Abstract:

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 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 one-parameter 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 Margulis.  

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Abstract:

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 2-torus admits a rich supply of Euclidean structures, which form an interesting moduli space which itself enjoys hyperbolic non-Euclidean 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.

 

 

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Abstract:

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.

Abstract:

At the heart of the Langlands program lies the reciprocity conjecture, which can be thought of as a non-abelian 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 Harris-Lan-Taylor-Thorne 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 p-adic interpolation techniques. Finally, I will mention some open problems.

Abstract:

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

Abstract:

The local Langlands correspondence (conjecture) relates (infinite-dimensional) representations of p-adic reductive groups with (finite-dimensional) 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 p-adic reductive groups by viewing arithmetic objects as "simpler" parameters. Then it will be natural to study representation-theoretic problems in terms of arithmetic invariants. In this talk, we explain some cases which can be answered in this setup: the Gan-Gross-Prasad conjecture (a restriction problem) and formal degrees (a generalization of dimensions).

Abstract:

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 low-dimensional topology: Khovanov's link homology. We will discuss advantages and applications of categorification in different settings, and open problems in the area.

Abstract:

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...

Abstract:

The notion of a p-adic modular form (a p-adic analogue of classical modular forms) was first introduced by Serre in 1973 via the q-expansion principle. Soon afterwards, Katz gave a new definition via the geometry of modular curves. The p-adic theory provides the appropriate framework for the study of congruences among classical forms, and p-adic interpolation (i.e. the construction of families of classical forms which converge p-adically) 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 p-adic automorphic forms. We will mostly focus on aspects of the p-adic theory related to Hida's Igusa tower, the q-expansion principle and their application to the construction of p-adic families of automorphic forms.

Abstract:

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 o-minimality. 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.

Abstract:

Motivic integration was invented by Kontsevich about 20 years ago for proving that Hodge numbers of Calabi-Yau 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.

Abstract:

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 non-derived 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 p-groups. 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.

Abstract:

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 number-theoretic 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 o-minimality, play a role.

Abstract:

An important generalization of Galois extensions of fields is to Hopf-Galois 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 Hopf-Galois 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 Hopf-Galois extensions, Grothendieck descent, and Koszul duality within the framework of Quillen model categories.

Abstract:

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 10-15 years, a new family of such applications directed at the notion of "Big Data" has been constructed.  We will discuss these developments with examples.

Abstract:

Topological cyclic homology is a topological refinement of Connes' cyclic homology. It was introduced twenty-five years ago by Bökstedt-Hsiang-Madsen who used it to prove the K-theoretic 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. 

Slides

Abstract:

Astronomical observations have long been interpreted to support a so-called 'cosmological principle' according to which the universe, viewed on a sufficiently large, coarse-grained scale, is spatially homogeneous and isotropic. Up to an overall, time-dependent scale factor, only three Riemannian manifolds are compatible with this principle, namely the constant curvature spaces S³, E³ and H³ or spherical, flat and hyperbolic space-forms respectively and these provide the bases for the standard Friedmann, Robertson-Walker, 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.

Abstract:

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 quasi-local concepts of mass and energy, and outline it's connection to the Penrose Conjecture.

Abstract:

In this talk I will review the history and use of an information-theoretical 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 large-dimension results.

Abstract:

A rectifiable metric space is a metric space (X,d) with a collection of bi-Lipschitz 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 Ambrosio-Kirchheim).  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 Gromov-Hausdorff and Intrinsic Flat sense to an integral current space (in particular the limit is rectifiable and the same dimension as the sequence).

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Abstract

Abstract:

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 Einstein-Vlasov 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.

Abstract:

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.

Abstract:

Swarming by Nature and by Design
Andrea Bertozzi, University of California, Los Angeles

The 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.

Abstract: The 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 deletion-contraction formula for the Grothendieck classes.

Abstract: 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 characteristic-free 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.

Abstract: Riemann surfaces, which we now think of abstractly as smooth algebraic
curves over the complex numbers, were described by Riemann as graphs of
multivalued holomorphic functions-in 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.

Abstract: 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 p-group, 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.

Abstract: 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 on-going. 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.

Abstract: 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.

Abstract: 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 Gromov-Witten 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 ...

Abstract: A 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 Fomin-Zelevinsky'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 Faddeev-Kashaev-Volkov's classical pentagon identity and the identities obtained by Reineke. Morally, the new identities follow from Kontsevich-Soibelman's theory of Donaldson-Thomas invariants. They can be proved rigorously using the theory linking cluster algebras to quiver representations.

Abstract: The 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.

Abstract: Sergey 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 Coxeter-Conway 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.

Abstract: The cluster algebras were introduced by Fomin-Zelevinsky 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 so-called cluster categories, which are modifications of certain module categories. For other types of quivers some stable categories of maximal Cohen-Macaulay modules over commutative Gorenstein rings are the relevant categories. We start the lecture with background material on quiver representations.

Abstract: Cluster 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) n-subsets called clusters. These generators and relations among them are constructed recursively and can be viewed as discrete dynamical systems on a n-regular tree. The interest to cluster algebras is caused by their surprising appearance in a variety of settings, including quiver representations, Poisson geometry, Teichmuller theory, non-commutative 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.

Abstract: Commutative 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.

Abstract: 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 self-avoiding walk and show how multiple perspectives are useful in its study. A (planar) self-avoiding walk is a lattice random walk path in the plane with no self-intersections. 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!

Abstract: 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 tile-based self-assembly, asymmetric exclusion processes, self-organized 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 non-uniform bias, in higher dimensions, or in weighted models with sufficiently high fugacity.

Abstract: 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.

Abstract: In this talk I plan to discuss some examples of percolation models in random environment. The goal is to show how multi-scale methods can be used to answer some basic questions related to such models in the presence of strong dependence.

Abstract: Symmetric 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.

Abstract: During 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 sub-critical and super-critical 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 multi-scale analysis which allows to reach some conclusions. At the end of the talk I will discuss applications and several open problems.

Abstract: I 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 three-dimensional 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.

Abstract: This 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.

Abstract: Mean 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.

Abstract: The 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 Eskin-Fisher-Whyte on quasi-isometries of the 3 dimensional solvable group and some subsequent generalization.

Abstract: A lot of good math starts by taking an existence theorem and asking ``How many?'' or ``How big?'' or ``How fast''. The best-known 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 simply-connected 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?

Abstract: 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.

Abstract: 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 non-differentiability 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.

Abstract: 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. Felli

Abstract: This 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.

Abstract: View abstract (PDF 30KB)

Abstract: Algebraic 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 scheme-related 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.

Abstract: The 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 wall-crossing phenomena. Following Reineke I will explain how this works in the down-to-earth setting of representations of quivers, where all the key ideas can already be seen.

Abstract: A very basic idea in algebraic geometry is to try to study a variety by considering all the hypersurfaces (and non-negative 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.