Caucher Birkar (University of Cambridge)
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Thursday, April 28, 2016, 16:15, room tba
Mikael Rordam (University of Copenhagen)
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Thursday, April 21, 2016, 16:15, room 17
Vassily Golyshev (Independent University of Moscow)
Around the gamma conjectures
We will state the gamma conjectures for Fano manifolds and explain how quantum cohomology makes it possible to enhance the classical Riemann-Roch-Hirzebruch theorem by relating the curve count on a variety to its characteristic classes. We will indicate how the gamma conjectures are proved in the known cases.
Thursday, March 17, 2016, 16:15, room Battelle
Nicolai Reshetikhin (UC Berkeley & U Amsterdam)
Limit shapes for the 6-vertex model in statistical mechanics
The phenomenon of limit shape formations in statistical
mechanics is similar in many ways to the semiclassical limit in quantum
mechanics. Its main feature is that for large systems random variable
can become deterministic at certain scales. Versions of this phenomenon are known as hydrodynamical limits. In probability theory counterparts of limit shape
formation are central limit theorems. The phenomenon was relatively well studied in dimer models where the corresponding variational principle is proven.
The first part of this talk will be an overview of the variational
principle for the limit shape formation in dimer models. In the second part
I will show that under certain assumptions, non-linear PDE describing
limit shapes in the 6-vertex model have infinitely many conserved quantities.
Thursday, March 10, 2016, 16:15, room 17
Martin Zirnbauer (University of Cologne)
Journée Georges de Rham 2016: Bott periodicity and the
Bott periodicity is said to be one the most surprising phenomena in topology. Perhaps even more surprising is its recent appearance in condensed matter physics. Building on work of Schnyder et al, Kitaev argued that symmetry-protected ground states of gapped free-fermion systems, also known as topological insulators and superconductors, organize into a kind of periodic table governed by a variant of the Bott periodicity theorem. In this colloquium, I will sketch the mathematical background, the physical context, and some new results of this ongoing story of mathematical physics.
Wednesday, March 02, 2016, 17:00, room Sciences II: Pictet - A100
Gregory F. Lawler (University of Chicago)
Journée Georges de Rham 2016: Self-avoiding motion
The self-avoiding walk (SAW) is a model for polymers that gives equal probability to all paths that do not return to places they have already been. The lattice version of this problem, while elementa- ry to define, has proved to be notoriously difficult and is still open. It seems initially more challeng- ing to construct a continuous limit of the lattice model which is a random fractal. However, in two dimensions this has been done and the continuous model (Schram-Loewner evolution) can be ana- lyzed rigorously and used to understand (nonrigorously) the predictions about SAWs. I will survey some results in this area and then discuss some recent work on this “continuous SAW’’ and well as a related model, the loop-erased random walk.
Wednesday, March 02, 2016, 15:30, room Sciences II: Pictet -- A100
Jürg Fröhlich (ETHZ)
Mirimanoff Lecture 2015: The Arrow of Time
Several examples of "effective quantum dynamics" of large open (but isolated) systems are discussed - among them:Heat transport between thermal reservoirs and heat engines, (derivation of the 2nd Law of Thermodynamics); quantum friction of particles moving through a gas, quantum Brownian motion; and the dynamics of quantum systems under repeated measurements. The theme common to these and other examples is the appearance of an "arrow of time", in the sense that they exhibit dynamical phenomena that are irreversible. Different sources of irreversibility are revealed and discussed.
Thursday, December 10, 2015, 16:00, room Stueckelberg Auditorium
Israel Michael Sigal (University of Toronto and ETHZ)
Mirimanoff Lecture 2015: Magnetic Vortices, vortex lattices and automorphic functions
The Ginzburg - Landau theory was first developed to understand behaviour of superconductors, but has had a profound influence on physics well beyond its original area. It had the first demonstration of the Higgs mechanism and it became a fundamental part of the standard model in the elementary particle physics. The theory is based on a pair of coupled nonlinear equations for a complex function (called order parameter or Higgs field) and a vector field (magnetic potential or gauge field). They are the simplest representatives of a large family of equations appearing in physics and mathematics.
Besides of their importance in physics, the equations contain beautiful mathematics (some of the mathematics was discovered independently by A. Turing in his explanation of patterns of animal coats). In this talk I will review recent results involving key solutions of these equations - the magnetic vortices (called Nielsen-Olesen or Nambu strings in the particle physics) and vortex lattices, their existence and stability and how they relate to the modified theta functions appearing in number theory and algebraic geometry. Certain automorphic functions play a key role in the theory described in the talk.
Thursday, December 10, 2015, 14:30, room Stueckelberg Auditorium
Andras Szenes (Université de Genève)
Basic notions: Enumerative Geometry
I take you on a tour around of one of the oldest subjects in Mathematics:
counting geometric objects.
Thursday, November 26, 2015, 16:15, room 17
Elvezio Ronchetti (University of Geneva)
Robustness: From Basic Concepts to Robust Filtering
Robust statistics deals with deviations from ideal models and develops statistical procedures which are still reliable and reasonably efficient in a small neighborhood of the model.
We first review some fundamental ideas developed in robust statistics which can be used to construct robust statistical procedures in fairly general settings.
We then adapt these ideas to filtering methods, which are powerful tools to estimate the hidden state of a state-space model, by defining a concept of robustness for a filter and by proposing robust filters which provide accurate state and parameter inference in the presence of model misspecifications.
Joint work with L. Calvet and V. Czellar.
Thursday, October 08, 2015, 16:15, room 17
Harald Helfgott (Göttingen / CNRS)
The ternary Goldbach conjecture
The ternary Goldbach conjecture (1742) asserts that every odd number greater than 5 can be written as the sum of three prime numbers. Following the pioneering work of Hardy and Littlewood, Vinogradov proved (1937) that every odd number larger than a constant C satisfies the conjecture. In the years since then, there has been a succession of results reducing C, but only to levels much too high for a verification by computer up to C to be possible (C>10^(1300)). My work proves the conjecture. We will go over the main ideas in the proof.
Tuesday, September 29, 2015, 16:15, room 17
Gerhard Wanner (Université de Genève)
Multiplicateurs de Lagrange et mécanique de Lagrange
The talk explains how
- A work of Archimedes (-287 to -212, 2300 anni fà),
- A thick book on statics (Varignon 1725)
- a letter by Johann Bernoulli to Varignon (1715, 300 anni fà)
- Euler's Methodus (1744, on variational calculus),
- and d'Alembert's Dynamique from 1743,
led to the famous Mécanique analytique (1788, 1811) by Lagrange, in which, in the first part, the advantage of the methods of multipliers is demonstrated at many examples and, in the second part, the equations of Lagrange dynamics are derived from the principle of least action.
In the last part of the talk we show the connection of the ideas of Euler and Lagrange with problems of optimal control (Carathéodory, Pontryagin).
Thursday, September 17, 2015, 16:15, room 17
Matthias Christandl (University of Copenhagen)
From Pauli's Principle to Fermionic Entanglement
The Pauli exclusion principle is a constraint on the natural occupation numbers of fermionic states. It has been suspected for decades, and only proved very recently, that there is a multitude of further constraints on these numbers, generalizing the Pauli principle. Surprisingly, these constraints are linear: they cut out a geometric object known as a polytope. This is a beautiful mathematical result, but are there systems whose physics is governed by these constraints?
In order to address this question, we studied a system of a few fermions connected by springs. As we varied the spring constant, the occupation numbers moved within the polytope. The path they traced hugs very close to the boundary of the polytope, suggesting that the generalized constraints affect the system. I will mention the implications of these findings for the structure of few-fermion ground states and then discuss the relation between the geometry of the polytope and different types of fermionic entanglement.
Thursday, April 16, 2015, 16:15, room 17
Vaughan Jones (University C Berkeley)
Block spin renormalisation, subfactors and the Thompson groups F and T
In a "naive'' attempt to create algebraic quantum field theories on the circle, we obtain a family of unitary representations of Thompson's groups T and F for any subfactor. The Thompson group elements are the "local scale transformations'' of the theory.
In a simple case, the coefficients of the representations are polynomial invariants of links. We show that all links arise and introduce new "oriented'' subgroups of Thompson's groups T and F, which allow us to produce all oriented knots and links.
Thursday, March 19, 2015, 16:15, room 17
Martin Hairer (University of Warwick)
Weak universality of the KPZ equation
The KPZ equation is a popular model of one-dimensional interface propagation. From heuristic consideration, it is expected to be "universal" in the sense that any "weakly asymmetric" or "weakly noisy" microscopic model of interface propagation should converge to it if one sends the asymmetry (resp. noise) to zero and simultaneously looks at the interface at a suitable large scale. The only microscopic models for which this has been proven so far all exhibit very particular that allow to perform a microscopic equivalent to the Cole-Hopf transform. The main bottleneck for generalisations to larger classes of models was that until recently it was not even clear what it actually means to solve the equation, other than via the Cole-Hopf transform. In this talk, we will see that there exists a rather large class of continuous models of interface propagation for which convergence to KPZ can be proven rigorously. The main tool for both the proof of convergence and the identification of the limit is the recently developed theory of regularity structures, but with an interesting twist.
Wednesday, March 18, 2015, 16:15, room Villa Battelle
Wesley Pegden (Carnegie Mellon University)
The Apollonian structure of the Abelian sandpile
The Abelian sandpile is a discrete diffusion process on
configurations of chips on the integer lattice, introduced in
1987 by Bak, Tang and Wiesenfeld. We will discuss recent
developments which connect the sandpile process with
Apollonian circle packings, and provide a mathematical
explanation for the fractal structures produced by this process.
Thursday, March 05, 2015, 16:15, room 17
Marco Gualtieri (University of Toronto)
Generalized geometry and its impact on physics
I will give a tour of the subject of generalized geometry, a subject introduced by Hitchin in 2002, with a special focus on generalized complex geometry, a kind of geometric structure which interpolates between complex manifolds and symplectic manifolds. It has been particularly useful for understanding mysterious geometric structures occurring in the physics of nonlinear sigma models. I will also describe how the subject has inspired advances in Dirac and Poisson geometry. Finally, I will outline a number of new developments and outstanding questions for the future.
Thursday, December 18, 2014, 16:15, room 17
Cristophe Garban (ENS Lyon)
Near-critical percolation and minimal spanning tree in the plane
Sample N points uniformly in a unit square of the plane. Among all the spanning trees which cover these N points, consider the tree with minimal euclidean length, called the Minimal Spanning Tree. In this talk, I will explain how to construct a continuous tree embedded in the plane which should be the scaling limit as N goes to infinity of the above tree. With such a Poissonian way of defining a planar minimal spanning tree, this limiting behaviour still remains conjectural, but if one considers instead the analogous model on a well-chosen planar graph, much more can be done: in a joint work started long ago with Gabor Pete and Oded Schramm and completed only recently, we prove that this continuous tree of a new kind is the scaling limit of the minimal spanning tree defined on the triangular lattice. Standard universality arguments suggest that the limiting tree should not depend on the microscopic structure of the model.
Wednesday, December 17, 2014, 16:15, room Villa Battelle
Dario Martelli (King's College, London)
Mathematical structures arising in string theory and the gauge/gravity duality
String theory has developed alongside different areas of modern mathematics. The gauge/gravity duality has strengthened this fruitful relationship, inspiring mathematical results across geometry and gauge theories. In this talk I will illustrate some results motivated by string theory and the gauge/gravity duality, with independent mathematical interest.
Thursday, December 11, 2014, 16:15, room 17
Alessio Figalli (University of Texas)
A transportation approach to random matrices
Optimal transport theory is an efficient tool to construct change of variables between probability densities. However, when it comes to the regularity of these maps, one cannot hope to obtain regularity estimates that are uniform with respect to the dimension except in some very special cases (for instance, between uniformly log-concave densities). In random matrix theory the densities involved (modeling the distribution of the eigenvalues) are pretty singular, so it seems hopeless to apply optimal transport theory in this context. However, ideas coming from optimal transport can still be used to construct approximate transport maps (i.e., maps which send a density onto another up to a small error) which enjoy regularity estimates that are uniform in the dimension. Such maps can then be used to show universality results for the distribution of eigenvalues in random matrices. The aim of this talk is to give a self-contained presentation of these results.
Wednesday, December 10, 2014, 16:15, room Villa Battelle
Vladimir Fock (Université de Strasbourg)
Flag configurations and integrable systems
The space of configurations of n-tuples of complete flags in a
finite-dimensional vector space admit a simple parameterisation and
serve as a building block for constructing canonical (cluster)
coordinates of such manifolds as character varieties, simple Lie groupes
and many others. We will show that a generalisation of this construction
for flags in infinite dimensions, but invariant with respect to two
commuting linear maps allows to describe the space of pairs (planar
algebraic curve, a line bundle on it) which is the phase space of many
known (and unknown) integrable systems and allow to simplify the study
of their properties.
Thursday, November 20, 2014, 16:15, room Villa Battelle
Emmanuel Kowalski (ETH-Zürich)
Geometry and probability of exponential sums
Exponential sums (and oscillatory integrals) play a very important role in analytic number theory. These sums and the problems that they raise have many fascinating aspects. The talk will explain some of these, focusing especially on geometric and probabilistic aspects.
Thursday, November 06, 2014, 16:15, room 17
Anatoly Libgober (University of Illinois at Chicago)
Elliptic genera in algebraic geometry
Elliptic genus of algebraic varieties is a topological invariant which is a modular form. It extends several classical numerical invariants including euler characteristic. I will discuss properties and significance of this invariant as well applications to study singularities, mirror symmetry and group theory.
Thursday, October 02, 2014, 16:15, room 17
Leonid Parnovski (University College London)
Multidimensional periodic and almost-periodic problems
I will make a survey of the recent developments in the spectral theory of multidimentional periodic and almost-periodic differential operators, concentrating mostly on Schroedinger type operators. In particular, I will discuss the Bethe-Sommerfeld Conjecture and the asymptotic expansion of the Integrated Density of States and other spectral characteristics.
Thursday, September 18, 2014, 16:15, room 17
Robin Pemantle (University of Pennsylvania)
Analytic Combinatorics in Several Variables: estimating coefficients of multivariate rational power series
The analytic framework for estimating coefficients of a generating function is the same in many variables as in one variable: evaluate Cauchy's integral by manipulating the contour into a "standard" position. That being said, the geometry when dealing with several complex variables can be much more complicated.
This talk surveys analytic methods for extracting asymptotics from multivariate generating functions. I will try to give an idea of the main pieces of the puzzle. At the logarithmic level, the rate of growth/decay can often be learned from the amoeba. Sharp asymptotics require an inverse Fourier transform. I will try to explain in pictures the roles of Morse theory, complex algebraic geometry and hyperbolicity in the evaluation of the integrals that arise.
Thursday, May 22, 2014, 16:15, room 17
Ivan Corwin (Clay Mathematics Institute, Columbia University, Institute Henri Poincare, MIT)
Integrable probability: beyond the Gaussian universality class
Methods originating in representation theory and integrable systems have led to detailed descriptions of new non-Gaussian statistical universality classes. This talk will focus on some of the probabilistic systems (ASEP, q-TASEP, the O'Connell-Yor polymer, and the KPZ equation) and methods (Schur / Macdonald processes and quantum integrable systems) which have played a prominent role in this story.
Thursday, May 15, 2014, 16:15, room 17
Lloyd N. Trefethen (Oxford University)
Six myths of polynomial interpolation and quadrature
Computation with polynomials is a powerful tool for all kinds of problems, but the subject has been held back by longstanding confusion on a number of points. In this talk I will focus on six widespread misconceptions. In each case I will explain how the myth took hold -- for all of them contain some truth -- and then give theorems and numerical demonstrations to explain why it is nevertheless mostly false. Along the way we will plot a polynomial interpolant of degree one million.
Thursday, March 27, 2014, 16:15, room 17
Mikhail Katsnelson (Radboud University Nimegen, The Netherlands)
Theory of graphene: CERN on the desk
Graphene, a recently (2004) discovered two-dimensional allotrope of carbon (this discovery was awarded a Nobel Prize in Physics in 2010), has initiated a huge activity in physics, chemistry and materials science, mainly, for three reasons. First, a peculiar character of charge carriers in this material makes it a “CERN on the desk”, allowing us to simulate subtle and difficult to achieve effects of high-energy physics. Second, it is the simplest possible membrane, an ideal testbed for statistical physics in two dimensions. Last, but not least, being the first truly two-dimensional material (just one atom thick) it promises brilliant perspectives for the next generation of electronics, which uses mainly only surface of materials.
In this talk, I will discuss the first aspect of graphene physics: some unexpected relations between materials science, quantum field theory and high-energy physics.
Electrons and holes in this material have properties similar to ultrarelativistic particles (a two-dimensional analog of massless Dirac fermions). This leads to some unusual and even counterintuitive phenomena, such as finite conductivity in the limit of zero charge carrier concentration (quantum transport by evanescent waves) or transmission of electrons through high and broad potential barriers with a high probability (Klein tunneling). This allows us to study subtle effects of relativistic quantum mechanics and quantum field theory in condensed-matter experiments, without accelerators and colliders. Some of these effects were considered as practically unreachable. Apart from the Klein tunneling, this is, for example, a vacuum reconstruction near supercritical charges predicted many years ago for collisions of ultra-heavy ions and recently experimentally discovered for graphene.
A huge recent progress in the sample quality makes many-body effects in electron spectrum of graphene near neutrality point observable. I will discuss various aspects of many-body theory of graphene such as realistic calculations of effective electron-electron interactions, possible exciton instability in freely suspended graphene (including the results of quantum Monte Carlo simulations), and defect-induced magnetism.
Thursday, March 06, 2014, 16:15, room 17
Martin Gander (UniGe)
Basic Notions: Euler, Lagrange, Ritz, Galerkin, Courant, Clough: On the Road to the Finite Element Method
The so-called Ritz-Galerkin method is one of the most fundamental tools of modern computing. Its origins lie in the variational calculus of Euler-Lagrange and in the thesis of Walther Ritz, who died a bit over 100 years ago at the age of 31 after a long struggle against tuberculosis. The thesis was submitted in 1902 in Goettingen, in a period of dramatic developments in Physics. Ritz tried to explain the phenomenon of Balmer series in spectroscopy, using eigenvalue problems
of partial differential equations on rectangular domains. While the physics of the model quickly turned out to be completely obsolete, his mathematics enabled him later to solve difficult problems in applied sciences. He thereby revolutionized the variational calculus and became one of the fathers of modern computational mathematics.
The Ritz method was immediately recognized by Russian mathematicians as a fundamental contribution, and put to use in the computational simulation of beams and plates, which led to the seminal paper of Galerkin in 1915. In Western Europe, however, especially in the mathematical center of that time, in Goettingen, it received very little attention, even though Ritz obtained a prize from the French Academy of Sciences, after having lost in the official competition for the Vaillant prize to Hadamard in 1907. It was only during the second world war, long after Ritz's death, in an address of Courant in front of the AMS, that the potential of Ritz's invention was fully recognized, and Courant presented what we now call the finite element method. This name was given to the method after Clough, working for Boeing, reinvented it in a seminal paper.
We will see in this talk that the path leading to modern computational methods and theory was a long struggle over three centuries, requiring the efforts of many great mathematicians.
Thursday, February 27, 2014, 16:15, room 17
Benny Sudakov (ETH)
Induced Matchings, Arithmetic Progressions and Communication
Extremal Combinatorics is one of the central branches of discrete mathematics which deals with the problem of estimating the maximum possible size of a combinatorial structure which satisfies certain restrictions. Often, such problems have applications to other areas, including Theoretical Computer Science, Additive Number Theory and Information Theory. In this talk, we will illustrate this fact by several closely related examples, focusing on a recent work with Alon and Moitra.
Thursday, November 28, 2013, 16:15, room 17
Pierre Pansu (Université Paris-Sud, Orsay)
Negative curvature pinching
A Riemannian manifold is q-pinched if its sectional curvatures are between q and 1. Closed Riemannian manifolds with curvatures close to 1 are spheres; more precisely, if such a manifold is more than 1/4-pinched, then it is a sphere. This bound is sharp: complex, quaternionic and octonionic projective spaces are 1/4-pinched. In this talk we discuss the analogous questions for the case of negative curvature. Complex, quaternionic and octonionic hyperbolic spaces have sectional curvatures between -1/4 and -1; we say they are -1/4 pinched. Do they admit metrics with pinching closer to -1? Avatars of this question have been studied using various tools : harmonic maps, Hodge theory, and Lp-cohomology.
Thursday, November 07, 2013, 16:15, room 17
Laure Saint-Raymond (ENS, Paris)
From Newton's dynamics to the heat equation
The goal of this lecture is to show how the Brownian motion can be derived rigorously from a deterministic system of hard spheres in the limit where the number of particles tends to infinity, and their diameter simultaneously converges to 0.
As suggested by Hilbert in his sixth problem, we will use the linear Boltzmann equation as an intermediate level of description for the dynamics of one tagged particle.
We will discuss especially the origin of irreversibility, which is a fundamental feature of both the Brownian motion and the Boltzmann equation, and which has no counterpart at the microscopic level.
Thursday, October 17, 2013, 16:15, room 17
David Cimasoni (UniGe)
Basic Notions: Ce que l'homologie peut faire
Le but de cet exposé est de partir de la théorie de l'homologie sous sa forme axiomatique, sans en démontrer l'existence, et d'en déduire un maximum de résultats topologiques: théorème du point fixe de Brouwer, invariance de la dimension, sphères poilues, algèbres à division, théorème du sandwich,...
Aucune connaissance préalable en topologie algébrique n'est requise.
Thursday, October 10, 2013, 16:15, room 17
Walter Gautschi (Purdue University)
Numerical integration: How to deal with singularities?
Integrating functions by means of a quadrature rule becomes problematic if the function has singularities close to the interval of integration. The closer they are, the worse the results. The problem can be resolved by suitably modifying the quadrature rule. This will be discussed in the case of Gaussian quadrature and illustrated by a number of examples, some of interest in quantum chemistry and solid state physics.
Thursday, May 30, 2013, 16:15, room 17
Francis Brown (IHES, Paris)
Multiple zeta values and rational associators
I will give an overview of some recent results and open conjectures on multiple zeta values, and explain how to define rational Drinfel'd associators as a consequence.
Thursday, May 23, 2013, 16:15, room 17
Mark Pollicott (Warwick)
Linkages and their behaviour
The study of mechanical linkages is a very classical subject, dating back to the Industrial Revolution. In this talk, we discuss the geometry of the configuration spaces in some simple idealized examples, in particular, their curvature and geometry. This leads to an interesting quantitative description of their dynamical behaviour.
Thursday, April 25, 2013, 16:15, room 17
Alexei Bondal (Steklov Institute, Moscow / Tokyo)
Mutually unbiased bases
Mutually unbiased bases are mathematical objects of basic importance in quantum information theory. We will explain how their study is motivated in quantum coding and in experiments of quantum teleportation.
Then we will discuss various aspects of the algebraic structure of mutually unbiased bases, their relations to representation theory of (certain quotients of) Hecke algebras of graphs, to discrete analysis of local systems on graphs, and to topology and algebraic geometry.
Wednesday, March 20, 2013, 16:15, room 17
Victor Kac (MIT)
An algebraic approach to integrable systems
TBA
Thursday, February 28, 2013, 16:15, room 17
Anders Karlsson (Université de Genève)
The magic of Poisson summation
In this talk I will describe the classical Poisson summation formula, dating from the early 19th century. This formula has been an important tool in many areas of mathematics, and has had a large number of extraordinary applications. I will describe some of these applications, including the Basel problem, quadratic reciprocity, and counting the number of ways an integer may be expressed as a sum of squares.
Time permitting, I will also give a brief indication of the wider context of so-called trace formulas, where a similar principle is at work.
The talk will be accessible to master students.
Wednesday, December 19, 2012, 16:15, room 17
Philippe Michel (EPFL)
Trace weights over the primes
For a prime p, consider the rational fractions {n^2/p, n=1,…,p} modulo 1. A consequence of Gauss's estimate for the Gauss sums is that, as p grows, this set of fractions becomes uniformly distributed modulo 1.
A natural question in analytic number theory is whether such statement persist when one restricts to fractions of the form q^2/p, for q ranging over the primes less that p. This is indeed the case and one can even replace the polynomial P(X)=X^2 by any rational fraction P(X)/Q(X) which is not a polynomial of degree < 2.
Such a statement, in fact, generalizes to a wide class of functions on Z/pZ, which we call "trace weights". These arise as "Frobenius traces" of "l-adic sheaves" on the affine line over F_p. It follows from Deligne's purity theorem, that these functions satisfy very pretty orthogonality properties, and as a consequence, look very much like "random" functions as p gets large. In particular, one can easily evaluate their Gowers norms.
Another consequence is that these functions do not correlate with all sorts of "arithmetic sequences" like Fourier coefficients of modular forms, the Moebius function or the primes.
We will explain this circle of ideas which are joint investigations with E. Fouvry and E. Kowalski
Notice that such concepts as "Frobenius traces", "l-adic sheaves" or "Deligne's purity theorem" will only be used as blackboxes and absolutely no knowledge of these or l-adic cohomology is required for this talk.
Thursday, December 06, 2012, 16:15, room 17
Katrin Wendland (Freiburg University)
A binary view on Kummer K3s and M24-Moonshine
Kummer K3s are examples of certain complex surfaces, the K3 surfaces,
which were originally discovered in physics and which have fascinated
mathematicians for almost two centuries. Quite recently, physicists have
observed the phenomenon of M24-Moonshine, which appears to relate
a geometric invariant of K3 surfaces, the elliptic genus, to a large, finite
group, the Mathieu group M24.
In this talk, we give an overview on the geometry of Kummer K3s and the
role of the Mathieu group M24 in describing their symmetries, using an
approach based on the binary field F2, the field with two elements.
Thursday, November 15, 2012, 16:15, room 17
Hugo Duminil-Copin (Université de Genève)
Phase transition and sharp threshold in planar statistical physics
In this talk, we will describe the concept of phase transition by giving several classical examples of planar lattice models of statistical physics. We will begin with demonstrating the connection between famous models such as Ising, Potts and percolation via coupling techniques. This will enable us to relate the phase transition in these different models. Then, we will justify the existence of such a phase transition. The key ingredients will be two sharp threshold theorems by Kahn-Kalai-Linial and Bourgain-Kahn-Kalai-Katznelson-Linial for Boolean functions.
The talk will be accessible to general mathematical audience.
Thursday, October 25, 2012, 16:15, room 17
Tobias Ekholm (Uppsala)
Lagrangian immersions with a single double point
We discuss exact Lagrangian immersions of a closed manifold M into complex n-space. By a classical result of Gromov such immersions cannot be injective and thus has at least one double point. We consider the simplest case of immersions with exactly one double point where rigidity and flexibility phenomena live side by side. For example, if n is even, n>4 and the Euler characteristic of M is not equal to -2 then M is necessarily diffeomorphic to the standard sphere. If on the other hand n is odd then any homotopy sphere admits a Lagrangian immersion with a single double point, but if a certain Maslov index of the double point is not equal to 1 then the homotopy sphere must bound a parallelizable manifold. The talk discusses joint work with Eliashberg, Murphy, and Smith.
Thursday, September 27, 2012, 16:15, room Villa Battelle
Yuri Tschinkel (Courant Institute, New York)
On the arithmetic of surfaces
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Thursday, March 22, 2012, 16:15, room 17
Anton Alekseev (UniGe)
Eigenvalues, interlacing inequalities and planar networks
The same set of inequalities comes up in two seemingly different problems. The first setup is the interlacing inequalities satisfied by the (generalized) eigenvalues of an n by n Hermitian matrix. By the classical result of Guillemin-Sternberg, they define a completely integrable system (named after Gelfand-Zeitlin who discovered it in the context of Representation Theory of unitary groups). The second setup is Boltzmann weights associated to (multi-) paths on a planar network with n sources and n sinks.In the talk, we explain the relation between the two theories.As an application, we present a new description of the Horn cone (spanned by eigenvalues of triples of Hermitian matrices adding up to zero).This is a joint work with M. Podkopaeva and A. Szenes.
Thursday, December 15, 2011, 16:15, room 17
Laszlo Erdos (University of Munich)
Universality of local spectral statistics of random matrices
The Wigner-Gaudin-Mehta-Dyson conjecture asserts that the local eigenvalue statistics of large random matrices exhibit universal behavior depending only on the symmetry class of the matrix ensemble.For invariant matrix models, the eigenvalue distributions are given by a statistical mechanical model with logarithmic interaction.Specific values of the inverse temperature $\beta = 1, 2, 4$ correspond to the orthogonal, unitary and symplectic ensembles.For $\beta \not \in \{1, 2, 4\}$, there is no matrix model behind this model, but the statistical physics interpretation of the log-gas is still valid for all $\beta > 0$ and universality holds.We demonstrate that the strong local ergodicity of the Dyson Brownian motion is the the intrinsic mechanism behind universality for both invariant and non-invariant ensembles.This is a joint work with H.-T. Yau, B. Schlein, J. Yin and Paul Bourgade.
Thursday, December 08, 2011, 16:15, room 17
Jorgen Ellegaard Andersen (Aarhus University)
Moduli space techniques in RNA and protein folding
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Thursday, November 17, 2011, 16:15, room 17
Wendelin Werner (Université Paris-Sud, Orsay)
Describing surface fluctuations
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Friday, November 11, 2011, 11:15, room 17
Balint Toth (Budapest University of Technology)
Scaling limits for self-repelling random walks and diffusions with long memory
Diffusion is a ubiquitous phenomenon in nature, but a satisfactory mathematical theory explaining it from microscopic principles of particle dynamics is a major open question of mathematically rigorous statistical physics. Standard „classroom” proofs heavily rely on loss of memory effect, i.e. on the assumed Markov property of the displacement process of the diffusing particle. In most "real life" cases, however, this assumption is simply wrong. In low dimensions it can happen that the naive approach leads to wrong results: so-called anomalous diffusion may occur, where the rate of the diffusion is different from what the naive approach would predict. Even in higher dimensions, where normal diffusive scaling is expected to hold, due to the long memory serious principial and technical difficulties arise.
I will survey recent results on scaling limits of a typical class of models where such anomalous behaviour occurs: self-repelling random walks and diffusions which are locally pushed towards domains less visited in the past. The typical examples are the so called 'true (or myopic) self-avoiding walk' or the 'self repelling Brownian polymer process' -- processes which got some notoriety in the theoretical physics and probabilistic context. It is proved that in three and more dimensions the processes scale diffusively (as predicted by the memoryless theory), in two dimensions (this is the the critical dimension of the phenomenon) multiplicative logarithmic corrections are valid, while in 1d the scaling order is robustly different from the simple random walk case.
Thursday, October 20, 2011, 16:15, room 17
Igor Pak (UCLA)
Finite tilings
Suppose we are given a finite set of tiles (think polyominoes). Can one use copies of these tiles to tile a given region? This problem is very hard in general, both mathematically and computationally, but special cases such as domino tilings are beautiful and well understood, with connections and applications from probability to commutative algebra to graph theory. The pioneer works by Conway, Lagarias, and Thurston, showed that for simply connected regions, the tileability problem is related and sometimes can be completely resolved via combinatorial group theory. In this talk I will give a broad survey of this approach and the state of art generally. I will also mention a few of my own recent results, notably a new hardness result (joint work with Jed Yang), and finish with some open problems.
Thursday, September 15, 2011, 16:15, room 17
Shahar Mozes (Hebrew University / ETHZ)
Stationary measures, stiffness and equidistribution on the torus
In a joint work with Jean Bourgain, Alex Furman and Elon Lindenstrauss we study stationary measures and equidistribution on the $d$-dimensional torus.
Let $\nu$ be a probability measure supported on a (finite) subset of $SL_d(\mathbb{Z})$. We show that if the group generated by the support of $\nu$ is large enough, in particular if this group is Zariski dense in $SL_d$, for any irrational point $x$ in the torus the probability measures $\nu^{*n} * \delta_x$ tend to the uniform measure on $\T^d$.
If in addition $x$ is Diophantine generic, we show this convergence is exponentially fast.
Thursday, May 12, 2011, 16:15, room 17
Oleg Viro (Stony Brook)
Dequantization of fields and hyperfields
The notion of hyperfield is a straightforward generalization of the notion of field. The difference between them is that addition in a hyperfield may be multivalued. Hyperfields with multivalued addition can be obtained from usual fields via factorization or degeneration. Hyperfields provide a natural base for tropical geometry. The degenerations (dequanitaztion) relating objects of the classical algebraic geometry to objects studied in the tropical geometry are induced by degenerations (dequantizations) of the ground fields to hyperfields. Hyperfields appear naturally in other contexts and are of general interest.
Thursday, April 14, 2011, 16:15, room 17
Francis Bonahon (Université de Californie Méridionale)
Variétés des caractères, et crochets de Kauffman
Je parlerai de deux notions qui sont apparemment très différentes. L'une est la variété des caractères formée des homomorphismes du groupe fondamental d'une surface dans un groupe de Lie G. L'autre est le crochet de Kauffman, qui est un invariant de nœuds et entrelacs dans l'espace et est une variante du polynôme de Jones. Touraev a montré que, quand G est SL_2(C), la variété des caractères peut être quantifiée par une généralisation du crochet de Kauffman aux surfaces. Je présenterai divers résultats et conjectures sur la classification des crochets de Kauffman sur une surface donnée, et décrirai une famille d'exemples particulièrement intéressants.
Thursday, March 17, 2011, 16:15, room 17
Hugo Duminil (UniGe)
Les travaux d'un médaillé Fields : Stanislav Smirnov
Dans cet exposé, nous présenterons une partie des travaux de Stanislav Smirnov. Nous nous concentrerons sur l'étude des modèles de physique statistique bi-dimensionels.
Smirnov s'est illustré en prouvant l'invariance conforme des deux plus connus d'entre eux: la percolation et le modèle d'Ising.
Nous expliquerons ce que ces résultats signifient concrètement et ce qu'ils impliquent en mathématiques et en physique. Nous introduirons également brièvement la notion d'analyticité discrète. Ce sujet (encore mal compris) est au coeur des preuves des théorèmes précédents. À ce titre, les travaux de Smirnov sont fondamentaux dans l'optique d'une meilleure compréhension de ce domaine.
Thursday, December 16, 2010, 16:15, room 17
Frank Kutzschebauch (University of Bern)
A solution to the Gromov-Vaserstein Problem
Any matrix in $Sl_n (\C)$ can (due to the Gauss elimination process) be written as a product of elementary matrices. If instead of the complex numbers (a field) the entries in the matrix are elements of a ring, this becomes a delicate question. In particular the rings of maps from a space $X \to \C$ are interesting cases. A deep result of Suslin gives an affirmative answer for the polynomial ring in $m$ variables in case the size of the matrix ($n$) is greater than 2. In the topological category the problem was solved by Thurston and Vaserstein. For holomorphic functions on $\C^m$ the problem was posed by Gromov in the 1980's. We report on a complete solution to Gromov's problem. A main tool is the Oka-Grauert-Gromov-h-principle in Complex Analysis. This is joint work with Björn Ivarsson.
Thursday, December 02, 2010, 16:15, room 17
Jean Pierre Eckmann (UniGe)
On some work of Cédric Villani
This semester there will be a few lectures explaining the works of the new Fields medalists to anyone with at least 3-4 years of university studies in mathematics. This talk is about the work of Cédric Villani, who has worked in problems of analysis. Given the breadth of his work, I will concentrate on hypoellipticity and hypocoercivity, which are closest to what I understand. These methods allow to study equilibrium states and the rates of convergence to them. I will mention some of the long-standing open problems he solved and will try to give an elementary glimpse on the basic ideas involved.
Thursday, November 25, 2010, 16:15, room 17
Markus Reineke (University of Wuppertal)
Simultaneous conjugacy of matrices
The classical unsolved linear algebra problem of describing tuples of matrices up to simultaneous conjugacy admits a geometric formulation as a moduli problem. We will review classical and recent results on the geometry of these moduli spaces and discuss relations to q-hypergeometric series, (quantized) Donaldson-Thomas type invariants and Higgs moduli.
Thursday, November 11, 2010, 16:15, room 17
Edward Frenkel (UC Berkeley and Fondation Sciences Mathématiques de Paris)
Geometrization of Trace Formulas
Trace formula is a powerful tool in the study of automorphic representations of reductive algebraic groups defined over number fields and the fields of functions of curves over finite fields. I will outline a conjectural framework of "geometric trace formulas" in the case that the curve is defined over the complex field, which exploits the categorical version of the geometric Langlands correspondence. This is joint work with Robert Langlands and Ngo Bao Chau (arXiv:1003.4578, arXiv:1004.5323).
Thursday, June 03, 2010, 16:15, room 17
Gautam Chinta (The City College of New York, CUNY)
Sums of two squares and three squares
Gauss established a formula for the number of ways to write a positive integer as a sum of three squares. We will recall this result, reinterpret it and present a generalization in the context of Eisenstein series on the group GL(3,Z). This is a joint work with Omer Offen.
Thursday, May 27, 2010, 16:15, room 17
Naichung Conan Leung (CUHK Hong Kong)
Geometry of special holonomy
In this talk, I will survey my recent work on the subject of geometry of special holonomy. I will give a unified description of all these geometries in term of normed division algebras. Another closely related description is also explained by using vector cross product. We also applied these approaches to obtain new results in these geometries.
Thursday, May 20, 2010, 16:15, room Villa Battelle
Jean-Pierre Demailly (Institut Fourrier Grenoble)
Inégalités de Morse holomorphes et cohomologie asymptotique
Champs magnétiques et inégalités de Morse pour la d''-cohomologie
Holomorphic Morse inequalities
Thursday, March 25, 2010, 16:15, room 17
04.03.2010
Jérôme Scherer (EPFL)
Groupes de Lie et espaces de lacets
Quelles propriétés topologiques caractérisent un groupe de Lie (compact) ? Le fait qu'il admette une multiplication conduit à définir les H-espaces et si l'on ajoute l'existence d'un inverse à homotopie près on se restreint aux espaces de lacets (finis). J'expliquerai quels problèmes on rencontre lors d'une tentative de classification de ces objets et comment on est amené à utiliser la complétion des espaces pour chaque nombre premier. A la fin de l'exposé je dirai quelques mots sur le cas non compact.
Thursday, March 04, 2010, 16:15, room 17
Nicolas Varopoulos (Paris & Genève)
A classification of Lie groups with prospects in combinatorial group theory
Already the notion of amenability for a group can be given in several equivalent but distinct ways.
- 1) Analytic: it is when we look at the decay, exponential or not, of the uniform norm of a convolution power of an appropriate measure.
- 2) Geometric: in terms of isoperimetric inequalities.
- 3) Algebraic: in terms of the Lie algebra in the case of Lie groups.
These are well-known facts. I propose a similar three-fold - Analytic-Geometric-Algebraic - classification different from the above, but very much in the same spirit, that I call the C and the NC (non-C) groups.
Just to illustrate the issue: an amenable group belongs to the NC class if the decay of the convolution powers if a measure is polynomial. If it is superpolynomial (it cannot be exponential!) we cay that it is a C-group.
This classification is done for Lie groups, where, just as for amenability, Lie algebras are considered first. The situation for discrete groups is not well understood, but it gives rise to a number of interesting problems. The main difficulty, as always for discrete groups, is that the algebra involved is far less tractable than a finite dimensional Lie algebra. The problems involved are truly fascinating, and I will make some propaganda for them.
Thursday, December 17, 2009, 16:15, room 17
Vadim Kaimanovich (Jakobs University, Bremen)
Random graphs, equivalence relations and stochastic homogenization
Random graphs arise in numerous areas of mathematics and applications. A natural new framework for their study is provided by the notion of a graphed equivalence relation (originally used in ergodic theory and operator algebras). "Stochastic homogenization" of a certain family of infinite graphs consists in finding a probability measure invariant with respect to an equivalence relation whose classes are endowed with graph structures from this family. The role of such a measure is then similar to the role of an invariant measure for a usual dynamical system or of a stationary measure for a Markov chain.
It is easy to construct stochastically homogeneous random graphs by random perturbations of Cayley graphs of discrete groups. However, there are also graphs whose origin has nothing to do with groups. In this talk we shall discuss the homogenization problem for trees and certain tree-like graphs.
Thursday, December 03, 2009, 16:15, room 17
Michael Farber (University of Durham and ETHZ)
Topology of random right angled Artin groups
The mathematical theory of random graphs was initiated by Paul Erdös and Alfréd Rényi in 1959. Nowadays it is a fast growing and well-developed branch of discrete mathematics at the crossroads with graph theory and probability theory; it has numerous applications in computer science and engineering. In the talk I will discuss properties of discrete groups and high-dimensional spaces associated to random graphs. These are random right angled Artin groups and I will be interested in their Betti numbers, cohomological dimension and topological complexity. The latter is a numerical homotopy invariant reflecting the complexity of motion planning algorithms in robotics. Our main result (joint work with A. Costa) states that the topological complexity of a random right angled Artin group assumes, with probability tending to one, at most three values, when n tends to infinity.
Thursday, November 26, 2009, 16:15, room 17
Phillipe Di Francesco (CEA Saclay)
Integrable Combinatorics
What is the link between Alternating Sign Matrices (ASM), Totally Symmetric Self-Complementary Plane Partitions (TSSCPP) and the equivariant cohomology of the variety of strictly upper triangular matrices with vanishing square?
Two-dimensional Integrable lattice models from statistical physics provide the natural framework for this missing link. We show in particular how a physical model, involving densely-packed loop configurations on an infinite surface, is connected to all three subjects above. This model turns out to be integrable, and we'll use this fact to reformulate all of the above in terms of polynomial solutions of the quantum Knizhnik-Zamolodchikov equation. Results include a proof of the Razumov-Stroganov sum rule, a new connection between ASM and TSSCPP, and the computation of the (multi)degree of the variety M^2=0. We also present generalizations to the commuting variety and to M^k=0.
Thursday, November 12, 2009, 16:15, room 17
Janos Pach (EPFL)
Separator theorems and geometric graph theory
What is the maximum number of edges that a graph with n vertices can have if it can be drawn in the plane without 3 pairwise crossing edges? This innocent looking question and its relatives raised by Avital, Erdos, Hanani, Kupitz, Perles, and others can be considered as the starting point of a new discipline: Geometric Graph Theory. We describe some basic problems and results in this area. We also show a couple of recent results, joint with Jacob Fox, illustrating how some far-reaching generalizations of the Lipton-Tarjan separator theorem for planar graphs can be used to answer extremal questions in Geometric Graph Theory.
Thursday, November 05, 2009, 16:15, room 17
Gian Michele Graf (ETHZ)
Transport in quantum devices and its geometry
This talk is about mesoscopic devices, which are driven slowly and periodically in time, and known as quantum pumps. In a first part I will present a discussion of transport properties, concerning charge, noise, and dissipation. Pump processes, which are "optimal" in this respect, may be characterized geometrically using the Hopf map. These considerations will be based on the scattering approach to pumping due to Buettiker. The system is viewed as consisting of a finite device connected to leads. It allows for scattering states at Fermi energy and is hence gapless. In a second part I will review an alternative approach due to Thouless, in which the system is idealized as being of infinite extent and gapped. The charge transported in a cycle is realized as a Chern number. It will be shown how to relate the seemingly disjoint approaches and in particular how they yield the same transported charge.
Thursday, October 08, 2009, 16:15, room 17
François Ledrappier (Université Paris VI)
Vitesse de fuite pour des revêtements de variétés compactes
Soit \tilde{M} une variété riemannienne qui est un revêtement régulier d'une variété compacte. La vitesse de fuite l et l'entropie de Kaimanovich h sont des invariants géométriques définis par les propriétés asymptotiques du mouvement brownien sur \tilde{M}. Nous discuterons ces invariants et leurs relations avec d'autres invariants. En particulier, un résultat général est que l^2 ≤ h
Thursday, October 01, 2009, 16:15, room 17
Stefano Marmi (Scuola Normale Superiore di Pisa)
Irrationality measures and stability of quasiperiodic orbits
It is customary in the theory of diophantine approximation to concentrate on the asymptotic behaviour of the accuracy of rational approximations of an irrational number. The study of quasiperiodic dynamics, and in particular of the problem of stability of quasiperiodic orbits, allows in some cases to reinterpret classical diophantine conditions in dynamical terms. Moreover it also sheds new light on the approximation problem since not only the asymptotics but also the history can be relevant.
This talk is based on joint work with Pierre Moussa and Jean-Christophe Yoccoz.
Thursday, June 04, 2009, 16:15, room 17
Mark Sapir (Vanderbilt University)
Polynomial maps over p-adics, random walks, and 1-related groups
This is a joint work with A. Borisov and I. Kozakova. We prove that almost all 1-related groups with at least 3 generators are residually (finite p-) groups and are coherent (that is all finitely generated subgroups are finitely presented). The proof involves methods from geometric group theory, algebraic geometry and the classical theory of Brownian motion.
Thursday, May 28, 2009, 16:15, room 17
Philippe Michel (EPFL)
La méthode ergodique de Linnik
Vers la fin des années 50, Yu. V. Linnik développe sa « méthode ergodique » pour obtenir (ainsi que son élève Skubenko) plusieurs résultats remarquables sur la répartition des représentations d'un entier par une forme quadratique en trois variables (par exemple les représentations d'un entier comme somme de trois carrés). Cette méthode avait plusieurs décades d'avance sur son temps et ce n'est qu'à la fin des années 80 que les résultats originaux de Linnik ont été significativement améliorés, comme conséquence de progrès fondamentaux en théorie des formes automorphes et en théorie analytique des nombres. Il faut dire aussi que cette méthode est restée largement confidentielle, la raison principale étant que le langage permettant de la formuler simplement n'existait pas à l'époque. Dans cette exposé, nous tâcherons d'exposer cette méthode en termes aussi élémentaires que possible et d'en présenter plusieurs applications. Nous montrerons qu'elle s'inscrit naturellement dans le cadre de la dynamique des espaces homogènes de rang 1 et nous présenterons des résultats récents, en rang supérieur, inspirés en partie par cette méthode. Il s'agit de travaux en commun avec M. Einsiedler, E. Lindenstrauss et A. Venkatesh.
Thursday, May 14, 2009, 16:15, room 17
Marc Burger (EPFZ)
Actions de groupes sur le cercle
Nous donnerons un panorama de résultats récents de rigidité d'actions de réseaux sur le cercle et proposerons un point de vue unificateur reposant sur la cohomologie bornée.
Thursday, May 07, 2009, 16:15, room 17
Felix Schlenk (UniNe)
Complexity of classical Hamiltonian Systems and Floer Homology
Classical Hamiltonian Systems describe dynamical processes without friction. One can therefore expect that such systems are complicated, since small oscillations never decay. A good measure for the complexity of a dynamical system is its topological entropy. One can therefore expect that the topological entropy of classical Hamiltonian Systems is positive. I will show how Floer Homology - a modern tool of symplectic geometry and Hamiltonian dynamics - can be used to confirm this expectation.
Thursday, April 30, 2009, 16:15, room 17
Geoffrey Grimmett (Cambridge)
The ferromagnet and stochastic geometry
Ising's 1925 thesis on one-dimensional ferromagnetism has since spawned a wealth of mathematics and physics. This includes two geometrical processes known as the random-cluster (or FK) model and the random-current representation. These two systems provide jointly a powerful set of tools. This will be illustrated in this talk, both for the classical Ising model, and the quantum model of Lieb, Schultz, Mattis (1961).
Thursday, March 26, 2009, 16:15, room 17
Nikita Nekrasov (IHES)
Supersymmetric vacua, quantum cohomology, and Bethe ansatz
The equivariant quantum cohomology, K-theory, and elliptic cohomology of various hyperkahler quotients are identified with the quantum integrable systems, such as spin chains or many-body systems. In particular, the cotangent bundle to the Grassmanian is mapped to the Heisenberg magnet. The spectrum of the operators of quantum multiplication is identified with the solutions of the Bethe equations. The generalized Donaldson theory on the product of a two-sphere and a Riemann surface lead to the instanton corrected Bethe equations.
Thursday, March 19, 2009, 16:15, room Salle A150, Sciences II
Alexander Molev (The University of Sydney)
Littlewood-Richardson polynomials
The classical Littlewood-Richardson coefficients are remarkable nonnegative integers which occupy a prominent place in combinatorics, representation theory and geometry. We review two versions of the original rule for their calculation then follow by a natural generalization of these coefficients called the Littlewood-Richardson polynomials and give a combinatorial rule for their calculation. Then we discuss two applications of this rule: we find the product of the Casimir elements for the general linear Lie algebra in the basis of the quantum immanants constructed by A. Okounkov and G. Olshanski. The same rule yields a positive and stable formula for the product of equivariant Schubert classes on the Grassmannian. The first positive formula for such a product was given by A. Knutson and T. Tao by using combinatorics of puzzles although the stability property was not apparent from their rule.
Thursday, March 12, 2009, 16:15, room 17
Michael Polyak (Technion)
Enumerative geometry and finite type invariants
Enumerative geometry deals with counting geometric objects (lines, curves, planes) subject to certain restrictions (passage through a collection of points, tangency to a curve, etc). I will discuss a wide class of such problems on a crossroads of a real enumerative geometry and smooth topology and explain its relation to the theory of finite type invariants. I will also present a simple approach to such problems using maps of configuration spaces and intersections. The talk does not assume any preliminary knowledge of the subject and should be easily accessible to students.
Thursday, March 05, 2009, 16:15, room 17
Edward Witten (CERN, on leave from IAS)
The problem of gauge theory
I sketch what it is supposed to mean to quantize gauge theory, and how this can be made more concrete in perturbation theory and also by starting with a finite-dimensional lattice approximation. Based on real experiments and computer simulations, quantum gauge theory in four dimensions is believed to have a mass gap. This is one of the most fundamental facts that makes the Universe the way it is. arXiv:0812.4512
Thursday, February 26, 2009, 16:15, room Auditoire Stueckelberg, Ecole de Physique
Jean-Michel Bismut (Université Paris-Sud XI, Département de Mathématiques d'Orsay)
Le Laplacien hypoelliptique
Si X une variété Riemannienne, le Laplacien hypoelliptique est un opérateur agissant sur l’espace total du fibré cotangent ou du fibré tangent de X, dépendant d’un paramètre b > 0, qui interpole de manière adéquate entre la théorie de Hodge usuelle de X, et le flot géodésique. Elle est produite à l’aide d’une théorie de Hodge exotique de l’espace total des fibrés tangents ou cotangents, par une déformation convenable des opérateurs d + d*. Le Laplacien hypoelliptique est essentiellement la somme de l’oscillateur harmonique le long des fibres et du champ de vecteurs engendrant le flot géodésique. Les motivations pour la construction de cette déformation sont multiples. En théorie de de Rham, on peut considérer le Laplacien hypoelliptique comme une version semiclassique d’une déformation de Witten du Laplacien sur l’espace des lacets, associé à la fonctionnelle d’énergie. Dans l’exposé, on donnera les motivations heuristiques pour la construction du Laplacien hypoelliptique, et on expliquera sa construction en théorie de de Rham. On exposera les résultats obtenus avec Gilles Lebeau sur l’analyse de cet opérateur. Enfin on donnera diverses applications du Laplacien hypoelliptique.
Thursday, November 20, 2008, 16:15, room 17
Pete Buhlmann (ETHZ)
Computationally Tractable Methods for High-Dimensional Data
Many applications nowadays involve high-dimensional data with p variables (or covariates), sample size n and the relation that p >> n. We focus on penalty-based estimation methods which are computationally feasible and have provable statistical and numerical properties. The Lasso (Tibshirani, 1996), an l_1-penalty method, became very popular in recent years for estimation in high-dimensional generalized linear models. Extensions to other models or data-types call for more flexible convex penalty functions, for example to handle categorical data or for improved control of smoothness in additive models. The Group-Lasso (Yuan and Lin, 2006) and a new sparsity-smoothness penalty are general and useful penalty functions for many high-dimensional models beyond GLM's. Fast coordinatewise descent algorithms can be used for solving the corresponding convex optimization problems which allow to easily deal with large dimensionality p (e.g. p ~ 10^6, n ~ 10^3). The talk includes: (i) a review of Lasso-type methods; (ii) new flexible penalty functions and fast algorithms (package "lasso"); and (iii) some illustrations for bio-molecular data.
Thursday, November 13, 2008, 16:15, room 17
Vladimir Fock (Strasbourg)
Variétés a clusters
La construction principale des variétés à clusters donne, à partir d'une matrice antisymétrisable d'entiers, deux variétés algébriques de même dimension. L'une (de type A) est munie d'une 2-forme fermée, l'autre (de type X) d'une structure de Poisson. Ces variétés sont duales l'une de l'autre en un certain sens. Les deux sont munie d'une action d'un groupe discret et la variété de type X admet une quantification canonique equivariante. Parmi les variétés cluster on trouve les groupes de Lie, les espaces de Teichmueller, les espaces des paramètres de Stokes et plusieurs autres variétés bien connues. On peut considérer la description cluster comme une structure supplémentaire sur ces variétés unifiant leur propriétés géométriques, algébriques, combinatoires, Poisson et arithmétiques.
Thursday, November 06, 2008, 16:15, room 17
Viviane Baladi (ENS Paris)
Réponse linéaire et violation de la réponse linéaire en dynamique
De nombreux systèmes dynamiques chaotiques possèdent une unique mesure "physique", décrivant les propriétés asymptotiques des orbites génériques pour la mesure de Lebesgue. Si l'on considère une famille f_t de systèmes dynamiques dépendant différentiablement d'un paramètre t, on peut se demander comment cette mesure \mu_t dépend du paramètre. D Ruelle a montré en 1997 que cette dépendance était différentiable dans le cadre hyperbolique différentiable, et il a calculé la dérivée ("formule de la réponse linéaire"). Il a ensuite suggéré que, dans un cadre assez vaste (plus forcément structurellement stable), cette dépendance pourrait rester différentiable (au sens de Whitney, si l'ensemble des bons paramètres n'est pas un voisinage) et a proposé un candidat pour la derivée, sous forme de série a priori divergente. Avec D Smania nous avons montré que dans le cadre des applications dilatantes par morceaux en dimension un, la mesure \mu_t est dérivable en 0 si et seulement si la famille f_t est tangente en 0 à la classe topologique de f_0. La valeur de la dérivée coïncide alors avec le candidat de Ruelle, resommé de façon appropriée. Si le temps le permet, nous mentionnerons aussi des conjectures et des résultats plus récents dans le cas non-uniformément hyperbolique.
Thursday, October 30, 2008, 16:15, room 17
Gerhard Wanner (UniGe)
Kepler, Newton et l'analyse numérique
Tous les collègues, étudiants, enseignants aux collèges, anciens étudiants de Gerhard Wanner sont cordialement invites. A la suite de ce colloque, la Section de Mathématiques invite tous les participants à un apéritif en l'honneur de l'orateur. Cet apéritif sera servi à la Salle 17 de la Section de Mathématiques
Thursday, October 16, 2008, 16:30, room 1S081 Sciences III
Allen Knutson (UC San Diego)
Why do matrices commute? Algebraic geometry meets statistical mechanics
The matrix equations M^2 = 0 are quadratic, so to derive the linear equation Trace(M)=0 from them requires nonalgebraic operations. Are there corresponding "surprising" equations implied by the matrix equation XY=YX? This question was posed in the '60s, and still nobody knows. Even the (normalized) volume of this space {(X,Y) : XY=YX} is very difficult to compute for large matrices, and until recently was only known to start 1,3,31,1145. I'll talk about a bunch of related spaces of matrices, some of which are probably harder and some easier to understand than the commuting scheme {(X,Y) : XY=YX}, and the volumes of these spaces. Then I'll explain how physicists came up with the same set of numbers from a statistical mechanical model (making them much easier to compute), and why they are indeed the same. Some of this work is joint with Paul Zinn-Justin.
Thursday, October 09, 2008, 16:15, room 17
Vincent Beffara (ENS Lyon, UniGe)
Universalité et invariance conforme en mécanique statistique
Le but de l'exposé est de présenter quelques résultats (hélas encore trop partiels) qui vont dans ce sens.
Thursday, October 02, 2008, 16:15, room 17
Christophe Pittet (Marseille)
Géométrie et analyse dans les groupes infinis, spectre du Laplacien et profil isospectral
Une formule simple permet d'estimer la distribution spectrale du Laplacien au voisinage de zéro à partir d'inégalités isoperimétriques.
Thursday, September 25, 2008, 16:15, room 17
Marcos Mariño (UniGe)
Instantons et problèmes asymptotiques en géométrie énumérative
Le but de la géométrie énumérative est de ``compter" le nombre des courbes holomorphes dans des variétés algébriques. Une question naturelle est: quel est le comportement asymptotique de ces nombres quand le dégrée ou le genre de la courbe devient très grands? Dans ce colloque on va étudier ce problème dans deux exemples très simples. Le premier exemple vient de la théorie de l'intersection dans l'espace des moduli des surfaces de Riemann, et le deuxième est encore plus élémentaire et vient de la théorie des recouvrements de Hurwitz. Les réponses (un théorème pour le premier exemple, une conjecture pour le deuxième) son intimement lies a l'étude des phénomènes non-perturbatives (instantons) en théorie de cordes.
Thursday, September 18, 2008, 16:15, room 17
Elvezio Ronchetti (Université de Genève)
T.B.A.
T.B.A.
Thursday, October 8, 2015, 16:15, room 17
Mikhail Katsnelson (Radboud University Nimegen, The Netherlands)
Theory of graphene: CERN on the desk
Graphene, a recently (2004) discovered two-dimensional allotrope of carbon (this discovery was awarded by Nobel Prize in physics 2010), has initiated a huge activity in physics, chemistry and materials science, mainly, for three reasons. First, a peculiar character of charge carriers in this material makes it a “CERN on the desk” allowing us to simulate subtle and hardly achievable effects of high energy physics. Second, it is the simplest possible membrane, an ideal testbed for statistical physics in two dimensions. Last not least, being the first truly two-dimensional material (just one atom thick) it promises brilliant perspectives for the next generation of electronics, which uses mainly only surface of materials.
I will tell about the first aspect of graphene physics, some unexpected relations between materials science and quantum field theory and high-energy physics.
Electrons and holes in this material have properties similar to ultrarelativistic particles (two-dimensional analog of massless Dirac fermions). This leads to some unusual and even counterintuitive phenomena, such as finite conductivity in the limit of zero charge carrier concentration (quantum transport by evanescent waves) or transmission of electrons through high and broad potential barriers with a high probability (Klein tunneling). This allows us to study subtle effects of relativistic quantum mechanics and quantum field theory in condensed-matter experiments, without accelerators and colliders. Some of these effects were considered as practically unreachable. Apart from the Klein tunneling, this is, for example, a vacuum
reconstruction near supercritical charges predicted many years ago for collisions of ultra-heavy ions and recently experimentally discovered for graphene.
A huge recent progress in the sample quality makes many-body effects in electron spectrum of graphene near neutrality point observable. I will discuss various aspects of many-body theory of graphene such as realistic calculations of effective electron-electron interactions, possible exciton instability in freely suspended graphene (including the results of quantum Monte Carlo simulations), and defect-induced magnetism.
Thursday, March 6, 2014, 16:15, room 17
Pierre Pansu (Université Paris-Sud, Orsay)
TBA
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Thursday, November 7, 2013, 16:15, room 17
David Cimasoni (UniGe)
Basic Notions: Ce que l'homologie peut faire
Le but de cet exposé est de partir de la théorie de l'homologie
sous sa forme axiomatique, sans en démontrer l'existence, et d'en
déduire un maximum de résultats topologiques:
théorème du point fixe de Brouwer,
invariance de la dimension, sphères poilues, algèbres à
division, théorème du sandwich,...
Aucune connaissance préalable en topologie algébrique n'est requise.
Thursday, October 3, 2013, 16:15, room 17
Philippe Michel (EPFL)
Trace weights over the primes
For a prime p, consider the rational fractions {n^2/p, n=1,…,p} modulo 1.
A consequence of Gauss's estimate for the Gauss sums is that, as p grows,
this set of fractions becomes uniformly distributed modulo 1.
A natural question in analytic number theory is whether such statement persist
when one restricts to fractions of the form q^2/p, for q ranging over the primes less that p. This is indeed the case and one can even replace the polynomial P(X)=X^2
by any rational fraction P(X)/Q(X) which is not a polynomial of degree <2.
Such a statement, in fact, generalizes to a wide class of functions on Z/pZ, which
we call "trace weights". These arise as "Frobenius traces"
of "l-adic sheaves" on the affine line over F_p. It follows from Deligne's purity theorem, that these functions satisfy very pretty orthogonality properties,
and as a consequence, look very much like "random" functions as p gets large. In particular, one can easily evaluate their Gowers norms.
Another consequence is that these functions do not correlate with all sorts of "arithmetic sequences" like, Fourier coefficients of modular forms,
the Moebius function or the primes.
We will explain this circle of ideas which are joint investigations with E. Fouvry and E. Kowalski
Notice that such concepts as "Frobenius traces", "l-adic sheaves" or "Deligne's purity theorem" will only be used as blackboxes and absolutely no knowledge of these
or l-adic cohomology is required for this talk.
Thursday, December 6, 2012, 16:15, room 17
