear in complex analysis as correct domains of multi-valued complex functions such as

Sqrt[z], or during the integration of vector fields as a suitable domain of potential functions. These and many other properties make them an effective tool in surface modeling as it has been demonstrated in recent surface parameterization algorithms. Starting from a beginner's perspective we will review the basic mathematical theory, provide novel visualization tools and hint at recent applications in surface modeling.

Abstract:

The Hasse square reconstructs the integers from its profinite completion and its rationalization, and it is not surprising that we can reconstruct the category of abelian groups on the same basis. Exactly the same idea can be used to great effect in some less familiar contexts, and the talk will describe some examples. The one from equivariant topology is closest to my heart, but shouldn't dominate the discussion.

*On the Question of Global Regularity for Three-dimensional Navier-Stokes Equations and Relevant Geophysical Models*by**Edriss S. Titi**

Abstract:

The basic problem faced in geophysical fluid dynamics is that a mathematical description based only on fundamental physical principles, the so-called the ``Primitive Equations'', is often prohibitively expensive computationally, and hard to study analytically. In this talk I will survey the main obstacles in proving the global regularity for the three-dimensional Navier-Stokes equations and their geophysical counterparts. Even though the Primitive Equations look as if they are more difficult to study analytically than the three-dimensional Navier-Stokes equations I will show in this talk that they have a unique global (in time) regular solution for all initial data.

Inspired by this work I will also provide a new global regularity criterion for the three-dimensional Navier-Stokes equations involving the pressure.

This is a joint work with Chongsheng Cao.

Abstract:

In 1986 Kardar, Parisi, and Zhang proposed a stochastic PDE for the motion of driven interfaces,in particular for growth processes with local updating rules. The solution to the 1D KPZ equation can be approximated through the weakly asymmetric simple exclusion process. Based on work of Tracy and Widom on the PASEP, we obtain an exact formula for the one-point generating function of the KPZ

equation in case of sharp wedge initial data. Our result is valid for all times, but of particular interest is the long time behavior, related to random matrices, and the finite time corrections. This is joint work with Tomohiro Sasamoto.

Abstract:

Topological spaces and manifolds are commonly used to model configuration spaces of systems of various nature. However, many practical tasks, such as those dealing with the modelling, control anddesign of large systems, lead to topological problems having mixedtopological-probabilistic character when spaces and manifolds depend on many random parameters. In my talk I will describe several models of stochastic algebraic topology. I will also mention some open problems and results known so far.

Abstract:

In three specific examples, we shall demonstrate how the theory of partial differential equations (PDEs) relates to pattern formation in nature: Spinodal decomposition and the Cahn-Hilliard equation, Rayleigh-Bénard convection and the Boussinesq approximation, rough crystal growth and the Kuramoto-Sivashinsky equation. These examples from different applications have in common that only a few physical mechanisms, which are modeled by simple-looking evolutionary PDEs, lead to complex patterns. These mechanisms will be explained, numerical simulation shall serve as a visual experiment. Numerical simulations also reveal that generic solutions of these deterministic equations have stationary or self-similar statistics that are independent of the system size and of the details of the initial data. We show how PDE methods, i. e. a priori estimates, can be used to understand some aspects of this universal behavior. In case of the Cahn-Hilliard equation, the method makes use of its gradient flow structure and a property of the energy landscape. In case of the Boussinesq equation, a “driven gradient flow”, the background field method is used. In case of the Kuramoto-Sivashinsky equation, that mixes conservative and dissipative dynamics, the method relies on a new result on Burgers’ equation.

Abstract:

The symplectic reduction of a Hamiltonian action of a Lie group on a symplectic manifold plays the role of a quotient construction in symplectic geometry. It has been understood for several decades that symplectic reduction is closely related to the quotient construction for complex reductive group actions in algebraic geometry provided by Mumford's geometric invariant theory (GIT). Symplectic implosion (the first version of which is due to Guillemin, Jeffrey and Sjamaar) is much more recent, and is related to a generalised version of GIT which provides quotients for non-reductive group actions in algebraic geometry. The aim of this talk is to give a brief survey of symplectic reduction and symplectic implosion and their relationship with GIT, and describe an application to Demailly's theory of jet differentials.

Abstract:

Free groups are very easy to define but their properties are surprisingly complicated. In particular, a lot of effort has gone into trying to understand their automorphisms groups, resulting in a theory that has strong links with hyperbolic geometry and dynamical systems. I will talk about one aspect: understanding the distortion of a free group induced by a single automorphism. Recently, several numerical characteristics have been introduced to quantify this distortion: generic stretch (Kaimanovich, Kapovich and Schupp), curl (Myasnikov and Shpilrain), conjugacy distortion spectrum (Kapovich). I shall describe how these may be unified by interpreting them in terms of an entropy function of a kind familiar in ergodic theory and the multifractal analysis of chaotic dynamical systems. I will try to explain all these concepts and, in particular, no knowledge of ergodic theory will be assumed.

## AUTUMN TERM

Abstract:

We will discuss recent result relating the perturbation determinants of the Birman-Schwinger operators familiar from quantum mechanics, and the Evans function, a Wronskian-type determinant, which is one of the main tools in detecting stability of traveling waves and other special solutions of PDEs.

Abstract:

If X is a compact Riemannian manifold, the hypoelliptic Laplacian is a natural second order operator acting on the total space of the cotangent bundle, which is supposed to interpolate in the proper sense between the ordinary Laplacian of X and the generator of the geodesic ow. It is essentially the weighted sum of the harmonic oscillator of the bres and of the generator of the geodesic ow. The hypoelliptic Laplacian is a Laplacian in the sense of Hodge theory. Its construction is obtained via an exotic deformation of classical Hodge theory, in de Rham or Dolbeault cohomology. The underlying stochastic process is a deformation of classical Brownian motion to a Langevin process. For locally symmetric spaces, the spectrum of the ordinary Laplacian is essentially preserved through the hypoelliptic deformation. One can exploit this fact to give an explicit evaluation of semisimple orbital integrals along lines which are formally similar to the proof of the Atiyah-Singer index theorem.

Abstract:

A standard approach, in studying the existence of positive solutions of boundary value problems (BVPs) for differential equations, is to rewrite the problem as an equivalent fixed-point problem for a Hammerstein integral operator of the form *Su(t)*:= ∫_{0}^{1} *G(t,s) f(s,u(s)) ds* in the space *C*[0, 1], where the nonlinearity *f *and the kernel *G* (the Green’s function of the problem, which incorporates the boundary conditions) are both nonnegative. One seeks fixed-points in a suitable cone of positive functions. Existence of one or of multiple fixed points can be shown using the theory of fixed-point index of compact nonlinear maps, which is related to Leray-Schauder degree.

We will give the main properties of the fixed point index and show how it can be used to obtain sharp existence results for (strictly) positive solutions of BVPs under easily checkable conditions on *f*. In particular we will use comparison with a closely related positive (cone invariant) linear operator. Therefore a good part of the talk will concern a class of positive linear operators, studied half a century ago by M. A. Krasnosel'skii (called *u*_{0}-positive).

Abstract:

The talk will explain how a geometric principle gave rise to a new variational description of information-theoretic entropy and led to the solution of a 50 year-old problem: is the central limit theorem driven by an analogue of the second law of thermodynamics?

Abstract:

Subject to what geometric constraints may one random family be found inside another? Several such open questions will be posed for one-dimensional families, including a third problem for a clairvoyant demon (the first two having been posed earlier by Peter Winkler). Certain positive and negative results will be summarised in higher-dimensional spaces. There are connections to earlier work on biLipschitz embeddings and quasi-isometries, and also to the Borsuk-Ulam theorem of topological combinatorics. (Joint work with Ander Holroyd.)

Abstract:

I will discuss the black hole stability problem in general relativity, and, in particular, recent results obtained in collaboration with Igor Rodnianski which definitively address the linear and scalar aspects of this

problem around an arbitrary Kerr black hole solution.

Abstract:

Many problems in number theory, such as Fermat's Last Theorem, amount to understanding the absolute Galois group of the rationals and where certain subgroups and elements sit inside it. We systematically develop this approach, leading to new conjectures and results. At the end, we use some related ideas to obtain the best result to date on a famous problem in control design.

Abstract:

We study a dynamical random network model in which at every construction step a new vertex is introduced and attached to every existing vertex independently with a probability proportional to a concave function of its current degree. We use approximation by branching random walks to find necessary amd sufficient criteria for the existence and robustness of a giant component in these networks. The talk is based on joint work with Steffen Dereich (Marburg).