Algebra and Geometry Seminar




Abstracts of talks 2015/2016

16
September 2015 – Aula Consiglio, 14:00
Rémy Coulon (Rennes 1)
Torsion group acting
on a CAT(0) cube complex
Abstract
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 YoungBaxter equation. In particular he showed that
Jacobson radical rings are in onetoone correspondence with twosided braces.
Recently, GatevaIvanova found that there is a
onetoone correspondence between braces and symmetric groups in the sense of
Takeuchi, that is braided groups with the involutive
YoungBaxter 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 YoungBaxter 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 padic 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
Abstract
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 noncompact
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 bdivisors,
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 6g6, 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 WeilPetersson 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 HarderNarasimhan
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 HarderNarasimhan 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 BruhatTits
tree associated with GL_{2}
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 GL_{2} to GL_{n},
opened the door for the construction of Ramanujan complexes as quotients of the
BruhatTits 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.
18 November 2015 –
Aula Consiglio, 15:45
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
noncomputability 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 DeligneMumford 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 C_{2}cofinite Walgebras 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 A_{1}^{(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 DeligneMumford
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)
KhovanovLaudaRouquier 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.
Thursday
7 January 2016 – Room B, 11:30
Andrea Appel (USC)
The Casimir connection of a KacMoody
algebra
In this talk, we
introduce the Casimir connection of a symmetrisable KacMoody 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 socalled singular weight. Holomorphic automorphic
products of singular weight seem to be very rare. The few known examples all
correspond to infinitedimensional 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.
20 January 2016 –
Aula Consiglio, 14:15
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 OrlikSolomon 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 nonunimodular
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).
Colloquium di geometria ed analisi
20 January 2016 –
Aula Consiglio, 15:45
Richard Melrose (MIT)
Lefschetz fibrations and the WeilPetersson metric
I will describe
recent work with Xuwen Zhu on the precise regularity
of the constantcurvature fibre metrics near the
singularities of a Lefschetz
map. Applied to the
universal curve over the KnudsenDeligneMumford
compactification of the moduli spaces of pointed Riemann surfaces this gives a
detailed picture of the iterative asymptotic behaviour
of the WeilPetersson and TahktajanZograf
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 GellRedmann
and Swoboda.
Guido Pezzini (Erlangen)
Sottogruppi sferici di gruppi di KacMoody
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 KacMoody, 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
KacMoody. 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 KacMoody, come i sottogruppi simmetrici.
Thursday
28 January 2016 – Aula B, 14:30
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
CliffordKlein forms
A compact CliffordKlein
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 semisimple groups, the action of G on G/H preserves a pseudoRiemannian
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 Ginvariant
form ω_{H} 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
(ClermontFerrand)
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 ArkhipovBezrukavnikovGinzburg,
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 FinkelbergMirkovic.
Paolo Stellari (Milano Statale)
Uniqueness of dg
enhancements in geometric contexts and FourierMukai functors
The quest for a
(possibly unique) lift of exact functors and
triangulated categories to dg analogues is highly nontrivial and rich of
subtle aspects, already in geometric contexts. This is made clear by the most
recent developments concerning FourierMukai 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 LuntsOrlov'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 quasicoherent
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 wellunderstood, 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 F_{q} and G a reductive group. On the automorphic side of Langlands
one studies the moduli space Y=Bun_{G}(X) of principal Gbundles on X. The classical theory of automorphic functions
studies the space of functions on Y(F_{q})  the set of F_{q}points of Y. The geometric theory studies the category of sheaves on Y defined of the algebraic closure of F_{q}.
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.
9 March 2016 – Aula
Consiglio, 14:00
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 FaberZagier
relations, the study of the kappa rings in the compact type case, and Pixton's calculus.
9
March 2016 – Aula Consiglio, 15:15
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 LabardiniFragoso (based on the work of DerksesWeymanZelevinsky)
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)
Supermaximal representations from fundamental groups
of punctured spheres to PSL(2,R)
In a recent work with
Nicolas Tholozan, aiming at refining the classical
MilnorWood inequality on character varieties, we have discovered a new class
of representations, that we have called supermaximal.
I will introduce them
and show some of their properties. We will see that
(i)
they are totally
nonhyperbolic, i.e. simple closed curves are mapped to nonhyperbolic 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 BenedettoGoldman.
Michel Coornaert (Strasbourg)
A Garden of Eden theorem for Anosov
diffeomorphisms on tori
Let f be an Anosov
diffeomorphism of the ndimensional
torus T^{n} and τ a continuous selfmapping of T^{n} 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 CeccheriniSilberstein
(see arXiv:1506.06945 and arXiv:1508.07553).
Julia Sauter (Bielefeld)
On quiver Grassmannians and
orbit closures for representationfinite algebras
We show that Auslander algebras have a unique tilting and cotilting module which is generated and cogenerated by a
projectiveinjective; its endomorphism ring is called the projective quotient
algebra. For any representationfinite 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 semialgebraic set formed by characters of
maximal representations. The talk will be accessible to a general audience.
Paolo Antonini (SISSA)
Nonintegrable
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:
 foliations
 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 CrainicFernandes
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 Ktheory, 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 Ktheory and Etheory, 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 Ktheory of
connective complex Ktheory ku, generalizing the Ktheory of 2vector 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 Tduality 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.
18 May 2016 – Aula Picone,
14:15
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.
18
May 2016 – Aula Picone, 15:45
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 PattersonSullivan theory. For
geometrically infinite manifolds or manifolds given by infinitely generated Kleinian groups, nonrecurrent 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.
25 May 2016 – Aula Consiglio, 14:15
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.
25 May 2016 – Aula Consiglio, 15:45
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 pseudoRiemannian 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
pseudoRiemannian 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.
Friday
10 June 2016  Aula Consiglio, 14:30
Ivan Losev (NEU)
Hecke algebras for
complex reflection groups
IwahoriHecke
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 MarinPfeiffer 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
Speaker (Institution)
Title
Abstract
22 June 2016 – Aula
Consiglio, 11:30
Donald King (NEU)
Spherical nilpotent
orbits and a conjecture of Vogan
22
June 2016 – Aula Consiglio, 14:30
Ron Donagi (UPenn)
Super Riemann
surfaces and some aspects of superstring perturbation theory
Super Riemann
surfaces exhibit many of the familiar features of ordinary Riemann surfaces,
and some novelties. They have moduli spaces and DeligneMumford
compactifications. One can integrate and construct measures on moduli spaces.
The punctures one can insert come in two varieties: Ramond and NeveuSchwarz. 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 DeligneMumford 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.
22
June 2016 – Aula Consiglio, 15:45
Robert Lazarsfeld (SUNY)
Measures of
irrationality for hypersurfaces of large degree
Given an ndimensional
smooth hypersurface X of degree d in projective nspace,
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
Speaker (Institution)
Title
Abstract
6 July 2016
Speaker (Institution)
Title
Abstract
13 July 2016
Speaker (Institution)
Title
Abstract
20 July 2016
Speaker (Institution)
Title
Abstract