Colloquium

Past sessions

Geometry and dynamics: towards a ${L}^{\infty }$-theory

Techniques of elliptic PDE's in the context of gauge theory as well as in symplectic topology have led to solutions of deep problems on the existence and behavior of periodic orbits of arbitrary Hamiltonian systems. They have also led to a much better knowledge of smooth and symplectic structures on manifolds, especially in dimension 4. Using Quantum homology and its relation with Floer homology, I will describe how this is now paving the way to the emergence of a new ${L}^{\infty }$-theory that has both geometric and dynamic interpretations.

Boundary Value Problems and the Green Formula

The Green formula plays an outstanding role in boundary value problems (BVPs) for linear partial differential equations (PDEs) in n-dimensional domains. With their help it is possible to establish existence and uniqueness of a solution, obtain a representation of this solution via layer potentials and to derive an equivalent boundary integral equation (BIE) on the boundary of the domain.

The simplest versions of the Green formula are readily obtained by “partial integration” (the Gauss formula). We describe how to get all possible versions of the Green formula by pure algebraic operations for BVPs when the basic equation in the domain is an arbitrary PDE with $$N \times N$$ matrix coefficients and the boundary conditions are prescribed by quasi-normal partial differential operators with vector $$1 \times N$$ coefficients.

If the basic system possesses a fundamental solution, a representation formula for the solution is derived. Exact boundedness properties of the relevant layer potentials, mapping function spaces on the boundary (Bessel potential, Besov, Zygmund spaces) into appropriate weighted function spaces in the domain, are established.

Some related topics, such as the Green formula for a BVP on a smooth open surface in the $$n$$-space, Calderon projections, Plemelji formulae etc. will be discussed as well.

Fundamental Problems in the Sedimentation of Particles in Newtonian and Viscoelastic Liquids

Over the last 40 years the study of the motion of small particles in a viscous liquid has become one of the main focuses of engineering research. The presence of the particles affects the flow of the liquid, and this, in turn, affects the motion of the particles, so that the problem of determining the flow characteristics is highly coupled. It is just this latter feature that makes any fundamental mathematical problem related to liquid-particle interaction a particularly challenging one.

Interestingly enough, even though the mathematical theory of the motion of rigid particles in a liquid is one of the oldest and most classical problems in fluid mechanics, owed to the seminal contributions of Stokes, Kirchhoff, Thomson (Lord Kelvin) and Tait, only very recently have mathematicians become interested in a systematic study of the basic problems related to liquid-particle interaction.

This lecture concentrates on the mathematical analysis of one of the several important and still not completely understood aspects of this fascinating subject, that is, the orientation of symmetric particles sedimenting in Newtonian and viscoelastic liquids. As is well known, the general phenomenon of particle orientation plays a crucial role in many engineering problems, like the manufacturing of short-fiber composites, separation of macromolecules by electrophoresis, flow-induced microstructures, and also in blood flow problems.

Some things Ramanujan may have had up his sleeve

In referring to the first letter he received from the then unknown Indian genius, Ramanujan, G. H. Hardy remarked that ...it soon became obvious that Ramanujan must possess much more general theorems and was keeping a great deal up his sleeve.

I hope to describe some of the overwhelming mathematical surprises in Ramanujan's Notebooks (including the Lost Notebook) and to offer a few thoughts on how Ramanujan might have discovered some of his incredible identities.

The Rehabilitation of Infinitesimals in Mathematics and Physics

One of the most remarkable recent developments in mathematics is the refounding, on a rigorous basis, of the idea of infinitesimal quantity, a notion which played a central role in the early development of the calculus and mathematical analysis. One of the most useful concepts of infinitesimal to have thus acquired rigorous status is that of a quantity so small (but not actually zero) that its square and all its higher powers can be set to zero. In my talk I will describe how infinitesimals of this kind can be used to develop the differential calculus and basic differential geometry. Time permitting, I will also describe the form that spacetime metrics assume in this approach.

Global and trajectory attractors for evolution equations and their homogenization

Global attractor for autonomous evolution equations. Examples.

Trajectory attractor for reaction-diffusion systems.

Trajectory attractor for evolution equation with rapidly oscillating coefficients. Examples.

Convergence of these attractors to the attractors of the averaged equations.

Quantitative homogenization of global attractors for reaction-diffusion systems with rapidly oscillating terms.

Five Remarkable Properties of Capillary Surfaces

A capillary surface is an interface $S$ separating two fluids that are in equilibrium adjacent to each other and do not mix. In the absence of external force fields, the mean curvature $H$ of such an interface is constant; in a gravity field, $H$ varies linearly with height. If $S$ extends to a rigid support surface, then it meets that surface in a “contact angle” that is determined physically by the materials. The behavior of capillary surfaces can under some conditions be counterintuitive. In the present talk, five examples will be discussed, all of which were predicted mathematically from the formal equations, and all of which contain features that were unexpected. The examples are:

1. discontinuous disappearance at critical data,
2. non-uniqueness and symmetry breaking
3. discontinuous behavior of liquid bridges,
4. existence and nonexistence of C-singular solutions,
5. discontinuous reversal of comparison relations at low gravity.

Results of drop tower and space experiments based on some of these predictions will be shown.

Quantales, $$C^\ast$$-algebras, and Computation

Quantales were introduced in order to provide a non-commutative generalisation of, on the one hand, the concepts of general topology, and, on the other, the ideas of intuitionistic logic. These considerations were motivated principally by the problems of extending Gelfand theory to non-commutative $$C^\ast$$-algebras, and of providing a constructive foundation for quantum mechanics. Given this field of interest, it is perhaps not surprising that connections began to be established at an early point with the linear logic being developed in theoretical computer science.

In this talk, we shall survey some of the progress that has been made on these diverse fronts, including the non-commutative generalisation of the Gelfand-Naimark theorem, the quantisation of the calculus of relations, and the semantics of computation. The closeness of these subjects becomes very apparent as the concepts of each become intertwined in the theory that is developed. In particular, the identification of the concept ofnon-commutative space provides a context in which these ideas interrelate critically.

Initial algebras, final coalgebras and specification of systems

The fact that natural numbers can be presented as an initial algebra of a very simple endofunctor $$F$$ of the category of sets (viz., $$F: X \mapsto X + 1 )$$, and the algebra of finite binary trees as an initial algebra of the endofunctor $$G: X \mapsto (X \times X) + 1$$ has inspired in the l970s the theory of abstract data types. A part of that theory has been concerned with infinite structures (e.g., infinite words and infinite trees), for which the formalism of complete partial orders (CPO) or complete metric spaces has been applied. It turns out that another very natural approach to these infinite structures is to consider the dual concept of intial algebra, namely, final coalgebra. As recently demonstrated by J. Rutten, coalgebras present a convenient formalism for describing systems. And final coalgebras often represent important structures, e.g. the (co-)algebra of finite and infinite binary trees is final for the above functor $$G$$. Moreover, each of these coalgebras carries a natural CPO structure which, thus, need not be assumed a priori.

Foundation and Application of Boundary Element Methods

First we consider some basic properties of integral operators as pseudodifferential operators. Boundary integral operators and non-local operators are mostly classical pseudodifferential operators defined either by the fundamental solution or a parametrix or Levi functions. The decisive properties of these operators are characterized by corresponding pseudo-homogeneous expansions of the kernels of corresponding general integral operators. These expansions are decisive for the mapping and smoothness properties of these non-local operators which one needs for appropriate discretizations. For stability and convergence of discrete schemes, strong ellipticity plays an essential role. For strongly elliptic boundary value problems, integral operators of the first kind inherit strong ellipticity under mild additional assumptions.

The lecture is based on: George C. Hsiao, Wolfgang L. Wendland: Boundary Integral Equations (in preparation).

Algebraic and Lagrangian Geometry

The technical formalisms and conceptual frameworks of present-day algebraic geometry and symplectic geometry are adequate for solving inner problems of each of these domains of mathematics.

However, the recently observed mirror symmetry phenomenon requires a fusion of them. Fortunately, this procedure began some time ago in the framework of the Geometric Quantization Theory. The aim of my lecture is to give an introduction to this “fusion” procedure.

Virtual Knot Theory

Virtual Knot theory is to classical knot theory as all graphs are to planar graphs. In virtual knot theory one studies Gauss codes representing "knots" that have an abstract existence, but require virtual crossings when drawn in the plane. Many new phenomena appear in this generalization of classical knot theory. The Jones polynomial, quantum link invariants and Vassiliev invariants generalize to virtuals as does the fundamental group, rack and quandle. There are non-trivial virtual knots with trivial Jones polynomial. There are non-trivial virtual knots with non-trivial Jones polynomial, but with the infinite cyclic fundamental group. A combinatorial theory of flat virtuals is interesting in its own right. Virtual braids can be analyzed and, surprisingly, virtuals have applications not only to knots and links in thickened surfaces but also to the homotopy theory of infinite loop spaces and to the embeddings of surfaces in four dimensional space.

Some Open Problems in Partial Differential Equations

The development of the theory of elliptic boundary value problems in the last century is concentrated mainly around the 19th and 20th Hilbert problems, dealing with existence, analyticity, and regularity properties of solutions. It is believed that both problems have been principally solved.

I intend to show that Hilbert’s program is far from being exhausted and still is able to give strong impetus to the theory of elliptic equations. Among the deep problems which are waiting to be explored are:

1. Description of classes of elliptic systems as well as higher order elliptic equations and variational problems with analytic nonlinearities which have only analytic solutions.
2. Characterization of local and boundary singularities of nonlinear elliptic equations.
3. The study of boundary behaviour of solutions to nonlinear and higher order linear equations in domains with unrestricted boundaries. In particular, a circle of questions concentrated around Wiener’s criterion for regularity of a boundary point.

The Role of Algebra in Computer Science

A strong a priori case can be made that so-called general (or universal) algebra should play a fundamental role in computer science. But in fact, it has yet to do so on any large scale, and moreover the best developed branches of modern algebra (e.g., group theory, algebraic geometry, homology) seem doomed to play at most tertiary roles. Set theory and logic of course are basic, but perhaps surprisingly, category theory has a significant, growing and diverse role. We will try to explain why this is, and will give some examples. We will also sketch some as yet little explored application areas for algebra and category theory, such as computability and complexity, modern component-based programming, automated theorem proving, semantics of programming, and user interface design.

Mathematical Perspectives in the Statistical Sciences, and the Current Trends

In modern interdisciplinary research encompassing a wider field of socio-economic, agricultural, biological, industrial, biomedical, environmental and public health disciplines, statistical planning, modeling and analysis are indispensible. Statistical science reflects this composite field where mathematical perspectives dominate the theory and methodology developments. Yet, the advent of modern computer science has opened the doors for highly computation-intensive statistical tools; in bioinformatics, for example, computational statistics plays the key role. In view of this apparent discordence between mathematical and computational approaches to statistical resolutions, a critical appraisal of statistical innovations is made with due respect to its mathematical heritage. Illustrations from diverse fields of application are included in the same vein.

The Role of Mathematics in Several Areas of Mechanical Engineering

A variety of mathematical tools are used in modeling the various processes that mechanical engineers are concerned with. In addition to classical ordinary and partial differential equations that occur naturally in problems involving the mechanics of fluids and solids, and integral-differential models for materials with memory, novel materials provide challenges in developing averaging methods and homogenization procedures. Control of mechanical systems have invoked the use of fuzzy sets and fuzzy logic and Banach algebras, randomness and indeterminacy inherent to certain problems have required sophisticated mathematical tools based on probability and statistics, and material symmetry consideration require appeal to group theory. Here, I shall discuss some of the mathematical tools used in the study of a variety of problems that confronts mechanical engineers.

Recent Advances in Asymptotic Spectral Theory

Consider a sequence of bounded linear operators acting in some Hilbert space and suppose that this sequence tends strongly to some operator. Then it is well-known that the limiting set of the spectra of the approximating operators has almost nothing to do with the spectrum of the limit operator. Even if the convergence is uniform, the picture is not changed (Kakutani, limpotent operators). Nevertheless, in applications frequently there occur problems where one has to relate spectral quantities of the approximating operators (traces, determinants, singular values, epsilon-pseudospectra and so on) with quantities of the limit operator or something else. Nowadays there is a variety of investigations and results in that direction concerning quite different classes of operators and their approximations. It is worthwhile noticing that Toeplitz and Wiener-Hopf operators have played an outstanding role in this context: they have served as some kind of generator of ideas. This talk is devoted to the very recent progress in the field.

Chaotic Dynamical Systems: a Probabilistic Viewpoint

The evolution of systems in Nature is, quite often, sensitive with respect to the initial data: small inaccuracies in the determination of the initial position give rise to gross errors in the long run. Does that mean that there is no hope to be able to understand and predict the evolution of such systems? Several recent developments indicate that, on the contrary, many chaotic systems admit a rich and rather complete description, in probabilistic terms. Building on this, one dares hope to develop a general theory for chaotic dynamics...

The Mathematics Colloquium is a series of monthly talks organized by the Department of Mathematics of IST, aiming to be a forum for the presentation of mathematical ideas or ideas about Mathematics. The Colloquium welcomes the participation of faculty, researchers and undergraduate or graduate students, of IST or other institutions, and is seen as an opportunity of bringing together and fostering the building up of ideas in an informal atmosphere.

Organizers: Conceição Amado, Lina Oliveira e Maria João Borges.