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1:05:21 Instituto de Ciencias Matemáticas (ICMAT) English 2006

Applications of Equivariant Cohomology

We will discuss the equivariant cohomology of a manifold endowed with the action of a Lie group. Localization formulae for equivariant integrals are explained by a vanishing theorem for equivariant cohomology with generalized coefficients. We then give applications to integration of characteristic classes on symplectic quotients and to indices of transversally elliptic operators. In particular, we state a conjecture for the index of a transversally elliptic operator linked to a Hamiltonian action. In the last part, we describe algorithms for numerical computations of values of multivariate spline functions and of vector-partition functions of classical root systems.
  • Published: 2006
  • Publisher: Instituto de Ciencias Matemáticas (ICMAT)
  • Language: English
1:02:58 Instituto de Ciencias Matemáticas (ICMAT) English 2006

Cardiovascular mathematics

We introduce some basic differential models for the description of blood flow in the circulatory system. We comment on their mathematical properties, their meaningfulness and their limitation to yield realistic and accurate numerical simulations, and their contribution for a better understanding of cardiovascular physio-pathology.
  • Published: 2006
  • Publisher: Instituto de Ciencias Matemáticas (ICMAT)
  • Language: English
56:50 Instituto de Ciencias Matemáticas (ICMAT) English 2006

Nevanlinna Prize Winner

Lecture of Jon Kleinberg, the Nevanlinna prize winner 2006.
  • Published: 2006
  • Publisher: Instituto de Ciencias Matemáticas (ICMAT)
  • Language: English
1:00:23 Instituto de Ciencias Matemáticas (ICMAT) English 2006

Energy-driven pattern formation

Many physical systems can be modelled by nonconvex variational problems regularized by higher-order terms. Examples include martensitic phase transformation, micromagnetics, and the Ginzburg–Landau model of nucleation. We are interested in the singular limit, when the coefficient of the higher-order term tends to zero. Our attention is on the internal structure of walls, and the character of microstructure when it forms. We also study the pathways of thermally-activated transitions, modeled via the minimization of action rather than energy. Our viewpoint is variational, focusing on matching upper and lower bounds.
  • Published: 2006
  • Publisher: Instituto de Ciencias Matemáticas (ICMAT)
  • Language: English
1:07:56 Instituto de Ciencias Matemáticas (ICMAT) English 2006

Moduli spaces from a topological viewpoint

This talk aims to explain what topology, at present, has to say about a few of the many moduli spaces that are currently under study in mathematics. The most prominent one is the moduli space Mg of all Riemann surfaces of genus g. Other examples include the Gromov–Witten moduli space of pseudo-holomorphic curves in a symplectic background, the moduli space of graphs and Waldhausen’s algebraic K-theory of spaces.
  • Published: 2006
  • Publisher: Instituto de Ciencias Matemáticas (ICMAT)
  • Language: English
1:01:50 Instituto de Ciencias Matemáticas (ICMAT) English 2006

Prime numbers and L-functions

The classical memoir by Riemann on the zeta function was motivated by questions about the distribution of prime numbers. But there are important problems concerning prime numbers which cannot be addressed along these lines, for example the representation of primes by polynomials. In this talk I will show a panorama of techniques, which modern analytic number theorists use in the study of prime numbers. Among these are sieve methods. I will explain how the primes are captured by adopting new axioms for sieve theory. I shall also discuss recent progress in traditional questions about primes, such as small gaps, and fundamental ones such as equidistribution in arithmetic progressions. However, my primary objective is to indicate the current directions in Prime Number Theory.
  • Published: 2006
  • Publisher: Instituto de Ciencias Matemáticas (ICMAT)
  • Language: English
50:33 Instituto de Ciencias Matemáticas (ICMAT) English 2006

Gauss prize lecture - On Kiyosi Itô’s work and its impact

Laudatio on the occasion of the Gauss Prize award for Applications of Mathematics to Kiyosi Itô “for laying the foundations of the theory of stochastic differential equations and stochastic analysis”.
  • Published: 2006
  • Publisher: Instituto de Ciencias Matemáticas (ICMAT)
  • Language: English
59:32 Instituto de Ciencias Matemáticas (ICMAT) English 2006

The dichotomy between structure and randomness, arithmetic progressions, and the primes

A famous theorem of Szemerédi asserts that all subsets of the integers with positive upper density will contain arbitrarily long arithmetic progressions. There are many different proofs of this deep theorem, but they are all based on a fundamental dichotomy between structure and randomness, which in turn leads (roughly speaking) to a decomposition of any object into a structured (low-complexity) component and a random (discorrelated) component. Important examples of these types of decompositions include the Furstenberg structure theorem and the Szemerédi regularity lemma. One recent application of this dichotomy is the result of Green and Tao establishing that the prime numbers contain arbitrarily long arithmetic progressions (despite having density zero in the integers). The power of this dichotomy is evidenced by the fact that the Green–Tao theorem requires surprisingly little technology from analytic number theory, relying instead almost exclusively on manifestations of this dichotomy such as Szemerédi’s theorem. In this paper we survey various manifestations of this dichotomy in combinatorics, harmonic analysis, ergodic theory, and number theory. As we hope to emphasize here, the underlying themes in these arguments are remarkably similar even though the contexts are radically different.
  • Published: 2006
  • Publisher: Instituto de Ciencias Matemáticas (ICMAT)
  • Language: English
52:38 Instituto de Ciencias Matemáticas (ICMAT) English 2006

P, NP and mathematics - a computational complexity perspective

The P versus NP question distinguished itself as the central question of theoretical computer science nearly four decades ago. The quest to resolve it, and more generally, to understand the power and limits of efficient computation, has led to the development of computational complexity theory. While this mathematical discipline in general, and the P vs. NP problem in particular, have gained prominence within the mathematics community in the past decade, it is still largely viewed as a problem of computer science.
  • Published: 2006
  • Publisher: Instituto de Ciencias Matemáticas (ICMAT)
  • Language: English
1:03:52 Instituto de Ciencias Matemáticas (ICMAT) English 2006

Deformation and rigidity for group actions and von Neumann algebras

We present some recent rigidity results for von Neumann algebras (II1 factors) and equivalence relations arising from measure preserving actions of groups on probability spaces which satisfy a combination of deformation and rigidity properties. This includes strong rigidity results for factors with calculation of their fundamental group and cocycle superrigidity for actions with applications to orbit equivalence ergodic theory.
  • Published: 2006
  • Publisher: Instituto de Ciencias Matemáticas (ICMAT)
  • Language: English
1:01:03 Instituto de Ciencias Matemáticas (ICMAT) English 2006

High dimensional statistical inference and random matrices

Multivariate statistical analysis is concerned with observations on several variables which are thought to possess some degree of inter-dependence. Driven by problems in genetics and the social sciences, it first flowered in the earlier half of the last century. Subsequently, random matrix theory (RMT) developed, initially within physics, and more recently widely in mathematics. While some of the central objects of study in RMT are identical to those of multivariate statistics, statistical theory was slow to exploit the connection. However, with vast data collection ever more common, data sets now often have as many or more variables than the number of individuals observed. In such contexts, the techniques and results of RMT have much to offer multivariate statistics. The talk reviews some of the progress to date.
  • Published: 2006
  • Publisher: Instituto de Ciencias Matemáticas (ICMAT)
  • Language: English
1:15:54 Instituto de Ciencias Matemáticas (ICMAT) English 2006

Knots and dynamics

The trajectories of a vector field in 3-space can be very entangled; the flow can swirl, spiral, create vortices etc. Periodic orbits define knots whose topology can sometimes be very complicated. In this talk, I will survey some advances in the qualitative and quantitative description of this kind of phenomenon. The first part will be devoted to vorticity, helicity, and asymptotic cycles for flows. The second part will deal with various notions of rotation and spin for surface diffeomorphisms. Finally, I will describe the important example of the geodesic flow on the modular surface, where the linking between geodesics turns out to be related to well-known arithmetical functions.
  • Published: 2006
  • Publisher: Instituto de Ciencias Matemáticas (ICMAT)
  • Language: English
58:28 Instituto de Ciencias Matemáticas (ICMAT) English 2006

Optimal computation

A large portion of computation is concerned with approximating a function u. Typically, there are many ways to proceed with such an approximation leading to a variety of algorithms. We address the question of how we should evaluate such algorithms and compare them. In particular, when can we say that a particular algorithm is optimal or near optimal? We shall base our analysis on the approximation error that is achieved with a given (computational or information) budget n. We shall see that the formulation of optimal algorithms depends to a large extent on the context of the problem. For example, numerically approximating the solution to a PDE is different from approximating a signal or image (for the purposes of compression).
  • Published: 2006
  • Publisher: Instituto de Ciencias Matemáticas (ICMAT)
  • Language: English
1:08:58 Instituto de Ciencias Matemáticas (ICMAT) English 2006

The Poincaré Conjecture

  • Published: 2006
  • Publisher: Instituto de Ciencias Matemáticas (ICMAT)
  • Language: English
1:01:17 Instituto de Ciencias Matemáticas (ICMAT) English 2006

Increasing and decreasing subsequences and their variants

We survey the theory of increasing and decreasing subsequences of permutations. Enumeration problems in this area are closely related to the RSK algorithm. The asymptotic behavior of the expected value of the length is(w) of the longest increasing subsequence of a permutation w of 1, 2,...,n was obtained by Vershik–Kerov and (almost) by Logan–Shepp. The entire limiting distribution of is(w) was then determined by Baik, Deift, and Johansson. These techniques can be applied to other classes of permutations, such as involutions, and are related to the distribution of eigenvalues of elements of the classical groups. A number of generalizations and variations of increasing/decreasing subsequences are discussed, including the theory of pattern avoidance, unimodal and alternating subsequences, and crossings and nestings of matchings and set partitions.
  • Published: 2006
  • Publisher: Instituto de Ciencias Matemáticas (ICMAT)
  • Language: English
1:00:44 Instituto de Ciencias Matemáticas (ICMAT) English 2006

Random, conformally invariant scaling limits in two dimensions

Many mathematical models of statistical physics in two dimensions are either known or conjectured to exhibit conformal invariance. Over the years, physicists proposed predictions of various exponents describing the behavior of these models. Only recently have some of these predictions become accessible to mathematical proof. One of the new developments is the discovery of a one-parameter family of random curves called stochastic Loewner evolution or SLE. The SLE curves appear as limits of interfaces or paths occurring in a variety of statistical physics models as the mesh of the grid on which the model is defined tends to zero. The main purpose of this article is to list a collection of open problems. Some of the open problems indicate aspects of the physics knowledge that have not yet been und erstood mathematically. Other problems are questions about the nature of the SLE curves themselves. Before we present the open problems, the definition of SLE will be motivated and explained, and a brief sketch of recent results will be presented.
  • Published: 2006
  • Publisher: Instituto de Ciencias Matemáticas (ICMAT)
  • Language: English
55:39 Instituto de Ciencias Matemáticas (ICMAT) English 2006

Advances in convex optimization: conic programming

During the last two decades, major developments in convex optimization were focusing on conic programming, primarily, on linear, conic quadratic and semidefinite optimization. Conic programming allows to reveal rich structure which usually is possessed by a convex program and to exploit this structure in order to process the program efficiently. We overview the major components of the resulting theory (conic duality and primal-dual interior point polynomial time algorithms), outline the extremely rich “expressive abilities” of conic quadratic and semidefinite programming and discuss a number of instructive applications.
  • Published: 2006
  • Publisher: Instituto de Ciencias Matemáticas (ICMAT)
  • Language: English
1:07:06 Instituto de Ciencias Matemáticas (ICMAT) English 2006

Perspectives in nonlinear diffusion: between analysis, physics and geometry

We review some topics in the mathematical theory of nonlinear diffusion. Attention is focused on the porous medium equation and the fast diffusion equation, including logarithmic diffusion. Special features are the existence of free boundaries, the limited regularity of the solutions and the peculiar asymptotic laws for porous medium flows, while for fast diffusions we find the phenomena of finite-time extinction, delayed regularization, nonuniqueness and instantaneous extinction. Logarithmic diffusion with its strong geometrical flavor is also discussed. Connections with functional analysis, semigroup theory, physics of continuous media, probability and differential geometry are underlined.
  • Published: 2006
  • Publisher: Instituto de Ciencias Matemáticas (ICMAT)
  • Language: English
1:06:56 Instituto de Ciencias Matemáticas (ICMAT) English 2006

Kähler manifolds and transcendental techniques in algebraic geometry

Our goal is to survey some of the main advances which took place recently in the study of the geometry of projective or compact Kähler manifolds: very efficient new transcendental techniques, a better understanding of the geometric structure of cones of positive cohomology classes and of the deformation theory of Kähler manifolds, new results around the invariance of plurigenera and in the minimal model program.
  • Published: 2006
  • Publisher: Instituto de Ciencias Matemáticas (ICMAT)
  • Language: English
59:34 Instituto de Ciencias Matemáticas (ICMAT) English 2006

Universality for mathematical and physical systems

All physical systems in equilibrium obey the laws of thermodynamics. In other words, whatever the precise nature of the interaction between the atoms and molecules at the microscopic level, at the macroscopic level, physical systems exhibit universal behavior in the sense that they are all governed by the same laws and formulae of thermodynamics. In this talk we describe some recent history of universality ideas in physics starting with Wigner’s model for the scattering of neutrons off large nuclei and show how these ideas have led mathematicians to investigate universal behavior for a variety of mathematical systems. This is true not only for systems which have a physical origin, but also for systems which arise in a purely mathematical context such as the Riemann hypothesis, and a version of the card game solitaire called patience sorting.
  • Published: 2006
  • Publisher: Instituto de Ciencias Matemáticas (ICMAT)
  • Language: English
1:00:36 Instituto de Ciencias Matemáticas (ICMAT) English 2006

Symplectic field theory and its applications

Symplectic field theory (SFT) attempts to approach the theory of holomorphic curves in symplectic manifolds (also called Gromov-Witten theory) in the spirit of a topological field theory. This naturally leads to new algebraic structures which seems to have interesting applications and connections not only in symplectic geometry but also in other areas of mathematics, e.g. topology and integrable PDE. In this talk we sketch out the formal algebraic structure of SFT and discuss some current work towards its applications.
  • Published: 2006
  • Publisher: Instituto de Ciencias Matemáticas (ICMAT)
  • Language: English
1:02:46 Instituto de Ciencias Matemáticas (ICMAT) English 2006

Emmy Noether Lecture

  • Published: 2006
  • Publisher: Instituto de Ciencias Matemáticas (ICMAT)
  • Language: English
1:06:50 Instituto de Ciencias Matemáticas (ICMAT) English 2006

Iwasawa theory and generalizations

Introduction to Iwasawa theory and its generalizations, discussion of some main conjectures and related subjects.
  • Published: 2006
  • Publisher: Instituto de Ciencias Matemáticas (ICMAT)
  • Language: English
53:33 Instituto de Ciencias Matemáticas (ICMAT) English 2006

Fields Medalist: Wendelin Werner

Lecture of Wendelin Werner, Fields medallist 2006.
  • Published: 2006
  • Publisher: Instituto de Ciencias Matemáticas (ICMAT)
  • Language: English
1:02:24 Instituto de Ciencias Matemáticas (ICMAT) English 2006

The Nature of Roughness in Mathematics, Science and Art

Special Lecture of Benoit Mandelbrot, ICM 2006.
  • Published: 2006
  • Publisher: Instituto de Ciencias Matemáticas (ICMAT)
  • Language: English
46:33 Instituto de Ciencias Matemáticas (ICMAT) English 2006

The Poincaré Conjecture

Special lecture on the recent spectacular developments concerning the Poincaré Conjecture.
  • Published: 2006
  • Publisher: Instituto de Ciencias Matemáticas (ICMAT)
  • Language: English
43:39 Instituto de Ciencias Matemáticas (ICMAT) English 2006

Fields Medalist: Andrei Okunkow

Lecture of Andrei Okunkow, Fields medallist 2006.
  • Published: 2006
  • Publisher: Instituto de Ciencias Matemáticas (ICMAT)
  • Language: English
23:44 Instituto de Ciencias Matemáticas (ICMAT) English 2006

The work of Wendelin Werner

Laudatio on the occasion of the Field medal award to Wendelin Werner.
  • Published: 2006
  • Publisher: Instituto de Ciencias Matemáticas (ICMAT)
  • Language: English
26:56 Instituto de Ciencias Matemáticas (ICMAT) English 2006

The work of Terence Tao

Laudatio on the occasion of the Fields medal award to Terence Tao.
  • Published: 2006
  • Publisher: Instituto de Ciencias Matemáticas (ICMAT)
  • Language: English
24:24 Instituto de Ciencias Matemáticas (ICMAT) English 2006

The work of Jon Kleinberg

Laudatio on the occasion of the Nevanlinna Prize award to Jon Kleinberg.
  • Published: 2006
  • Publisher: Instituto de Ciencias Matemáticas (ICMAT)
  • Language: English
21:44 Instituto de Ciencias Matemáticas (ICMAT) English 2006

The work of Andrei Okounkov

Laudatio on the occasion of the Fields medal award to Andrei Okounkov.
  • Published: 2006
  • Publisher: Instituto de Ciencias Matemáticas (ICMAT)
  • Language: English
2:00:45 Instituto de Ciencias Matemáticas (ICMAT) English 2006

Closing Round Table - Are pure and applied mathematics drifting apart?

Closing Round Table discussion of ICM 2006.
  • Published: 2006
  • Publisher: Instituto de Ciencias Matemáticas (ICMAT)
  • Language: English
18:11 Instituto de Ciencias Matemáticas (ICMAT) English 2006

The work of Grigory Perelman

Laudatio on the occasion of the Fields medal award to Grigory Perelman.
  • Published: 2006
  • Publisher: Instituto de Ciencias Matemáticas (ICMAT)
  • Language: English
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