
MSRI Evans Talk: Topological rigidity theorems and Homogeneous dynamics
Location: 60 Evans Hall Speakers: Hee Oh (Yale University)In the hyperbolic plane H^2, horocycles are given by Euclidean circles tangent to the boundary. In 1936, Hedlund showed that in a compact hyperbolic surface Gamma\H^2, every horocycle is dense; this is one of the first theorems on orbit closures in homogeneous spaces.
Hedlund’s approach is based on the study of minimal sets and long time behavior of horocyclic flows. This approach was significantly generalized by Margulis in his proof of Oppenheim conjecture on values of quadratic forms in 1989.
In 1991, Ratner gave a complete description of all possible orbit closures on a finite volume homogeneous space under the action of a subgroup generated by unipotent oneparameter subgroups, based on her measure classification theorem.
We will discuss certain topological rigidity theorems in infinite volume hyperbolic manifolds of dimension 2 or 3 which can be approached by generalizing the ideas of Hedlund and Margulis.
Updated on Apr 17, 2015 11:14 AM PDT
Past MSRI/Evans Lectures

MSRI Evans Talk: Dynamics on moduli spaces of flat surfaces  questions and new directions
Location: 60 Evans Hall Speakers: Maryam Mirzakhani (Stanford University)In this talk, we will discuss various basic problems regarding the topology, geometry and dynamical systems on surfaces: all these questions are closely related to the study of closed Riemann surfaces endowed with a holomorphic oneform. This is the same as having a flat metric on the surface with finitely many conetype singularities.The moduli space of holomorphic oneforms on a closed surface of genus $g$ has a natural piecewise linear structure and carries an action of GL(2,R).One can investigate the geometry and dynamics of an individual flat surface by studying its orbit under this linear group action.One famous example is how understanding a billiard path on a rational billiard is related to studying the orbits of the action of GL(2,R) on the moduli spaces of flat surfaces.Many of the results in this area are motivated by statements in homogenous dynamics. On the other hand, they have strong connections with other topics in algebraic geometry and number theory. In this talk, I will give an overview of some known results and open questions regarding these moduli spaces.Updated on Mar 30, 2015 12:00 PM PDT 
MSRI Evans Talk: Moduli of Geometric Structures
Location: 60 Evans Hall Speakers: William Goldman (University of Maryland)Given a topology S, how many ways (if any) are there of putting some kind of classical geometry on S? For example, the sphere has no compatible system of coordinates with Euclidean geometry. (There is no metrically accurate atlas of the world.) On the other hand, the 2torus admits a rich supply of Euclidean structures, which form an interesting moduli space which itself enjoys hyperbolic nonEuclidean geometry. For other geometric structures, the moduli spaces are much more complicated and are best described by a dynamical system. This talk will survey some of the interesting dynamical systems which arise for simple examples of geometries on surfaces.
Updated on Feb 18, 2015 08:58 AM PST 
MSRI Evans Talk: Dynamics and Integer Points on the Sphere
Location: 60 Evans Hall Speakers: Elon Lindenstrauss (Hebrew University)A theorem of Gauss and Legendre states that every integer n that is not congruent to 0,4 or 7 mod 8 can be presented as a sum of three relatively prime squares. In fact, we know (ineffectively) that under these conditions there are n^{1/2+o(1)} such representations of n.Rescaling the points to the unit sphere, we get as n increases more and more points on this sphere, and some 50 years ago Linnik, in a book with the intriguing name "Ergodic properties of number fields", used ergodic theory to study the distribution of these points, at least for sequences of n satisfying an auxiliary congruence condition  for instance if n is in addition assumed to be congruent to 2 (mod 3).An alternative route, using bounds on Fourier coefficients of certain automorphic forms, was given by Duke and Iwaniec. This alternative route has several advantages, e.g. does not need a congruence condition and gives much more quantitative information. But the highly original work of Linnik on the subject is still very relevant, and had motivated much recent work.I will explain why and how ergodic theory  more specifically, homogeneous dynamics  can be used to study integer points on the sphere, and time permitting present some recent work in this direction.Updated on Jan 23, 2015 10:38 AM PST 
MSRI Evans Talk: Finite dimensional Banach Spaces
Location: 60 Evans Hall, UC Berkeley Speakers: Pierre Colmez (Institut de Mathématiques de Jussieu)The algebraic closure of the field ${\bf Q}_p$ of $p$adic number is infinite dimensional over ${\bf Q}_p$, hence its completion ${\bf C}_p$ is also infinite dimensional. I will describe objects that can be thought of as finite dimensional vector spaces over ${\bf C}_p$ up to finite dimensional ${\bf Q}_p$vector spaces and explain what kind of properties they have. I will end up with some recent applications.
Updated on Nov 24, 2014 10:00 AM PST 
MSRI Evans Talk: Locally symmetric spaces and Galois representations
Location: 60 Evans Hall, UC Berkeley Speakers: Ana Caraiani (Princeton University)At the heart of the Langlands program lies the reciprocity conjecture, which can be thought of as a nonabelian generalization of class field theory. An example is the correspondence between modular forms and representations of the absolute Galois group of Q. This can be realized geometrically in the cohomology of modular curves, making essential use of their structure as algebraic curves.
In this talk, I will describe some techniques involved in the recent work of HarrisLanTaylorThorne and Scholze, who construct Galois representations associated to systems of Hecke eigenvalues occurring in the cohomology of locally symmetric spaces for GL_n. These are real manifolds which generalize modular curves, but lack the structure of algebraic varieties. I will then focus on a very specific property of these Galois representations: the image of complex conjugation, which can be identified by combining Hodge theory with padic interpolation techniques. Finally, I will mention some open problems.
Updated on Nov 20, 2014 12:24 PM PST 
Chern Lecture 4/MSRIEvans Talk: Geometry related the beyond endoscopy proposal
Location: 60 Evans Hall, UC Berkeley Speakers: Ngô Bảo Châu (University of Chicago)Certain part of Langlands's beyond endoscopy proposal can be formulated in terms of geometry of certain fibration, similar to Hitchin's. In my talk, I will discuss about its construction and new geometric problems it offers.
http://math.berkeley.edu/about/events/lectures/chern
Updated on Nov 07, 2014 11:10 AM PST 
MSRI Evans Talk: Representation theory and arithmetic invariants
Location: 60 Evans Hall, UC Berkeley Speakers: Atsushi Ichino (Kyoto University)The local Langlands correspondence (conjecture) relates (infinitedimensional) representations of padic reductive groups with (finitedimensional) Galois representations. This can be viewed in many ways, and from the viewpoint of the representation theory, this gives us a "classification" of representations of padic reductive groups by viewing arithmetic objects as "simpler" parameters. Then it will be natural to study representationtheoretic problems in terms of arithmetic invariants. In this talk, we explain some cases which can be answered in this setup: the GanGrossPrasad conjecture (a restriction problem) and formal degrees (a generalization of dimensions).
Updated on Oct 16, 2014 09:22 AM PDT 
MSRI Evans Talk: Stacks in Representation Theory. What is a continuous representation of an algebraic group?
Location: 60 Evans Hall, UC Berkeley Speakers: Joseph Bernstein (Tel Aviv University)Updated on Oct 06, 2014 01:37 PM PDT 
MSRI Evans Talk: The What, Why and How of Categorification
Location: 60 Evans Hall, UC Berkeley Speakers: Zsuzsanna Dancso (MSRI  Mathematical Sciences Research Institute)Categorification has been a central topic in the past two decades in several areas of mathematics including representation theory and low dimensional topology. In broad terms, categorification introduces richer structure by replacing algebraic structures (groups, rings, etc) by categories, in a way that lifts the important properties of the original structure.
In this talk I will explain the general idea of categorification and illustrate it with an example from lowdimensional topology: Khovanov's link homology. We will discuss advantages and applications of categorification in different settings, and open problems in the area.
Updated on Sep 16, 2014 10:29 AM PDT 
MSRI Evans Talk: The beginning of a local Langlands correspondance modulo $p$
Location: 60 Evans Hall, UC Berkeley Speakers: MarieFrance Vigneras (Université de Paris VII (Denis Diderot))The local Langlands correspondence is a small part of a vast land discovered by Langlands. The general feeling is that a local Langlands correspondance should exist modulo $p$. But it remains mysterious...
Updated on Sep 16, 2014 01:25 PM PDT 
MSRI Evans Talk: An introduction to padic automorphic forms
Location: 60 Evans Hall, UC Berkeley Speakers: Elena Mantovan (California Institute of Technology)The notion of a padic modular form (a padic analogue of classical modular forms) was first introduced by Serre in 1973 via the qexpansion principle. Soon afterwards, Katz gave a new definition via the geometry of modular curves. The padic theory provides the appropriate framework for the study of congruences among classical forms, and padic interpolation (i.e. the construction of families of classical forms which converge padically) is a crucial tool behind many important arithmetic results on classical modular forms.
Automorphc forms are a vast generalization of modular forms. Yet, in recent years, many arithmetic properties of modular forms were extended to automorphic forms, with stunning consequences. In this talk we will introduce the notions of classical and padic automorphic forms. We will mostly focus on aspects of the padic theory related to Hida's Igusa tower, the qexpansion principle and their application to the construction of padic families of automorphic forms.
Updated on Aug 29, 2014 11:03 AM PDT 
MSRI/Evans Lecture: The (un)reasonable effectiveness of model theory in mathematics
Location: 60 Evans Hall Speakers: Carol Wood (Wesleyan University)The talk will be built around examples of how model theory informs our understanding in areas of mathematics such as algebra, number theory, algebraic geometry and analysis. The model theory behind these applications includes concepts such as compactness, definability, stability and ominimality. However, we assume no special expertise in model theory, but rather aim to illustrate the kinds of mathematical questions for which a model theoretical perspective has proven to be useful.
Updated on Apr 15, 2014 11:58 AM PDT 
MSRI/Evans Lecture: A journey through motivic integration
Location: 60 Evans Hall Speakers: François Loeser (Université de Paris VI (Pierre et Marie Curie))Motivic integration was invented by Kontsevich about 20 years ago for proving that Hodge numbers of CalabiYau are birational invariants and has developed at a fast pace since. I will start by presenting informally a device for defining and computing motivic integrals due to Cluckers and myself. I will then focus on some recent applications. In particular I plan to explain how motivic integration provides tools for proving uniformity results for $p$adic integrals occuring in the Langlands program and how one may use "motivic harmonic analysis" to study certain generating series counting curves in enumerative algebraic geometry.
Updated on Apr 11, 2014 03:43 PM PDT 
MSRI/Evans Lecture: Sullivan's conjecture and applications to arithmetic
Location: 60 Evans Hall Speakers: Kirsten Wickelgren (Georgia Institute of Technology)One can phrase interesting objects in terms of fixed points of group actions. For example, class numbers of quadratic extensions of Q can be expressed with fixed points of actions on modular curves. Derived functors are frequently better behaved than their nonderived versions, so it is useful to consider the associated derived functor, called the homotopy fixed points. Sullivan's conjecture is an equivalence between appropriately completed spaces of fixed points and homotopy fixed points for finite pgroups. It was proven independently by H. Miller, G. Carlsson, and J. Lannes. This talk will present Sullivan's conjecture and its solutions, and discuss analogues for absolute Galois groups conjectured by Grothendieck.
Updated on Apr 01, 2014 03:43 PM PDT 
MSRI/Evans Lecture: Bounding the density of rational points on trascendental hypersurfaces via model theory
Location: MSRI: Simons Auditorium Speakers: Alex Wilkie (University of Manchester)I shall begin by describing the solution (by Bombieri and Pila) to the following problem of Sarnak: prove that if f is a real analytic, but not algebraic, function defined on the closed interval [0, 1], then the equation f(s)=q has rather few solutions in rational numbers.
Once the terms here have been made precise, one can then formulate a natural conjecture for analytic functions f of several variables. It turns out that the numbertheoretic part of the argument in the one variable case generalises easily. However, the difficulty comes in the analysis, and it is here that techniques from model theory, specifically ominimality, play a role.
Updated on Mar 28, 2014 09:53 AM PDT 
MSRI/Evans Lecture: Galois, Hopf, Grothendieck, Koszul, and Quillen
Location: 60 Evans Hall Speakers: Kathryn Hess (École Polytechnique Fédérale de Lausanne (EPFL))An important generalization of Galois extensions of fields is to HopfGalois extensions of associative rings, which Schneider proved can be characterized in terms faithfully flat Grothendieck descent. I will begin by recalling this classical theory and then sketch recent homotopical generalizations, motivated by Rognes' theory of HopfGalois extensions of structured ring spectra. In particular, I will present a homotopical version of Schneider's theorem, which describes the close relationships among the notions of HopfGalois extensions, Grothendieck descent, and Koszul duality within the framework of Quillen model categories.
Updated on Mar 11, 2014 03:24 PM PDT 
MSRI/Evans Lecture: New Applications of Algebraic Topology
Location: 60 Evans Hall Speakers: Gunnar Carlsson (Stanford University)Algebraic Topology is a library of techniques for, in an appropriate sense, measuring shapes. It has long been understood, via the work of Weil, Quillen, Deligne, and many others that analogues of these techniques can give very interesting information about problems in other disciplines. Deligne's solution of a family of conjectures of Weil in the 1970's is a prime example. Over the last 1015 years, a new family of such applications directed at the notion of "Big Data" has been constructed. We will discuss these developments with examples.
Updated on Mar 06, 2014 12:15 PM PST 
MSRI/Evans Lecture: Around Approximate Subgroups
Location: 60 Evans Hall Speakers: Ehud Hrushovski (Hebrew University)Updated on Jan 20, 2014 09:31 AM PST 
MSRI/Evans Lecture: Topological cyclic homology
Location: 60 Evans Hall Speakers: Lars Hesselholt (Nagoya University)Topological cyclic homology is a topological refinement of Connes' cyclic homology. It was introduced twentyfive years ago by BökstedtHsiangMadsen who used it to prove the Ktheoretic Novikov conjecture for discrete groups all of whose integral homology groups are finitely generated. In this talk, I will give an introduction to topological cyclic homology and explain how results obtained in the intervening years lead to a short proof of this result in which the necessity of the finite generation hypothesis becomes transparent. In the end I will explain how one may hope to remove this restriction and discuss number theoretic consequences that would ensue.
Updated on Jan 15, 2014 09:05 AM PST 
MSRI/Evans Lecture: Could the Universe have an Exotic Topology?
Location: UC Berkeley, 60 Evans Hall Speakers: Vincent Moncrief (Yale University)Astronomical observations have long been interpreted to support a socalled 'cosmological principle' according to which the universe, viewed on a sufficiently large, coarsegrained scale, is spatially homogeneous and isotropic. Up to an overall, timedependent scale factor, only three Riemannian manifolds are compatible with this principle, namely the constant curvature spaces S^{3}, E^{3} and H^{3} or spherical, flat and hyperbolic spaceforms respectively and these provide the bases for the standard Friedmann, RobertsonWalker, Lemaîtra (FRWL) cosmological models. But astronomers only observe a fraction of the universe and the possibility remains open that its actual topology could be more 'exotic' and perhaps only locally compatible with isotropy and homogeniety. In this talk I shall discuss some mathematical evidence suggesting that the Einstein field equations, formulated on much more general manifolds than the FRWL ones listed above, contain a dynamical mechanism according to which the universe will, in the direction of cosmological expansion, be volume dominated by regions that are asymptotically homogeneous and isotropic.
Updated on Nov 25, 2013 01:05 PM PST 
MSRI/Evans Lecture: Inverse mean curvature flow, black holes and quasilocal mass
Location: UC Berkeley, 60 Evans Hall Speakers: Kristen Moore (Stanford University)Inverse mean curvature flow is a parabolic geometric evolution equation that expands a hypersurface at a rate given by the reciprocal of the mean curvature. In recent years it has proven to be an effective tool in the study problems in general relativity. In this talk I will discuss the role of inverse mean curvature flow in the study of black hole horizons and quasilocal concepts of mass and energy, and outline it's connection to the Penrose Conjecture.
Updated on Nov 19, 2013 10:57 AM PST 
MSRI/Evans Lecture: On an informationtheoretical interpolation inequality
Location: UC Berkeley, 60 Evans Hall Speakers: VILLANI Cedric (Institute Henri Poincare)In this talk I will review the history and use of an informationtheoretical inequality introduced by Otto and I at the end of the nineties, which we called the HWI inequality; how it was used recently in some largedimension results.
Updated on Oct 09, 2013 02:38 PM PDT 
MSRI/Evans Lecture: A Compactness Theorem for Sequences of Rectifiable Metric Spaces
Location: UC Berkeley, 60 Evans Hall Speakers: Christina Sormani (CUNY, Graduate Center)A rectifiable metric space is a metric space (X,d) with a collection of biLipschitz charts that cover all but a set of Hausdorff measure 0 of the space. Such a space can be endowed with an orientation and viewed as a rectifiable current space (X,d,T) where the T is called current structure and uses the charts to capture the notion of integration of forms (rigorously defined in work of AmbrosioKirchheim). If the boundary of T, defined via Stoke's theorem, is also rectifiable, then (X,d,T) is called an integral current space. This notion is defined in joint work with Stefan Wenger.
Riemannian manifolds of finite volume with cone singularities are examples of integral current spaces. If the manifold has a cusp singularity, the corresponding integral current space has the cusp removed. Here we will present the Tetrahedral Compactness Theorem which assumes certain uniform distance estimates on tetrahedra in a sequence of integral current spaces (or Riemannian manifolds) and a uniform upper bound on diameter and concludes that a subsequence converges in the GromovHausdorff and Intrinsic Flat sense to an integral current space (in particular the limit is rectifiable and the same dimension as the sequence).
Updated on Oct 31, 2013 12:01 PM PDT 
MSRI/Evans Lecture: Polyconvex integrands in the calculus of variations.
Location: UC Berkeley, 60 Evans Hall Speakers: Wilfrid Gangbo (Georgia Institute of Technology)Updated on Oct 16, 2013 03:43 PM PDT 
MSRI/Evans Lecture: On the topology and future stability of the universe
Location: UC Berkeley, 60 Evans Hall Speakers: Hans Ringström (Royal Institute of Technology (KTH))The current standard model of the universe is spatially homogeneous, isotropic and spatially flat. Furthermore, the matter content is described by two perfect fluids (dust and radiation) and there is a positive cosmological constant. Such a model can be well approximated by a solution to the EinsteinVlasov equations with a positive cosmological constant. As a consequence, it is of interest to study stability properties of solutions in the Vlasov setting. The talk will contain a description of recent results on this topic. Moreover, the restriction on the global topology of the universe imposed by the data collected by observers will be discussed.
Updated on Sep 11, 2013 01:02 PM PDT 
MSRI/Evans Lecture: On the topology of black holes and beyond.
Location: UC Berkeley, 60 Evans Hall Speakers: Greg Galloway (University of Miami)In recent years there has been an explosion of interest in black holes in higher dimensional gravity. This, in particular, has led to questions about the topology of black holes in higher dimensions. In this talk we review Hawking's classical theorem on the topology of black holes in 3+1 dimensions (and its connection to black hole uniqueness) and present a generalization of it to higher dimensions. The latter is a geometric result which imposes restrictions on the topology of black holes in higher dimensions. We shall also discuss recent work on the topology of space exterior to a black hole. This is closely connected to the Principle of Topological Censorship, which roughly asserts that the topology of the region outside of all black holes (and white holes) should be simple. All of the results to be discussed rely on the recently developed theory of marginally outer trapped surfaces, which are natural spacetime analogues of minimal surfaces in Riemannian geometry. This talk is based primarily on joint work with Rick Schoen and with Michael Eichmair and Dan Pollack.
Updated on Aug 19, 2013 10:38 AM PDT 
MSRI/Evans Lecture: Swarming by Nature and by Design
Location: UC Berkeley, 60 Evans Hall Speakers: Andrea Bertozzi (University of California, Los Angeles)Swarming by Nature and by Design
Andrea Bertozzi, University of California, Los AngelesThe cohesive movement of a biological population is a commonly observed natural phenomenon. With the advent of platforms of unmanned vehicles, such phenomena have attracted a renewed interest from the engineering community.
This talk will cover a survey of the speakers research and related work in this area ranging from aggregation models in nonlinear partial differential equations to control algorithms and robotic testbed experiments.
We conclude with a discussion of some interesting problems for the mathematics community.
Updated on Aug 12, 2013 02:46 PM PDT 
A motivic approach to Potts models
Location: UC Berkeley, 60 Evans Hall Speakers: Matilde MarcolliThe use of motivic techniques in Quantum Field Theory has been widely explored in the past ten years, in relation to the occurrence of periods in the computation of Feynman integrals. In this lecture, based on joint work with Aluffi, I will show how some of these techniques can be extended to a motivic analysis of the partition function of Potts models in statistical mechanics. An estimate of the complexity of the locus of zeros of the partition function, can be obtained in terms of the classes in the Grothendieck ring of the affine algebraic varieties defined by the vanishing of the multivariate Tutte polynomial, based on a deletioncontraction formula for the Grothendieck classes.Updated on Apr 15, 2013 06:42 AM PDT 
From Linear Algebra to Noncommutative Resolutions of Singularities (No April Fools\\' Joke!)
Location: UC Berkeley, 60 Evans Hall Speakers: RagnarOlaf Buchweitz (University of Toronto)Ten years ago, G.Bergman asked: "Can one factor the classical adjoint of a generic matrix?"
With G.Leuschke we showed that, yes, sometimes you can. We arrived at this by translating the question into one on morphisms between matrix factorizations of the determinant.
Understanding all such morphisms lead through joint work with Leuschke and van den Bergh to a description of a noncommutative desingularization of determinantal varieties in a characteristicfree way and we are just about to put the finishing touches on this work here at MSRI.
I will desribe the highlights of this journey and mention some resulting, and remaining open questions.Updated on Sep 18, 2013 02:29 PM PDT 
Geometry of Hurwitz Spaces
Location: UC Berkeley, 60 Evans Hall Speakers: Joe HarrisRiemann surfaces, which we now think of abstractly as smooth algebraic
curves over the complex numbers, were described by Riemann as graphs of
multivalued holomorphic functionsin other words, branched covers of the
Riemann sphere $\P^1$. The Hurwitz spaces, varieties parametrizing the
set of branched covers of $\P^1$ of given degree and genus, are still
central objects in the study of curves and their moduli. In this talk, we'll
describe the geometry of Hurwitz spaces and their compactifications, leading
up to recent work of Anand Patel and others.Updated on Mar 18, 2013 05:18 AM PDT 
Modules for elementary abelian groups and vector bundles on projective space
Location: UC Berkeley, 60 Evans Hall Speakers: David Benson (University of Aberdeen)I shall begin with a gentle introduction to modular representation theory of finite groups. Many questions about these reduce to the case of an elementary abelian pgroup, so I shall spend most of the talk on these. In particular, I shall talk about modules of constant Jordan type, and what they have to do with algebraic vector bundles on projective space.Updated on May 24, 2013 10:08 AM PDT 
An Introduction to Noncommutative Algebraic Geometry
Location: UC Berkeley, 60 Evans Hall Speakers: Toby Stafford (University of Manchester)In recent years a surprising number of insights and results in noncommutative algebra have been obtained by using the global techniques of projective algebraic geometry. Many of the most striking results arise by mimicking the commutative approach: classify curves, then surfaces, and we will use this approach here. As we will discuss, the noncommutative analogues of (commutative) curves are well understood while the study of noncommutative surfaces is ongoing. In the study of these objects a number of intriguing examples and significant techniques have been developed that are very useful elsewhere. In this talk I will discuss several of them.Updated on May 24, 2013 11:18 AM PDT 
Multiplicities of graded families of ideals
Location: UC Berkeley, 60 Evans Hall Speakers: Steven Cutkosky (University of Missouri)The multiplicity of a local ring R is its most fundamental invariant. For example, it tells us how singular the ring is. The multiplicity is computed from the limit as n goes to infinity of the length of R modulo the nth power of its maximal ideal. Many other multiplicity like invariants naturally occur in commutative algebra. We discuss a number of naturally occurring limits of this type, and show that in very general rings, such limits always exist.Updated on May 13, 2013 04:20 PM PDT 
(Quantum) fusion versus (quantum) intersection
Location: UC Berkeley, 60 Evans Hall Speakers: Catharina Stroppel (Hausdorff Research Institute for Mathematics, University of Bonn)Intersection theory grew out of very basic questions like: "given four generic lines in \mathbb{P}^3  how many lines intersect all four of them?" Questions like that are one of the simplest examples of an application of Schubert Calculus, a very important tool in combinatorial representation theory. It is used to describe cohomology rings, but also to construct categories which describe representation theoretic problems geometrically. It serves as the basis of more sophisticated methods of enumerative geometry, like GromovWitten theory and Quantum Cohomology. An amazing fact is that the same combinatorics also occurs when decomposing tensor products of representations of a semisimple complex Lie algebra.
This talk will describe some of these basic ideas and use them to explain a connection between quantum fusion products and quantum cohomology, relating Verlinde algebras and quantum cohomology rings. All this is related to questions arising in commutative and noncommutative algebraic geometry, integrable systems, representation theory, combinatorics ...
Updated on Dec 19, 2014 12:00 PM PST 
Quiver mutation and quantum dilogarithm identities
Location: UC Berkeley, 60 Evans Hall Speakers: Bernhard KellerA quiver is an oriented graph. Quiver mutation is an elementary operation on quivers which appeared in physics in Seiberg duality in the 1990s and in mathematics in FominZelevinsky's definition of cluster algebras in 2002. In this talk, I will show how, by comparing sequences of quiver mutations, one can construct identities between products of quantum dilogarithm series. These identities generalize FaddeevKashaevVolkov's classical pentagon identity and the identities obtained by Reineke. Morally, the new identities follow from KontsevichSoibelman's theory of DonaldsonThomas invariants. They can be proved rigorously using the theory linking cluster algebras to quiver representations.Updated on Nov 16, 2012 02:29 AM PST 
Secant Varieties, Symbolic Powers, Statistical Models
Location: UC Berkeley, 60 Evans Hall Speakers: Seth SullivantThe join of two algebraic varieties is obtained by taking the closure of the union of all lines spanned by pairs of points, one on each variety. The secant varieties of a variety are obtained by taking the iterated join of a variety with itself. The symbolic powers of ideals arise by looking at the equations that vanish to high order on varieties. Statistical models are families of probability distributions with special structures which are used to model relationships between collections of random variables. This talk will be an elementary introduction to these topics. I will explain the interrelations between these seemingly unrelated topics, in particular, how symbolic powers can shed light on equations for secant varieties, and how theoretical results on secant varieties shed light on properties of statistical models including mixture models and the factor analysis model. Particular emphasis will be placed on combinatorial aspects including connections to graph theory.Updated on Oct 19, 2012 02:31 AM PDT 
Recurrence relations and cluster algebras
Location: UC Berkeley, 60 Evans Hall Speakers: Pierre Guy PlamondonSergey Fomin and Andrei Zelevinsky defined cluster algebras by recursively constructing their generators via a process called mutation. This process is closely related to various sequences of integers, arising for instance from CoxeterConway friezes, whose terms can be seen as specializations of the generators of specific cluster algebras. Although the fact that these sequences contain only integers is sometimes surprising from their definition, the theory of cluster algebras provides a common explanation for it: the Laurent Phenomenon. In this talk, we will first list some examples of recurrence relations of integers, then we will try to understand them from the point of view of cluster algebras.Updated on Mar 24, 2015 02:25 PM PDT 
Categorification of quiver mutation
Location: UC Berkeley, 60 Evans Hall Speakers: Idun ReitinThe cluster algebras were introduced by FominZelevinsky in a paper which appeared 10 years ago. There are connections to many different areas of mathematics, including quiver representations. One direction of research has been to try to model the ingredients in the definition of cluster algebras in “nice” categories, like module categories or related categories. In this lecture we illustrate the idea and use of “categorification” by concentrating on only one ingredient in the definition of cluster algebras. This is the operation of quiver mutation, which we define. For finite quivers (i.e. directed graphs) without oriented cycles this leads to the socalled cluster categories, which are modifications of certain module categories. For other types of quivers some stable categories of maximal CohenMacaulay modules over commutative Gorenstein rings are the relevant categories. We start the lecture with background material on quiver representations.Updated on Oct 05, 2012 01:41 AM PDT 
Introduction to cluster algebras
Location: UC Berkeley, 60 Evans Hall Speakers: Andrei ZelevinskyCluster algebras are a class of commutative rings discovered by Sergey Fomin and the speaker about a decade ago. A cluster algebra of rank n has a distinguished set of generators (cluster variables) grouped into (possibly overlapping) nsubsets called clusters. These generators and relations among them are constructed recursively and can be viewed as discrete dynamical systems on a nregular tree. The interest to cluster algebras is caused by their surprising appearance in a variety of settings, including quiver representations, Poisson geometry, Teichmuller theory, noncommutative geometry, integrable systems, quantum field theory, etc. We will discuss the foundations of the theory of cluster algebras, with the focus on their algebraic and combinatorial structural properties.Updated on Sep 11, 2013 09:10 AM PDT 
(Multigraded) Hilbert functions
Location: UC Berkeley, 60 Evans Hall Speakers: Diane MaclaganCommutative algebra often abstracts geometric problems into simple questions about algebraic invariants. I will illustrate this with some open problems on the Hilbert function (a simple algebraic invariant which measures the dimensions of graded pieces of a graded ring). Geometry enters the picture when the ring is the projective coordinate ring of a variety. When the ring has a multigrading we also get some interesting combinatorics. I will emphasize the computational and combinatorial sides of this story.Updated on Sep 10, 2013 10:21 AM PDT 
QUASI PERIODIC ORBITS: THE CASE OF THE NON LINEAR SCHRÖDINGER EQUATION
Location: UC Berkeley, 60 Evans Hall Speakers: Claudio ProcesiUpdated on Aug 24, 2012 05:52 AM PDT 
SelfAvoiding Walk
Location: UC Berkeley, 60 Evans Hall Speakers: Gregory Lawler (University of Chicago)The participants at the Random Spatial Processes program come from many different areas: combinatorics, probability, complex analysis, theoretical physics, computer science, representation theory. Although these give different perspectives, they all arise in the analysis of critical processes in statistical physics. I will discuss a simple (to state, not necessarily to analyze!) model, the selfavoiding walk and show how multiple perspectives are useful in its study. A (planar) selfavoiding walk is a lattice random walk path in the plane with no selfintersections. It can be viewed as a simple model for polymers. I will show how we now in one sense understand this model very well, and in another sense we still know very little!Updated on Jan 24, 2015 08:59 AM PST 
Sampling Paths, Permutations and Lattice Structures
Location: UC Berkeley, 60 Evans Hall Speakers: Dana Randall (Georgia Institute of Technology)Random sampling is ubiquitous throughout mathematics, computing and the sciences as a means of studying very large sets. In this talk we will discuss simple, classical Markov chains for efficiently sampling paths and permutations. We will look at various natural generalizations with some surprising results. First, we show how to extend these Markov chain algorithms to sample biased paths, with applications to tilebased selfassembly, asymmetric exclusion processes, selforganized lists, and biased card shuffling. Next, we show how generating random configurations with mutliple paths allows us to sample planar tilings and colorings. Using insights from statistical physics, however, we will see why these methods break down and may be inefficient in models with nonuniform bias, in higher dimensions, or in weighted models with sufficiently high fugacity.Updated on Sep 11, 2013 02:04 PM PDT 
New Extremes for Random Walk on a Graph
Location: UC Berkeley, 60 Evans Hall Speakers: Peter Winkler (Dartmouth College)Random walk on a graph is a beautiful and (viewed from today) classical subject with elegant theorems, multiple applications, and a close connection to the theory of electrical networks. The subject seems to be livelier now than ever, with lots of exciting new results. We will discuss recent progress on some extremal problems. In particular, how long can it take to visit every edge of a graph, or to visit every vertex a representative number of times, or to catch a random walker? Can random walks be scheduled or coupled so that they don't collide? Can moving targets be harder to hit than fixed targets? Mentioned will be work by or with Omer Angel, Jian Ding, Agelos Georgakopoulos, Ander Holroyd, Natasha Komarov, James Lee, James Martin, Yuval Peres, Perla Sousi, and David Wilson.Updated on Sep 11, 2013 09:50 AM PDT 
Multiscale tools and dependent percolation
Location: UC Berkeley, 60 Evans Hall Speakers: Maria VaresIn this talk I plan to discuss some examples of percolation models in random environment. The goal is to show how multiscale methods can be used to answer some basic questions related to such models in the presence of strong dependence.Updated on Feb 22, 2012 12:09 PM PST 
Counting tricks with symmetric functions
Location: UC Berkeley, 60 Evans Hall Speakers: Greta PanovaSymmetric functions and Young tableaux arose from the representation theory of the symmetric and general linear groups, but found their central place in algebraic combinatorics and also ventured to other fields like algebraic geometry and statistical mechanics. In this talk we will review some of their combinatorial properties and then we will employ their power and various tricks to solve a counting problem: we will find a product formula for the number of standard Young tableaux of unusual, truncated shapes and we will pass through truncated plane partitions.Updated on Feb 06, 2012 03:42 AM PST 
From random interlacements to coordinate percolation
Location: UC Berkeley, 60 Evans Hall Speakers: vladas sidoraviciusDuring the past few years, several percolation models with long (infinite) range dependencies were introduced. Among them Random Interlacements (introduced by A.S. Sznitman) and Coordinate Percolation (introduced by P. Winkler). During the talk I will focus on the connectivity properties of these models. The latter model has polynomial decay in subcritical and supercritical regime in dimension 3. I will explain the nature of this phenomenon and why it is difficult to handle these models technically. In the second half of the talk I will present key ideas of the multiscale analysis which allows to reach some conclusions. At the end of the talk I will discuss applications and several open problems.Updated on Aug 19, 2014 10:40 AM PDT 
Combinatorics of Donaldson–Thomas and Pandharipande–Thomas invariants
Location: UC Berkeley, 60 Evans Hall Speakers: Benjamin YoungI will discuss a combinatorial problem which comes from algebraic geometry. The problem, in general, is to show that two theories for "counting" curves in a complex threedimensional space X (Pandharipande–Thomas theory and reduced Donaldson–Thomas theory) give the same answer. I will prove a combinatorial version of this correspondence in a special case (X is toric Calabi–Yau), where the difficult geometry reduces to a study of the ``topological vertex\'\' (a certain generating function) in these two theories. The combinatorial objects in question are plane partitions, perfect matchings on the honeycomb lattice and related structures.
There will be many pictures. This is a combinatorics talk, so no algebraic geometry will be used once I explain where the problem is coming from.
Updated on May 29, 2013 09:25 AM PDT 
The Second Law of Probability: Entropy Growth in the Central Limit Theorem
Location: UC Berkeley, 60 Evans Hall Speakers: Keith BallThis talk will explain how a geometric principle led to the solution of a 50 year old problem: to prove an analogue of the second law of thermodynamics for the central limit process.Updated on Oct 13, 2011 10:58 AM PDT 
Mean curvature flow
Location: UC Berkeley, 60 Evans Hall Speakers: Tobias ColdingMean curvature flow is the negative gradient flow of volume, so any hypersurface flows through hypersurfaces in the direction of steepest descent for volume and eventually becomes extinct in finite time. Before it becomes extinct, topological changes can occur as it goes through singularities. Thus, in some sense, the topology is encoded in the singularities. In this lecture I will discuss new and old results about singularities of mean curvature flow focusing on very results about generic singularities.Updated on Jan 12, 2015 02:34 PM PST 
Differentiation at large scales
Location: UC Berkeley, 60 Evans Hall Speakers: Irine PengThe usual notion of differentiation is a way of understanding the infinitesimal behaviour of a map, i.e. the map restricted to ever decreasing scales. However there are elements of the differentiation process that make sense at larger scales as well. I will discuss the utility of this view point in the work of EskinFisherWhyte on quasiisometries of the 3 dimensional solvable group and some subsequent generalization.Updated on Oct 13, 2011 10:54 AM PDT 
Dimension reduction in discrete metric geometry
Location: UC Berkeley, 60 Evans Hall Speakers: William JohnsonUpdated on Oct 03, 2011 11:11 AM PDT 
Simple connectivity is complicated: an introduction to the Dehn function
Location: UC Berkeley, 60 Evans Hall Speakers: Robert YoungA lot of good math starts by taking an existence theorem and asking ``How many?'' or ``How big?'' or ``How fast''. The bestknown example may be the Riemann hypothesis. Euclid proved that infinitely many primes exist, and the Riemann hypothesis describes how quickly they grow. I'll discuss what happens when you apply the same idea to simple connectivity. In a simplyconnected space, any closed curve is the boundary of some disc, but how big is that disc? And what can that tell you about the geometry of the space?Updated on Aug 19, 2014 11:15 AM PDT 
Graph Sparsification
Location: UC Berkeley, 60 Evans Hall Speakers: Nikhil Srivastava (University of California, Berkeley)We consider the following type of question: given a finite graph with nonnegative weights on the edges, is there a sparse graph on the same set of vertices (i.e., a graph with very few edges) which preserves the geometry of G? The answer of course depends on what we mean by preserves and geometry. It turns out that if we are interested in preserving (1) pairwise distances between all pairs of vertices or (2) weights of boundaries of all subsets of vertices, then the answer is always yes in a certain strong sense: every graph on n vertices admits a sparse approximation with O(nlogn) or O(n) edges. We discuss some of the ideas around the proof of (2), which turns out to be a special case of a more general theorem regarding matrices. The original motivation for this problem was in the design of fast algorithms for solving linear equations, but lately the ideas have found other uses, for instance in metric embeddings and probability. Joint work with J. Batson and D. Spielman.Updated on Nov 04, 2014 11:52 AM PST 
MSRI Evans Lecture
Location: UC Berkeley, 60 Evans Hall Speakers: Marianna Csornyei (University College)Differentiability of Lipschitz functions and tangents of sets
Differentiability of Lipschitz functions and tangents of sets
We will show how elementary product decompositions of measures can detect directionality in sets, and show how this can be used to describe nondifferentiability sets of Lipschitz functions on R^n, and to understand the phenomena that occur because of behaviour of Lipschitz functions around the points of null sets.
In order to prove this we will need to prove results about the geometry of sets of small Lebesgue measure: we show that sets of small measure are always contained in a "small" collection of Lipschitz surfaces.
The talk is based on a joint work
with G. Alberti, P. Jones and D. Preiss.
Updated on Sep 18, 2013 03:19 PM PDT 
Boundary singular solutions associated with connecting thin tubes
Location: UC Berkeley, 60 Evans Hall Speakers: Susanna TerraciniRefreshments after lecture at La Val's Pizza.
Consider two domains connected by a thin tube so that the mass of a given eigenfunction (linear or nonlinear) concentrates in only one of the two domains. The restriction on the other domain develops a singularity at the junction of the tube, as the section of the channel shrinks to zero. The asymptotics for this type of solutions can be precisely described. This is a result obtained in collaboration with L. Abatangelo and V. FelliUpdated on Apr 19, 2011 04:28 AM PDT 
Counting Points on Curves over Finite Fields
Location: UC Berkeley, 60 Evans Hall Speakers: Melanie WoodRefreshments after lecture at La Val's Pizza
Updated on Apr 05, 2011 09:14 AM PDT 
Linearization Techniques in Free Boundary Problems
Location: UC Berkeley, 60 Evans Hall Speakers: John AnderssonRefreshments after lecture at La Val\\\\\\'s Pizza.
Updated on Mar 17, 2011 04:25 AM PDT 
SSL group, course "Space Weather"
Group will visit the first floor terrace to catch a view of the satellite dish.
Created on Mar 10, 2011 01:27 AM PST 
MSRIEvans Lecture Henryk Iwaniec
Location: UC Berkeley, 60 Evans Hall Speakers: Henryk IwaniecRefreshments after lecture at La Val's Pizza.
Updated on Jan 15, 2011 08:15 AM PST 
MSRI/Evans Talk: "Hamiltonian group actions"
Location: UC Berkeley, 60 Evans Hall Speakers: Dr. Yael KarshonThis talk is about symplectic manifolds equipped with compact group actions that are generated by moment maps. Such structures model symmetries in classical mechanics and often arise in purely mathematical contexts. The moment map encodes manifold information into polytopes and graphs. We will show how to obtain moment map ``pictures” and how to use these pictures to extract symplectic geometric information.Updated on May 13, 2013 11:01 PM PDT 
MSRI/Evans Lecture Series: "Boundedness of varieties of general type "
Location: UC Berkeley, 10 Evans Hall Speakers: Christopher HaconView abstract (PDF 30KB)Updated on May 13, 2013 11:01 PM PDT 
MSRI/Evans Lecture Series
Location: UC Berkeley, 60 Evans Hall Speakers: Dr. Ravi VakilUpdated on May 13, 2013 11:01 PM PDT 
MSRI/Evans Lecture Series: "Algebraic stacks without schemes"
Location: UC Berkeley, 60 Evans Hall Speakers: Dr. Barbara FantechiAlgebraic stacks are about 40 years old, and play a growing role both within algebraic geometry and in its relation with high energy physics; increasingly, they are seen as geometric objects, interesting in their own right. In this talk we'll give an outline of the relevant definitions, focusing on the case of smooth stacks over the complex numbers so as to avoid schemerelated complications. We'll then outline a few classical applications and a selection of current research. The focus will be on examples rather than on detailed rigor. Prerequisites for the talk are some familiarity with the basics of complex manifolds, or at least differentiable manifolds; a nodding acquaintance with categories and functors would be helpful. No algebra or algebraic geometry is needed.Updated on May 13, 2013 11:01 PM PDT 
MSRI/Evans Lecture Series
Location: UC Berkeley, 60 Evans Hall Speakers: Dr. William FultonUpdated on May 13, 2013 11:01 PM PDT 
MSRI/Evans Lecture Series: "Hall algebras and wallcrossing"
Location: MSRI: Simons Auditorium Speakers: Tom BridgelandThe Hall algebra construction goes back to work of Steinitz in the early years of the last century. In the 1990s Ringel used the same construction as an approach to quantum groups. More recently, Joyce, Kontsevich, Soibelman and others have used Hall algebras as a tool for studying wallcrossing phenomena. Following Reineke I will explain how this works in the downtoearth setting of representations of quivers, where all the key ideas can already be seen.Updated on May 13, 2013 11:01 PM PDT 
MSRI/Evans Lecture Series: "Which powers of holomorphic functions are integrable?"
Location: UC Berkeley, Evans Hall Speakers: Dr. János KollárUpdated on May 13, 2013 11:01 PM PDT 
MSRI/Evans Lecture Series  Positivity Properties of Divisors and Higher Codimension Cycle
Location: UC Berkeley, Evans Hall Speakers: Dr. Robert LazarsfeldA very basic idea in algebraic geometry is to try to study a variety by considering all the hypersurfaces (and nonnegative linear combinations thereof) inside it. This allows one to construct various interesting cones of cohomology classes, whose structure often reflects the geometry of the underlying variety. Remarkably, the precise shape of these cones is unknown even for some quite simple surfaces. After quickly reviewing the classical theory, I will survey more contemporary developments concerning the codimension one situation. Then I will present some questions and conjectures about what one might expect for cycles of higher codimension.Updated on May 13, 2013 11:00 PM PDT 
January 1416, 2009: Kickoff Presentations
Updated on Apr 01, 2011 05:06 AM PDT