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1:01:17 Instituto de Ciencias Matemáticas (ICMAT) Englisch 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.
  • Erscheinungsjahr: 2006
  • Herausgeber: Instituto de Ciencias Matemáticas (ICMAT)
  • Sprache: Englisch
58:28 Instituto de Ciencias Matemáticas (ICMAT) Englisch 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).
  • Erscheinungsjahr: 2006
  • Herausgeber: Instituto de Ciencias Matemáticas (ICMAT)
  • Sprache: Englisch
56:50 Instituto de Ciencias Matemáticas (ICMAT) Englisch 2006

Nevanlinna Prize Winner

Lecture of Jon Kleinberg, the Nevanlinna prize winner 2006.
  • Erscheinungsjahr: 2006
  • Herausgeber: Instituto de Ciencias Matemáticas (ICMAT)
  • Sprache: Englisch
43:39 Instituto de Ciencias Matemáticas (ICMAT) Englisch 2006

Fields Medalist: Andrei Okunkow

Lecture of Andrei Okunkow, Fields medallist 2006.
  • Erscheinungsjahr: 2006
  • Herausgeber: Instituto de Ciencias Matemáticas (ICMAT)
  • Sprache: Englisch
1:07:06 Instituto de Ciencias Matemáticas (ICMAT) Englisch 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.
  • Erscheinungsjahr: 2006
  • Herausgeber: Instituto de Ciencias Matemáticas (ICMAT)
  • Sprache: Englisch
55:39 Instituto de Ciencias Matemáticas (ICMAT) Englisch 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.
  • Erscheinungsjahr: 2006
  • Herausgeber: Instituto de Ciencias Matemáticas (ICMAT)
  • Sprache: Englisch
2:00:45 Instituto de Ciencias Matemáticas (ICMAT) Englisch 2006

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

Closing Round Table discussion of ICM 2006.
  • Erscheinungsjahr: 2006
  • Herausgeber: Instituto de Ciencias Matemáticas (ICMAT)
  • Sprache: Englisch
1:02:58 Instituto de Ciencias Matemáticas (ICMAT) Englisch 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.
  • Erscheinungsjahr: 2006
  • Herausgeber: Instituto de Ciencias Matemáticas (ICMAT)
  • Sprache: Englisch
59:32 Instituto de Ciencias Matemáticas (ICMAT) Englisch 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.
  • Erscheinungsjahr: 2006
  • Herausgeber: Instituto de Ciencias Matemáticas (ICMAT)
  • Sprache: Englisch
1:03:52 Instituto de Ciencias Matemáticas (ICMAT) Englisch 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.
  • Erscheinungsjahr: 2006
  • Herausgeber: Instituto de Ciencias Matemáticas (ICMAT)
  • Sprache: Englisch
1:00:44 Instituto de Ciencias Matemáticas (ICMAT) Englisch 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.
  • Erscheinungsjahr: 2006
  • Herausgeber: Instituto de Ciencias Matemáticas (ICMAT)
  • Sprache: Englisch
46:33 Instituto de Ciencias Matemáticas (ICMAT) Englisch 2006

The Poincaré Conjecture

Special lecture on the recent spectacular developments concerning the Poincaré Conjecture.
  • Erscheinungsjahr: 2006
  • Herausgeber: Instituto de Ciencias Matemáticas (ICMAT)
  • Sprache: Englisch
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