Abstracts of talks 2015/2016
Rémy Coulon (Rennes 1)
Torsion group acting on a CAT(0) cube complex
Agata Smoktunowicz (Edinburgh)
On the Jacobson radical of noncommutative rings
The Jacobson radical of a ring was introduced in 1945 by Jacobson, and has been studied ener since; consequently its structure is well understood. Around 2007, Rump showed some surprising connections between Jacobson radical rings and solutions to the Young-Baxter equation. In particular he showed that Jacobson radical rings are in one-to-one correspondence with two-sided braces. Recently, Gateva-Ivanova found that there is a one-to-one correspondence between braces and symmetric groups in the sense of Takeuchi, that is braided groups with the involutive Young-Baxter operator. Hence Jacobson radical rings can be used to construct examples of braided groups and braces. Moreover, every Jacobson radical ring yields a solution to the Young-Baxter equation. In this talk we will look at some old and new methods of constructing Jacobson radical rings and nil rings (recall that a nil ring is a ring in which every element to some power is zero, and that every nil ring is Jacobson radical). We then answer a question of Amberg and Sysak from 1997 by constructing a nil ring whose adjoint group is not an Engel group. We also mention some other examples of Jacobson radical rings related to differential polynomial rings.
Wolfgang Soergel (Freiburg)
Motives in representation theory
I will try to explain how newly developed methods from motives allow a streamlined treatment of Koszul duality in representation theory.
Alexander Premet (Manchester)
Rigid nilpotent orbits and sheets in good characteristic
In my talk, based on joint work with David Stewart, I'm going to discuss the theory of sheets in Lie algebras of reductive groups over fields of characteristic p>0. It turns out that Borho's classification of sheets and the distribution of nilpotent orbits among them remain the same as in characteristic 0 under very mild assumptions on p.
Michael Rapoport (Bonn)
The combinatorics and geometry of the (arithmetic) fundamental lemma
The fundamental lemma is a conjectural identity between two orbital integrals. I will explain that it comes down to an explicit combinatorial statement that compares numbers of certain lattices in related p-adic vector spaces. The arithmetic fundamental lemma gives a conjectural interpretation of the derivative of an orbital integral. I will explain that also this conjecture comes down to a very concrete statement, this time in geometry.
Maxim Kontsevich (IHES)
On microlocal support, Stokes structures and cluster coordinates
Jürg Kramer («Humboldt» Berlin)
Mumford’s theorem on mixed Shimura varieties: the case of a line bundle on the universal elliptic curve
In the case of a Shimura variety X of non-compact type, by a theorem of Mumford, automorphic vector bundles equipped with the natural invariant metric have a unique extension to vector bundles over a toroidal compactification of X equipped with a logarithmically singular Hermitian metric. This result is crucial to define arithmetic Chern classes for these vector bundles. It is natural to ask whether Mumford’s result remains valid for “automorphic” vector bundles on mixed Shimura varieties.
In our talk we will examine the simplest case, namely the Jacobi line bundle on the universal elliptic curve, whose sections are Jacobi forms. We will show that Mumford’s result cannot be extended directly to this case since a new type of metric singularity appears. By using the theory of b-divisors, we show however that an analogue of Mumford’s extension theorem can be obtained in this setting.
Andrea Seppi (Pavia)
Tensori di Codazzi su una superficie iperbolica e variazioni infinitesime nello spazio di Teichmüller
Data una superficie di Riemann S di genere maggiore o uguale a 2, è noto che lo spazio di Teichmüller di S è una varietà di dimensione 6g-6, e il suo spazio tangente in un punto è identificato sotto diversi punti di vista con un primo gruppo di coomologia di S. In questo seminario mostrerò una costruzione che permette di associare ad un tensore di Codazzi su una superficie iperbolica un elemento nello spazio tangente allo spazio di Teichmüller, e discuterò due tipi di applicazioni. La prima riguarda la classificazione delle varietà lorentziane piatte globalmente iperboliche, già compresa da Mess negli anni ’90. La seconda permette di ridimostrare un teorema di Goldman che esprime la forma di Weil-Petersson in termini della varietà dei caratteri. Questi risultati possono essere estesi al caso dello spazio di Teichmüller di una superficie puntata, ed a varietà con singolarità coniche.
Victoria Hoskins (Freie Universität Berlin)
Stratifications for representations of quivers and sheaves
Many moduli spaces are constructed via geometric invariant theory and in this talk we focus on moduli of coherent sheaves and moduli of quiver representations. In both cases, we introduce two stratifications: a Harder-Narasimhan stratification and a stratification coming from the GIT construction. For quiver representations, we show that both stratifications coincide. However, this is not quite true for sheaves. We explain why this is the case and we construct an asymptotic GIT stratification which agrees with the Harder-Narasimhan stratification for sheaves.
Alex Lubotzky (Hebrew University of Jerusalem)
From Ramanujan graphs to Ramanujan complexes
Ramanujan graphs are optimal expanders (from spectral point of view). Explicit constructions of such graphs were given in the 80's as quotients of the Bruhat-Tits tree associated with GL2 over a local field F, by suitable congruence subgroups. The spectral bounds were proved using works of Hecke, Deligne and Drinfeld on the "Ramanujan conjecture" in the theory of automorphic forms. The work of Lafforgue, extending Drinfeld from GL2 to GLn, opened the door for the construction of Ramanujan complexes as quotients of the Bruhat-Tits buildings. This gives finite simplical complxes which on one hand are "random like" and at the same time have strong symmetries. Recently various applications have been found in combinatorics, coding theory and in relation to Gromov's overlapping properties.
We will describe these developments and some recent applications. In particular, we will present a joint work with Tali Kaufman and David Kazhdan and another one by S. Evra and T. Kaufman in which these complexes are uses to answer a question of Gromov.
Matteo Cavaleri (Roma « Sapienza »)
Quantification and computation of soficity
In the class of finitely generated groups we introduce equivalent definitions for sofic groups, amenable groups, word problem and a quantification of soficity, the sofic dimension growth. We present the Minsky machines in order to define computability and to construct Kharlampovich group, a finitely presented solvable group with unsolvable word problem. We prove the computability of Følner sets and the non-computability of sofic approximations for Kharlampovich group. More generally we study the classes of groups with computable Følner sets, with subrecursive sofic dimension growth and their stability properties.
Gavril Farkas (Berlin Humboldt)
Compact moduli of Abelian differentials
The moduli space of holomorphic differentials (with prescribed zeros and poles) on nonsingular curves is not compact since the curve may degenerate. I will discuss a compactification of these strata in the moduli space of Deligne-Mumford stable pointed curves, which includes the
space of canonical divisors as an open subset. The theory leads to geometric/combinatorial constraints on the closures of the strata of holomorphic differentials and as a consequence, one can determine the cohomology classes of the strata. This is joint work with Rahul Pandharipande.
Drazen Adamovic (Zagreb)
On explicit realization of affine vertex algebras and superconformal algebras
We shall present explicit realization of certain affine and superconformal vertex algebras and their modules. These realizations show that admissible affine vertex algebras are closely related with C2-cofinite W-algebras appearing in logarithmic conformal field theory.
We shall also present a solution of the irreducibility problem for the Wakimoto modules and Whittaker modules for the affine Lie algebra A1(1).
Samuel Grushevsky (SUNY)
The closure of the loci of meromorphic differentials with prescribed zeroes and poles
In the total space of the Hodge bundle of stable differentials over the Deligne-Mumford compactification of the moduli space of curves with marked points, we describe precisely the closure of the locus of differentials with prescribed multiplicities of zeroes and poles at these points. Our focus will be on necessary and sufficient conditions for a stable curve to lie in this closure. Based on joint work with M. Bainbridge, D. Chen, Q. Gendron, and M. Moeller.
Alexander Kleshchev (Oregon)
Khovanov-Lauda-Rouquier algebras and PBW bases in quantum groups
We discuss standard module theory for KLR algebras of finite and affine types, its connections with PBW bases in quantum groups, and affine highest weight categories.
Andrea Appel (USC)
The Casimir connection of a Kac-Moody algebra
In this talk, we introduce the Casimir connection of a symmetrisable Kac-Moody algebra, whose monodromy gives rise to a representation of the corresponding generalized braid group. We then prove that this representation is equivalently described by the action of the quantum Weyl group operators of the corresponding quantum group (joint with V. Toledano Laredo).
Nils Scheithauer (Darmstadt)
Automorphic products of singular weight
Borcherds' singular theta correspondence is a map from modular forms for the Weil representation to automorphic forms on orthogonal groups. Since these automorphic forms have nice product expansions at the cusps they are called automorphic products. The zeros and poles of these functions lie on rational quadratic divisors and their orders are explicitly known. The smallest possible weight of a holomorphic automorphic form on an orthogonal group is the so-called singular weight. Holomorphic automorphic products of singular weight seem to be very rare. The few known examples all correspond to infinite-dimensional Lie superalgebras describing compactified superstrings. In this talk we show that holomorphic automorphic products of singular weight on lattices of prime level exist only in small signatures and we give a complete classification of reflective automorphic products of singular weight on lattices of prime level.
Filippo Callegaro (Pisa)
The integer cohomology algebra of toric arrangements
The topic of this talk is to compute the integer cohomology ring of the complement of a toric arrangement, giving a description of the toric analogous of the Orlik-Solomon algebra.
We begin recalling some basic combinatorial invariants and we investigate the dependency of the cohomology ring from the arrangement's combinatorial data. To this end, we first consider the real complexified case and we study a morphism of spectral sequences associated to certain combinatorially defined subcomplexes of the toric Salvetti category. Then we use a technical argument in order to extend the results to full generality.
In the case of a non-unimodular arrangement, it is still an open problem to find a purely combinatorial description of the integer cohomology ring.
This is a joint work with Emanuele Delucchi (Univ. of Fribourg, CH).
20 January 2016 – Aula Consiglio, 15:45
Richard Melrose (MIT)
Lefschetz fibrations and the Weil-Petersson metric
I will describe recent work with Xuwen Zhu on the precise regularity of the constant-curvature fibre metrics near the singularities of a Lefschetz
map. Applied to the universal curve over the Knudsen-Deligne-Mumford compactification of the moduli spaces of pointed Riemann surfaces this gives a detailed picture of the iterative asymptotic behaviour of the Weil-Petersson and Tahktajan-Zograf metrics. As time permits I will explain the anticipated boundary behaviour of harmonic forms for this metric, now known in the two simplest cases from results due to Gell-Redmann and Swoboda.
Guido Pezzini (Erlangen)
Sottogruppi sferici di gruppi di Kac-Moody
Le varietà sferiche sono una generalizzazione delle varietà toriche, simmetriche e delle bandiere. La classificazione degli spazi omogenei sferici equivale alla classificazione dei sottogruppi sferici dei gruppi algebrici riduttivi connessi. E' stata portata a termine negli ultimi anni, in termini di oggetti combinatorici chiamati dati sferici omogenei, definiti mediante assiomi proposti da D. Luna nel 2001.
Nel seminario presenteremo brevemente questa teoria, ed una generalizzazione ai gruppi infinito dimensionali di Kac-Moody, nell'ambito di un progetto di ricerca in collaborazione con F. Knop e B. Van Steirteghem. Nella prima parte del progetto abbiamo definito una classe di sottogruppi detti sferici di tipo finito di un gruppo di Kac-Moody. Per essi si possono definire invarianti combinatorici in completa analogia con il caso classico di dimensione finita, e gli oggetti risultanti soddisfano gli stessi assiomi di Luna. Nel seminario discuteremo anche altre possibili generalizzazioni, che includono sottogruppi particolarmente interessanti di gruppi di Kac-Moody, come i sottogruppi simmetrici.
Matteo Casati (SISSA)
Algebraic methods for the theory of deformations of Poisson brackets of hydrodynamic type
I will present the theory of multidimensional Poisson Vertex Algebras as the setting for studying Hamiltonian d+1 dimensional, n component systems of PDEs, focussing on the notion of the PVA cohomology and its relation with the deformation theory for brackets of hydrodynamic type. I will discuss a few explicit examples for the n=1, d>1 case and for the n=d=2 case. If time allows that, I will also present the results of a joint work with G. Carlet and S. Shadrin about the full cohomology for the n=1 case.
Nicolas Tholozan (Luxembourg)
Volume of compact Clifford-Klein forms
A compact Clifford-Klein
form of a homogeneous space G/H is a quotient of this homogeneous
space by a discrete subgroup Γ of G acting properly
discontinuously and cocompactly on G/H. When both G and H
are semi-simple groups, the action of G on G/H preserves a pseudo-Riemannian
metric, and in particular a volume form.
I will prove that the volume of the compact CliffordKlein form Γ\G/H is the integral over a certain fundamental class of Γ of some G-invariant form ωH on the Riemannian symmetric space G/K. As a consequence, one obtains in many cases that this volume is rigid. Moreover, this provides a new obstruction to the existence of compact CliffordKlein forms of certain homogeneous spaces.
Simon Riche (Clermont-Ferrand)
The geometry of modular representations of reductive algebraic groups
In this talk we will present a joint work with P. Achar which provides some equivalences of categories allowing to describe the principal block of a connected reductive algebraic group over an algebraically closed field of positive characteristic in terms of geometry (in fact both in terms of coherent sheaves and of constructible sheaves). This geometric picture is analogous to a similar picture for representations of Lusztig's quantum groups at a root of unity due to Arkhipov-Bezrukavnikov-Ginzburg, and is expected to help solving important questions in the representation theory of reductive groups. More directly, it allows to prove a "graded analogue" of a conjecture by Finkelberg-Mirkovic.
Paolo Stellari (Milano Statale)
Uniqueness of dg enhancements in geometric contexts and Fourier-Mukai functors
The quest for a (possibly unique) lift of exact functors and triangulated categories to dg analogues is highly non-trivial and rich of subtle aspects, already in geometric contexts. This is made clear by the most recent developments concerning Fourier-Mukai functors, which will be reviewed in this talk. On the other hand, it was a general belief and a formal conjecture by Bondal, Larsen and Lunts that the dg enhancement of the bounded derived category of coherent sheaves or the category of perfect complexes on a (quasi-)projective scheme is unique. This was proved by Lunts and Orlov in a seminal paper.
In this talk we will explain how to extend Lunts-Orlov's results to several interesting geometric contexts. Namely, we care about the category of perfect complexes on noetherian separated schemes with enough locally free sheaves and the derived category of quasi-coherent sheaves on any scheme.
This is a joint work with A. Canonaco.
Oliver Lorscheid (IMPA)
Quiver Grassmannians of extended Dynkin type D
Since cluster algebras have been introduced in the early 2000's, they receive growing attention from researchers in representation theory and other other areas of mathematics and physics. A fundamental problem in this subject are explicit descriptions of cluster algebras in terms of generators and relations. While cluster algebras of extended Dynkin type A are well-understood, there are only partial results for type D. In collaboration with Thorsten Weist, we have established formulas for the Euler characteristics of the associated quiver Grassmannians, which leads to explicit formulas for the cluster variables in terms of the initial cluster variables.
In this talk, we will give an overview over the methods that entered the lengthy proof of these formulas. The central result of this work is that all quiver Grassmannians of extended Dynkin type admit decompositions into affine spaces.
Dennis Gaitsgory (Harvard)
Classical and geometric Langlands for functions fields
Let X be a curve over a finite field Fq and G a reductive group. On the automorphic side of Langlands one studies the moduli space Y=BunG(X) of principal G-bundles on X. The classical theory of automorphic functions studies the space of functions on Y(Fq) - the set of Fq-points of Y. The geometric theory studies the category of sheaves on Y defined of the algebraic closure of Fq. Langlands correspondence aims to relate both of these problems to the study of homomorphisms of the fundamental group of X to the Langlands dual group of G.
In the talk we will explain what Langlands correspondence actually says in each of these contexts and how the two problems, the classical and the geometric, are related.
Rahul Pandharipande (ETH)
The algebra of tautological classes on the moduli of curves
Mumford opened the door to the study of tautological classes on the moduli of curves in his 1983 paper.
I will attempt to explain the state of the field as I see it now with an emphasis on the advances in the past 5 years: the proof of the Faber-Zagier relations, the study of the kappa rings in the compact type case, and Pixton's calculus.
Daniel Huybrechts (Bonn)
The K3 category of a cubic fourfold
The bounded derived category of coherent sheaves on a smooth cubic fourfold contains a category that behaves in many respect like the derived category of a K3 surface.
We will survey what is known about this category and state some open problems and conjectures.
Alessandra Iozzi (ETH)
Irreducible lattices and bounded cohomology
We show some of the similarities and some of the differences between irreducible lattices in product of semisimple Lie groups and their siblings in product of locally compact groups. In the case of product of trees, we give a concrete example with interesting properties, among which some in terms of bounded cohomology and quasimorphisms.
Alexander Kuznetsov (Steklov)
Fano fourfolds of K3 type
A Fano fourfold is said to be of K3 type if its derived category of coherent sheaves has a semiorthogonal decomposition consisting of several exceptional objects and a derived category of a noncommutative K3 surface. Cubic fourfolds form a family of very interesting examples of this
sort. They are interesting because of their intriguing birational behavior, and of their relation to hyperkähler geometry, and most probably the noncommutative K3 category is responsible for both features. I will talk about other examples of Fano fourfolds of K3 type.
Laurent Demonet (Nagoya)
Algebras of partial triangulations
We introduce a class of finite dimensional algebras coming from partial triangulations of marked surfaces. A partial triangulation is a subset of a triangulation. This class contains Jacobian algebras of triangulations of marked surfaces studied by Labardini-Fragoso (based on the work of Derkses-Weyman-Zelevinsky) and Brauer graph algebras (introduced by Wald and Waschbüsch). We generalize properties which are known or partially known for Brauer graph algebras and Jacobian algebras of marked surfaces. In particular, these algebras are symmetric when the considered surface has no boundary, they are at most tame, and we give a combinatorial generalization of flips or Kauer moves on partial triangulations which induces (in most cases) derived equivalences between the corresponding algebras. Notice that we also give an explicit formula for the dimension of the algebra.
This is based on the preprint arXiv:1602.01592 (2016).
Bertrand Deroin (ENS Paris)
Super-maximal representations from fundamental groups of punctured spheres to PSL(2,R)
In a recent work with Nicolas Tholozan, aiming at refining the classical Milnor-Wood inequality on character varieties, we have discovered a new class of representations, that we have called super-maximal.
I will introduce them and show some of their properties. We will see that
(i) they are totally non-hyperbolic, i.e. simple closed curves are mapped to non-hyperbolic elements of PSL(2,R);
(ii) they are geometrizable in a very strong sense by conical metrics;
(iii) they define compact components in certain relative character varieties, hence generalizing a construction of Benedetto-Goldman.
Michel Coornaert (Strasbourg)
A Garden of Eden theorem for Anosov diffeomorphisms on tori
Let f be an Anosov diffeomorphism of the n-dimensional torus Tn and τ a continuous self-mapping of Tn commuting with f.
We prove that τ is surjective if and only if the restriction of τ to each homoclinicity class of f is injective.
This is an analogue of the celebrated Garden of Eden theorem of Moore and Myhill in symbolic dynamics.
This is joint work with Tullio Ceccherini-Silberstein (see arXiv:1506.06945 and arXiv:1508.07553).
Julia Sauter (Bielefeld)
On quiver Grassmannians and orbit closures for representation-finite algebras
We show that Auslander algebras have a unique tilting and cotilting module which is generated and cogenerated by a projective-injective; its endomorphism ring is called the projective quotient algebra. For any representation-finite algebra, we use the projective quotient algebra to construct desingularizations of quiver Grassmannians, orbit closures in representation varieties, and their desingularizations. This generalizes results of Cerulli Irelli, Feigin and Reineke.
Marc Burger (ETH)
Maximal representations, non Archimedean Siegel spaces, and buildings
Maximal representations of the fundamental group G of a compact surface S into a real symplectic group Sp(V) are natural generalisations of the holonomy representations of G into SL(2,R) associated to hyperbolic structures on S. Maximal representations are injective, have discrete image, and their images in Sp(V) can be seen as higher rank analogues of Kleinian groups. A lot of activity in the last decade has been devoted to understand which features of classical hyperbolic geometry and Teichmüller spaces generalize to this setting.
In this talk I will report on joint work with Beatrice Pozzetti, were we study the structure of representations associated to points in the real spectrum compactification of the semi-algebraic set formed by characters of maximal representations. The talk will be accessible to a general audience.
Paolo Antonini (SISSA)
Non-integrable Lie algebroids
In this seminar we report on work in progress with Iakovos Androulidakis concerning the integrability problem of Lie algebroids.
In many constructions in non commutative geometry the passage from a singular space to a C*-algebra involves the use of a Lie groupoid as an intermediate desingularization space. The infinitesimal datum of a Lie groupoid is a Lie algebroid and they appear independently for example as:
- Poisson manifolds
- objects that describe the connections on principal bundles.
However in general is not possible to integrate a Lie algebroid to a Lie groupoid (in contrast to the theory of Lie algebras).
The first part of the talk will be concerned with the discussion of Lie algebroids: basic definitions, examples, the integration problem, the obstructions to the integrability of Crainic-Fernandes and the discussion of an important non integrable example given by Molino.
In the last part we will explain our idea of removing the obstructions of a transitive algebroid, passing to a suitable extension, and discuss generalizations. In these cases one can still perform some of the basic constructions in index theory and non commutative geometry.
Hisham Sati (Pittsburgh and NYU)
Twisted generalized cohomology theories and applications
Twists for various cohomology theories play an important role in various areas of mathematics as well as in mathematical physics. I will survey this area, starting with twisted de Rham cohomology, twisted K-theory, and then considering higher twists from the point of view of generalized cohomology theories of higher chromatic and categorical levels. I will describe joint work with Craig Westerland on twists of Morava K-theory and E-theory, descending from complex cobordism, at all chromatic levels. Then I will describe recent work with John Lind and Craig Westerland on a new periodic form of the iterated algebraic K-theory of connective complex K-theory ku, generalizing the K-theory of 2-vector bundles K(ku) to all categorical degrees, as well as a natural twisting of this cohomology theory by higher gerbes. As an application, this leads to topological T-duality for sphere bundles oriented with respect to this theory as an isomorphism of the twisted theories, generalizing various existing results.
Maxim Nazarov (York)
Cherednik algebras and Zhelobenko operators
We study canonical intertwining operators between modules of the trigonometric Cherednik algebra, induced from the standard modules of the degenerate affine Hecke algebra. We show that these operators correspond to the Zhelobenko operators for the affine Lie algebra of series A. To establish the correspondence, we use the functor of Arakawa, Suzuki and Tsuchiya which maps certain modules of the affine Lie algebra to modules of the Cherednik algebra.
This is a joint work with Sergey Khoroshkin.
Ruggero Bandiera (Roma « Sapienza »)
Modelli algebrici della mappa dei periodi e accoppiamento di Yukawa
Dopo aver richiamato l'approccio alla teoria delle deformazioni tramite DGLAs, e più in generale algebre L∞, descriveremo alcuni modelli L∞ della mappa dei periodi locale di una varietà Kähleriana X, e tramite questi costruiremo l'algebra L∞ che controlla le deformazioni di X dove l'associata variazione di struttura di Hodge soddisfa certi vincoli. Come conseguenza, otteniamo una interpretazione dell'accoppiamento di Yukawa nel contesto della teoria delle deformazioni.
Il seminario è basato su una collaborazione con M. Manetti.
Kurt Falk (Bremen)
Dimensions of limit sets of Kleinian groups
The dynamics of geometrically finite hyperbolic manifolds, where recurrence and ergodicity play a central role, is well understood by means of Patterson-Sullivan theory. For geometrically infinite manifolds or manifolds given by infinitely generated Kleinian groups, non-recurrent dynamics becomes the "thick part" of dynamics, not only in measure but often also in (Hausdorff) dimension. I will discuss the Hausdorff and Minkowski dimension of limit sets of Kleinian groups and the connection between these notions and dynamics within the convex core of the associated hyperbolic manifold.
Ozren Perse (Zagreb)
On conformal embeddings of affine vertex algebras and branching rules
In this talk we study branching rules for conformal embeddings of simple affine vertex algebras. We discuss finiteness properties of these decompositions.
The talk is based on joint work with Drazen Adamovic, Victor G. Kac, Pierluigi Moseneder Frajria and Paolo Papi.
Simon G. Gindikin (Rutgers University)
Complex horospheres for real symmetric spaces
Gel'fand and Graev found that the horospherical transform is a perfect way to construct harmonic analysis on complex semisimple Lie groups or Riemannian symmetric spaces. However, for real semisimple Lie groups (starting from SL(2,R)) or, more generally, for pseudo-Riemannian symmetric spaces the horospherical transform has a kernel corresponding to the discrete series. Gelfand’s problem is the following: is it possible to find a version of the horospherical transform that works for discrete series?
We will consider the idea to extend the set of real horospheres by complex horospheres without real points. We define a horospherical Cauchy transform on a real symmetric space with singularities on complex horospheres and it completely reproduces the harmonic analysis on pseudo-Riemannian symmetric spaces.
Sergey Fomin (UMich)
Computing without subtracting (and/or dividing)
Algebraic complexity of a rational function can be defined as the minimal number of arithmetic operations required to compute it. Can restricting the set of allowed arithmetic operations dramatically increase the complexity of a given function (assuming it is still computable in the restricted model)? In particular, what can happen if we disallow subtraction and/or division?
This is joint work with D. Grigoriev and G. Koshevoy.
Ivan Losev (NEU)
Hecke algebras for complex reflection groups
algebras are classical objects in representation theory. An important basic
property is that these algebras are flat deformations of the group algebra of
the corresponding real reflection group. In 1998 Broue, Malle and Rouquier have
extended the definition of a Hecke algebra to the case of complex reflection
groups. They conjectured that the Hecke algebras are still flat deformations of
the group algebras. Recently, the proof of this conjecture was completed by
myself and Marin-Pfeiffer in the case when the base field has characteristic 0.
In my talk I will introduce Hecke algebras for complex reflection groups and explain some ideas of the proof of the BMR conjecture.
15 June 2016
Donald King (NEU)
Spherical nilpotent orbits and a conjecture of Vogan
Ron Donagi (UPenn)
Super Riemann surfaces and some aspects of superstring perturbation theory
surfaces exhibit many of the familiar features of ordinary Riemann surfaces,
and some novelties. They have moduli spaces and Deligne-Mumford
compactifications. One can integrate and construct measures on moduli spaces.
The punctures one can insert come in two varieties: Ramond and Neveu-Schwarz. I
will survey some of the expected and unexpected features, including some
aspects of non splitness: for genus g>4, the moduli space of super Riemann
surfaces is not projected (and in particular is not split); it cannot be
holomorphically projected to its underlying reduced manifold. Physically, this
means that certain approaches to superstring perturbation theory that are very
powerful in low orders have no close analog in higher orders. Mathematically,
it means that the moduli space of super Riemann surfaces cannot be constructed
in an elementary way starting with the moduli space of ordinary Riemann
surfaces. It has a life of its own.
When we examine the Deligne-Mumford compactification of moduli space, and especially the Ramond boundary divisors, we find that the interesting new phenomena start already in genus one. This is interpreted as the mechanism that allows supersymmetry to remain unbroken at tree level in certain models of superstring perturbation theory, but to be spontaneously broken at one loop.
Robert Lazarsfeld (SUNY)
Measures of irrationality for hypersurfaces of large degree
Given an n-dimensional
smooth hypersurface X of degree d in projective n-space,
it is elementary that X cannot be rational when d>n+1, but it
is interesting to ask “how irrational” such a hypersurface can be. We discuss
various measures of irrationality, and show that they are governed by
positivity properties of the canonical bundle. Among other things, we prove a
conjecture of Bastianelli, Cortina and De Poi concerning the least degree with
which X can be expressed as a rational covering of projective space.
This is joint work with Ein and Ullery.
29 June 2016
6 July 2016
13 July 2016
20 July 2016