Algebra and Geometry
Seminar




Abstracts of talks
2014/2015

Markus Reineke (Wuppertal)
Topology and
arithmetic of matrix invariants
We consider the
action of a general linear group on tuples of matrices via simultaneous
conjugation. The quotient by this action is called a space of matrix
invariants, and we consider global invariants of these spaces. Closed formulas
for their numbers of rational points over finite fields, and for their
intersection cohomology, are derived.
Salvatore Stella (NCSU)
dvector fans per cluster algebra di tipo finito ed affine
Ad ogni algebra cluster, grazie al fenomeno
di Laurent, si può associare in maniera naturale una famiglia di vettori
a coordinate intere (i vettori d
delle variabili cluster).
Sorprendentemente questa famiglia codifica
molte delle proprietà combinatoriche
dell'algebra di partenza.
In questo seminario, dopo aver ricordato le
definizioni di base, spiegherò come costruire l'insieme dei vettori d nel caso di algebre cluster di tipo
finito ed affine utilizzando l'azione di una particolare simmetria del
corrispondente sistema di radici.
Matthieu Gendulphe (Roma « Sapienza »)
A theorem of Maryam
Mirzakhani: the asymptotic of the growth of simple closed geodesics on
hyperbolic surfaces
Last summer, Maryam
Mirzakhani received the Fields medal for "her outstanding contributions to
the dynamics and geometry of Riemann surfaces and their moduli spaces."
We will describe one
of her most famous results, which illustrates the sentence above.
This talk is intended
for nonexperts.
Dmitri Panov
(King’s College of London)
Definite connections
and real symplectic Fano
manifolds
A definite connection
is an SO(3)connection over a
4manifold, whose curvature is nonzero on every tangent 2plane. Such
geometric objects were first considered by Allan Weinstein and were called fat bundles. Given such a connection,
the associated S^{2}bundle
is naturally a symplectic manifold. Surprisingly the symplectic manifold is either a symplectic Fano or a symplectic CalabiYau.
Important examples of
such connections come from differential geometry. Namely, the LeviCevita connection on the twistor
bundle of a Riemannian 4manifold with sufficiently pinched curvature is
definite. In particular we have examples such as the unit S^{4} that give rise to positive definite connections and
examples such as hyperbolic 4manifolds that give rise to negative definite
connections.
In this talk I will
discuss the following spheretype theorem: the only four manifolds that
admit an S^{1}invariant definite connection are S^{4} and CP^{2}. This work is joined
with Joel Fine.
Gregory Sankaran (Bath)
Fundamental groups of
toroidal compactifications
We give a precise
unified description of the fundamental group of a toroidal compactification of
a locally symmetric variety, and explain some consequences in special cases.
This is joint work
with A.K. Kasparian (Sofia).
Michael Bulois (SaintÉtienne)
Slice induction and
sheets
Let G be an algebraic group acting on a
variety V. Sheets of V are defined as irreducible components of
the sets of elements with fixed orbit dimension. Knowing the made up and
geometry of sheets is important to understand moduli spaces and some other
varieties. In the talk, I will focus on the adjoint
action and its generalization to symmetric Lie algebras. In this setting, many
information follow by considering the Slodowy slices
of the sheet. This allows to rebuilt important part of the theory of induction
of nilpotent orbits.
Giovanni Cerulli Irelli (Roma « Sapienza »)
Realizzazione, come sottomoduli
di Demazure, dei moduli abelianizzati
per algebre di Lie di tipo A e C
In questo seminario presenterò un
progetto che sto portando avanti con M. Lanini e P. Littelmann: sia L
un'algebra di Lie di tipo A o C e sia V una sua rappresentazione irriducibile
di dimensione finita. La filtrazione PBW dell'algebra inviluppante di L, induce una filtrazione di V il cui graduato associato V^{a} si dice l'abelianizzato
associato a V. Si noti che V^{a} è un modulo per
l'algebra simmetrica S(n), dove n è la sottoalgebra
delle matrici triangolari strettamente inferiori di L. Motivati da una congettura di Vinberg,
i moduli abelianizzati hanno ricevuto un discreto interesse
negli ultimi anni, grazie ai lavori di Littelmann,
Fourier, Finkelberg e Feigin.
In questo seminario dimostriamo che V^{a}
è isomorfo (come S(n)modulo)
ad un opportuno sottomodulo di Demazure
di un opportuno modulo irriducibile per un'algebra di Lie
dello stesso tipo di L, ma di rango
doppio.
Indrava Roy (Roma « Sapienza »)
Analytic surgery
exact sequence and L^{2} spectral invariants
For a discrete group Γ, the analytic surgery exact
sequence of Higson and Roe characterizes the failure of the BaumConnes assembly map μ:
K_{Û}(BΓ)¦K_{Û}(C^{Û}Γ) to be an isomorphism. In
particular, there are obstruction structure groups S_{Û}(Γ) which vanish precisely when the
assembly map μ is bijective.
Higson and Roe showed that the certain spectral invariants associated with
finitedimensional representations of Γ
called relative ηinvariants
(a.k.a. ρinvariants), defined
in the seminal work of Atiyah, Patodi
and Singer, appear naturally as morphisms from these structure groups to R. This allowed them to conceptualize and reprove classical deep results
of Mathai, Weinberger, Keswani, and PiazzaSchick on
the vanishing of the AtiyahPatodiSinger
rhoinvariants for metrics of positive scalar curvature for spin Dirac
operators and their homotopy invariance for signature
operators, when Γ is
torsionfree and the assembly map μ
is an isomorphism. We shall extend the utility of the analytic surgery exact
sequence to the semifinite case associated with Galois Γcoverings, by introducing L^{2}structure groups. These structure groups appear in an
exact sequence which is compatible with the one studied by Higson and Roe. We
thus prove the corresponding vanishing results for L^{2 }ρinvariants, first introduced and studied by Cheeger and Gromov.
(Joint work with
M.T. Benameur).
Henning Krause (Bielefeld)
Koszul, Ringel, and Serre duality for strict polynomial functors
Strict polynomial functors were introduced by Friedlander and Suslin in their work on the cohomology
of finite group schemes. More recently, a Koszul
duality for strict polynomial functors has been
established by Chalupnik and Touze.
In my talk I will give a gentle introduction to strict polynomial functors (via representations of divided powers) and will
explain the Koszul duality, making explicit the
underlying monoidal structure which seems to be of independent interest. Then I
connect this to Ringel duality for Schur algebras and describe Serre
duality for strict polynomial functors.
Tuesday 25
November 2014 – Room TBA – 15:00
Francesco Vaccarino (Politecnico di Torino)
Homological scaffold
of the psychedelic brain
We will introduce our recent results on the use of
persistent homology in the field of complex networks. In particular we will
show how persistence homology can be used to discriminate the variation of the
brain functional connectome under the influence of psilocybin extracted from
magic mushrooms. We furthermore give account of our theorem which shows that a
persistence module on a finite poset P can be obtained from the filtration of
a graph weighted over P.
Silvana Bazzoni (Padova)
tstrutture
e equivalenze indotte da moduli tilting classici e
infinitamente generati
Ricorderò i risultati classici di
BrennerButler, HappelRingel e Rickard
riguardanti le equivalenze indotte da moduli tilting
finitamente generati (e complessi tilting).
Presenterò la generalizzazione naturale a moduli tilting
infinitamente generati, le proprietà delle classi tilting
e le equivalenze indotte. Accennerò ad un recente problema posto da Saorin riguardante le proprietà della categoria
abeliana “cuore” della tstruttura
associata a un modulo tilting.
Margherita LelliChiesa (Centro « De Giorgi », Pisa)
Varietà di Severi e teoria di BrillNoether di curve su superfici abeliane
Le varietà di Severi e la teoria di
BrillNoether di curve su superfici K3 sono ben
comprese; al contrario, poco è noto riguardo le curve su superfici
abeliane.
Data una superficie abeliana generale S con polarizzazione H di tipo (1,n), dimostreremo la non vuotezza e la regolarità della
varietà di Severi che parametrizza curve δnodali C nel
sistema lineare H per 0≤δ≤n1. Studieremo poi
le serie lineari di tipo g^{1}_{d}
sulla normalizzazione di C;
già nel caso liscio, la gonalità delle
curve in H risulterà non
costante. Tali risultati sono ottenuti per degenerazione ad una superficie semiabeliana. Nell'ultima parte del seminario,
proporrò un approccio per studiare le serie lineari di tipo g^{r}_{d}
con r≥2.
Questo lavoro è in collaborazione con
A. L. Knutsen e G. Mongardi.
Giovanna Carnovale (Padova)
Un analogo del teorema di Katsylo
per sheet di classi di coniugio sferiche
Le sheet per
l'azione di un gruppo algebrico G su
una varietà X sono le
componenti irriducibili degli insiemi di punti di X la cui orbita ha dimensione fissata. Nel caso dell'azione di G su se stesso per coniugio, esse sono
le componenti irriducibili delle parti di una partizione introdotta
recentemente da G. Lusztig in termini di
corrispondenza di Springer. Mostreremo come
descrivere le parti e le sheet nel caso in cui esse contengano
una classe di coniugio sferica e dimostreremo che per queste sheet e parti esiste un quoziente geometrico, isomorfo ad
una sottovarietà affine di G
modulo l'azione di un gruppo abeliano finito.
Martina Balagovic (Newcastle)
Irreducible modules
for degenerate double affine Hecke algebras of type A
Double affine Hecke algebras and their degenerate versions have been an
active area of study since their original definition in 1993, when they were
used by Ivan Cherednik to solve Macdonald
conjectures. When studying their representation theory, one usually focuses on Category O. Here one defines Verma
modules M as certain induced modules,
shows that with appropriate assumptions these modules always have an
irreducible quotient L=M/K, and that
all irreducible modules in Category O can be uniquely realized in this
way as quotients of Verma modules. However, the
description of this maximal submodule K,
and consequently of the irreducible module L,
are not known in general. This is why alternative characterizations of
irreducible modules are interesting.
In this talk, I will
focus on the question “Can irreducible Category O modules for degenerate double affine Hecke
algebras of type A always be
realized as submodules of Verma modules?”
The corresponding
answer for affine Hecke algebras, as discovered by Guzzi, Nazarov and Papi, is always yes. The answer for double affine Hecke algebras is more involved.
I will describe both
the modules which do not allow such a realization and the imbedding of the
modules which do allow it in terms of the combinatorics
of the affine symmetric group and periodic skew Young diagrams.
7
January 2015 – Aula Consiglio, 14:00
Shifra Reif
(University of Michigan)
The KacWakimoto character formula
Finding a character formula for finite dimensional
representations has been a central problem in the theory of Lie superalgebras in the past few decades. In 1994, Kac and Wakimoto conjectured an
elegant character formula for modules admitting certain choices of simple
roots. We shall give several characterizations for these modules and show how
they can be used to prove the formula for the general linear superalgebra. Joint with Crystal Hoyt and Michael Chmutov.
14 January 2015
Speaker (Institution)
Title
Abstract
Marco Zambon (Leuven)
Multisymplectic manifolds, homotopy moment
maps, and conserved quantities
Multisymplectic structures are
higher generalizations of symplectic structures, where
forms of higher degree are considered. We introduce a notion of moment map for
such structures, based on Stasheff’s notion of
L_{∞}algebra, and by relating it to equivariant
cohomology we are able to produce examples. Even
though moment maps are quite involved, they admit a simple cohomological
description. Finally, we discuss some geometric implications of our moment maps
(the existence of conserved quantities).
Fosco Loregian (SISSA)
A characterization of
quasicategorical tstructures
After having introduced the basic definition of ∞category
and stable ∞category, we characterize tstructures in stable ∞categories as suitable quasicategorical factorization
systems. More precisely we show that a tstructure
t
on a stable ∞category C is
equivalent to a normal torsion theory
F
on C, i.e. to a factorization system F=(E,M) where both classes satisfy the
3for2 cancellation property, and a certain compatibility with pullbacks/pushouts.
Luca Vitagliano (Salerno)
Homotopy Algebras and PDEs
The jet space approach is a powerfool
tool to study invariant properties of PDEs, i.e. properties independent of the
choice of coordinates (symmetries, conservation laws, recursion operators,
etc.). The highest outcome of this approach is that the space of solutions of a
PDE is a "derived manifold", which roughly means that vector fields,
differential forms, etc. on it form "algebras up to homotopy".
The aim of the talk is providing a smooth introduction to the homotopical algebra of PDEs.
Angela Ortega (Humboldt Universität Berlin)
The BrillNoether curve and PrymTyurin varieties
It is a result of L. Masiewicki that the Jacobian of a non hyperelliptic
curve of genus 5 is isomorphic to the Prym variety
associated to the natural involution on the singular locus of the Θdivisor of the curve. We will
show that this result can be generalized to a curve C of arbitary odd genus g=2a+1. More precisely, the Jacobian of C can be realized as the PrymTyurin variety for the BrillNoether
curve W^{1}_{a+2}(C) of pencils of minimal degree on C. As a consequence we obtain an application to the enumerative
geometry of the secants to C.
Steve Karam (Lille)
Growth of balls in
the universal cover of graphs and surfaces
We prove that if the area of a closed Riemannian
surface M of genus at least two is
sufficiently small with respect to its hyperbolic area, then for every radius R≥0 the universal cover of M contains an Rball with area at least the area of a cRball in the hyperbolic plane,
where c is a universal positive
constant in (0,1). In particular
(taking the area of M smaller if
needed), we prove that for every radius R≥1,
the universal cover of M contains an Rball with area at least the area of a
ball with the same radius in the hyperbolic plane.
This result answers positively a question of L. Guth for surfaces. We also prove an analog result for
graphs. Specifically, we prove that if G
is a connected metric graph of first Betti number b≥2 and of length sufficiently
small with respect to the length of a connected trivalent graph G_{b} of the same Betti number where the length of each edge is 1, then for every radius R≥0 the universal cover of G_{b} contains an Rball with length at least c times the length of an Rball in the universal cover of G_{b} where c is a universal constant in (½,1).
Gianluca Pacienza (Strasbourg)
Curve razionali e gruppi di Chow di
varietà irriducibili simplettiche
Mori
e Mukai hanno dimostrato che ogni sistema lineare
ampio su di una superficie K3 contiene un elemento somma di curve razionali. Beauville e Voisin hanno
osservato come da ciò discenda l’esistenza di uno 0ciclo «canonico» sulla
superficie K3, dato dalla classe di un punto qualsiasi su una qualunque curva
razionale sulla K3.
Nel
seminario parlerò di un lavoro in corso e in collaborazione con
François Charles nel quale generalizziamo i risultati di MoriMukai e di BeauvilleVoisin alle
varietà irriducibili simplettiche di dimensione arbitraria che sono
deformazioni di schemi di Hilbert di punti su una K3.
Lidia Angeleri Hügel (Verona)
Moduli silting
I
moduli silting generalizzano una nozione introdotta
recentemente da Adachi, Iyama
e Reiten nello studio di mutazioni in teoria cluster.
Sono i moduli che corrispondono a complessi silting
non necessariamente compatti formati da due termini, e sono strettamente legati
a certe tstrutture nella categoria
derivata. Ci soffermeremo in particolare sul ruolo della teoria silting nello studio di epimorfismi
di anelli e localizzazioni, mostrando che in alcuni casi le localizzazioni di
un anello sono parametrizzate da moduli silting.
Questo lavoro è in collaborazione con Frederik
Marks e Jorge Vitória.
Valentina Di Proietto (Strasbourg)
On the homotopy exact sequence for the algebraic fundamental group
There is a strong link between the fundamental group
of a variety and the linear differential equations we can define on it. The
definition of the fundamental group given in terms of homotopy
classes of loops does not generalize easily to algebraic varieties defined over
an arbitrary field. But exploiting this link we can give another definition
that makes sense in very general contexts: it is called the algebraic
fundamental group. We prove the homotopy exact
sequence for the algebraic fundamental group for a fibration
with singularities with normal crossing.
This is a joint work with Atsushi Shiho.
Tuesday
24 March 2015 – Aula di Consiglio, 14:00
Jesus Juyumaya (Valparaiso)
Knot invariants from
the YokonumaHecke algebras
In this talk we define invariants for classical knots,
singular knots and framed knots. These invariant are constructed by using the YokonumaHecke algebras in the Jones recipe.
Frédéric Patras (Université de SophiaAntipolis)
B_{∞}
structures and finite topologies
Models (simplicial, cellular...) of topological spaces
are usually equipped with various operations: products, dualities, cohomological operations, and so on. One can also consider
topological spaces from another point of view: instead of looking at the
internal structure of each space, one can consider their collective structure.
In the particular case of finite spaces, they are in bijection with
quasiorders, and it is natural to investigate their linear span using
techniques of algebraic combinatorics. New operations
and structure theorems arise from this point of view, as well as connexions with Hopf algebras
such as the one of quasisymmetric functions.
Joint work with L. Foissy
and C. Malvenuto.
Gabriele Mondello (« Sapienza »
Roma)
On the existence of
spherical metrics with conical singularities on a 2sphere
A celebrated result by Poincaré
states that a compact connected Riemann surface has a conformal metric of
constant curvature, unique up to rescaling and biholomorphisms.
Clearly, the case of genus 0 is not so exciting, being trivially solved by any FubiniStudy metric on the complex projective line.
The problem becomes more interesting if we require
such metrics to have conical singularities of prescribed angles at a finite
subset of n marked points. The case of negative and zero curvature was settled
by McOwen and Troyanov in
19891991: they established the existence and uniqueness of such a metric in
each conformal class.
The case of positive curvature is more delicate:
existence and uniqueness results are known for small angles (Troyanov), whereas nonuniqueness results are known in
positive genus (BartolucciDe MarchisMalchiodi).
In a joint work with D.Panov
(still in progress), we determine for which angle assignment there exists a
surface of genus 0 with a metric of curvature 1 and conical singularities of
such prescribed angles (and noncoaxial holonomy).
Stéphane Launois (Kent)
On the quadratic
Poisson Gel’fandKirillov problem
We study the field of fractions of certain Poisson
polynomial algebras.
For a large class of Poisson polynomial algebras, we
prove that their field of fractions is isomorphic to a field of rational
functions endowed with a socalled quadratic Poisson brackets. The result
extends to homogeneous Poisson prime quotients. Examples include Poisson matrix
varieties and Poisson determinantal varieties. This
is joint work with Cesar Lecoutre.
Elena Martinengo (Hannover)
Singolarità di spazi di moduli di
fasci su K3
Negli
anni Ottanta Mukai dimostra che le singolarità
dello spazio dei moduli di fasci su una superficie K3 sono contenute nel luogo
dei fasci strettamente semistabili. Se la polarizzazione rispetto a cui si
considera la stabilità è generica e il vettore di Mukai è primitivo tutti i fasci semistabili sono
stabili e lo spazio dei moduli è liscio e ha una struttura simplettica.
Nel caso in cui il vettore di Mukai è non
primitivo, Kaledin, Lehn e
Sorger congetturano che la dgalgebra che controlla le deformazioni di fasci su
una K3 sia formale, che implicherebbe una completa descrizione delle possibili
singolarità dello spazio di moduli. La congettura è stata
dimostrata in alcuni casi da KaledinLehn e
successivamente da Zhang. Le tecniche utilizzate sono simili e consistono
nell'analizzare i sollevamenti dei fasci sulla K3 alla famiglia twistor e poi applicare il teorema di formalità in
famiglie di Kaledin.
In
un lavoro in progress in collaborazione con Manfred Lehn
vogliamo completare la dimostrazione della congettura. Nel seminario
parlerò della nostra dimostrazione per un caso mancante e del tentativo
di estendere la nostra costruzione ad hoc a una dimostrazione generale.
Olivier Debarre (ENS Paris + Paris 7)
Fano varieties and EPW sextics
We explore a connection between smooth projective
varieties X of dimension n with an ample divisor H such that
H^{n}=10 and
K_{X}=(n2)H
and a class of sextic
hypersurfaces of dimension 4 considered by Eisenbud, Popescu, and Walter (EPW sextics).
This connection makes possible the construction of moduli spaces for these
varieties and opens the way to the study of their period maps. This is work in
progress in collaboration with Alexander Kuznetsov.
Alessio Fiorentino (Roma « Sapienza »)
La matrice delle bitangenti di una quartica
piana non singolare da un punto di vista modulare
È noto che una quartica piana ammette
28 bitangenti e può essere ricostruita a partire da uno qualunque dei
suoi 288 sistemi di Aronhold. Sfruttando questo
teorema classico e un risultato di Hesse sull'esistenza di 36 rappresentazioni determinantali non equivalenti per la quartica piana non
singolare, Plaumann, Sturmfels
e Vinzant hanno recentemente definito una matrice 8x8
simmetrica di rango 4 che parametrizza le bitangenti di una quartica piana non
singolare. Nel seminario verrà presentato un lavoro scritto in
collaborazione con Francesco Dalla Piazza e Riccardo Salvati Manni, in cui si
è determinata l'espressione della matrice in termini di funzioni Θ
di Riemann; tale approccio permette di considerare la
matrice delle bitangenti da un punto di vista modulare, come funzione della
curva.
6 May 2015
Speaker (Institution)
Title
Abstract
Claus Michael Ringel (Bielefeld)
Representations of
quivers over the algebra of dual numbers
The representations
of a quiver Q over a field k (the kQmodules, where kQ is the path algebra of Q over
k) have been studied for a long
time, and one knows quite well the structure of the module category Mod(kQ). It seems to be worthwhile to
consider also representations of Q
over arbitrary finitedimensional kalgebras A.
The lecture will draw
the attention to the case when A=k[ε] is the algebra of dual numbers
(the factor algebra of the polynomial ring k[T] in one variable T modulo the ideal generated by T^{2}), thus to the Amodules, where A=kQ[ε]=kQ[T]/<T2>.
The algebra Λ is a 1Gorenstein algebra, thus
the torsionless modules are known to be of special
interest (as the Gorensteinprojective or maximal
CohenMacaulay modules). They form a Frobenius
category L, thus the corresponding stable category L is a
triangulated category. Actually, this category L is the category of
perfect differential kQmodules and L is the corresponding homotopy category. As we will see, the homology functor H: Mod → Mod(kQ) yields a bijection between the indecomposables in L and those in Mod(kQ) and the kernel of H is a finitely generated ideal of the
category L which will be described explicitly.
This is a report on
joint investigations with Zhang Pu (Shanghai).
13 May 2015 –
Aula Consiglio, 15:45
Ulrike Rieß (Bonn)
On Beauville's conjectural weak splitting property
We present a recent result on the Chow ring of
irreducible symplectic varieties.
The main object of interest is Beauville's
conjectural weak splitting property, which predicts the injectivity
of the cycle class map restricted to a certain subalgebra
of the rational Chow ring (the subalgebra generated
by divisor classes).
For special irreducible symplectic
varieties we relate it to a conjecture on the existence of rational Lagrangian fibrations.
After deducing that this implies the weak splitting
property in many new cases, we present parts of the proof.
20 May 2015  Aula
Consiglio, 14:0015:00
Alessandro Valentino (Bonn)
Central extensions of
mapping class groups from characteristic classes
I will discuss a functorial
construction of extensions of mapping class groups of smooth manifolds which
are induced by extensions of (higher) diffeomorphisms groups via the group
stack of automorphisms of manifolds equipped with
higher degree topological structures. The problem of constructing such
extensions arises naturally in the study of topological quantum field theories,
in particular in 3d ChernSimons theory. Joint work
with Domenico Fiorenza and Urs
Schreiber.
20
May 2015 – Aula Consiglio, 15:1516:15
Ghislain Fourier (Glasgow)
Quantum PBW
filtrations and monomial ideals
Our aim is to define a PBWtype filtration for quantum
groups, following the framework from recent years in the nonquantum setup. For
this we recall main properties of this classical filtration which guideline us
towards a quantum PBW filtration. In type A_{n}
we obtain an approriate degree function via
dimensions of homomorphism space of the corresponding oriented quiver.
By specializing to q=1,
we obtain a new filtration on the universal enveloping algebra and hence on any
simple module. We can give a basis for the associated graded module and see
that the annihilating ideal is monomial, in contrast to the standard PBW
filtration. We finish with exploring the relations to flag varieties, Schubert
varieties, quiver Grassmanian and their degenerations
to toric varieties.
This is joint work with X.Fang
and M.Reineke.
Simone Cecchini (Northeastern University)
Von Neumann algebra
valued differential operators over noncompact manifolds
We provide criteria for selfadjointness
and τFredholmness
of first and second order differential operators acting on sections of infinite
dimensional bundles, whose fibers are modules of finite type over a von Neumann
algebra A endowed with a trace τ. We extend the Calliastype index to operators acting on sections of such
bundles and show that this index is stable under compact perturbations. (Joint work with Maxim Braverman).
3
June 2015  Aula di Consiglio, 14:3015:30
Pierre Mathieu (AixMarseille)
Random walks on
(hyperbolic) groups
Abstract
3 June 2015 –
Aula di Consiglio, 15:3016:30
Fabio Trova (Roma « Tor Vergata »)
Quantization of local
systems over finite homotopy types
In Topological Quantum Field Theories from
Compact Lie Groups D.S. Freed, M.J. Hopkins, J. Lurie, and C. Teleman have suggested the existence of an ndimensional Extended Topological
Quantum Field Theory arising from representations of finite homotopy
types. Unfortunately, no real indication is given on how this TQFT should be
defined: only a few hints concerning the need of equivalences between certain
limits and colimits and a sketchy account of the case
of trivial representations can be found in the article.
I will make this
conjecture rigorous in dimension 1,
under the sole assumption of the existence of duals, and explain how the
construction could be promoted to that of an ndimensional theory. As an aside result, I will show how certain
notions from the representation theory of finite groups can be stated in the
general framework of categories with duals; in particular, the Nakayama
isomorphism will be recovered.
Time permitting,
related problems will be discussed.
Jan Schröer (Bonn)
Enveloping algebras
of KacMoody algebras as convolution algebras
This is joint work with Christof
Geiss and Bernard Leclerc. The positive part of a
symmetric KacMoody Lie algebra can be realized as a
convolution algebra of constructible functions on representation varieties of
acyclic quivers.
We will explain how to generalize this from the
symmetric to the symmetrizable case. In this more
general setup we need quivers with loops and relations.
Ryan Kinser
(Iowa)
Combinatorial formulas for type A orbit closures
This talk is based on joint work with Allen Knutson
and Jenna Rajchgot. Orbit closures of quivers come up in several
of areas of mathematics, for example: representations of finitedimensional
algebras, Lusztig's construction of the canonical
basis, generalizations of determinantal varieties,
and degeneracy loci for maps of vector bundles. This talk is most related to the last
one, where "formulas" for orbit closures means their equivariant Kclasses and cohomology
classes.
Previous work of many people (e.g. Buch,
Fulton, Feher, Rimanyi,
Knutson, Miller, Shimozono) produced such formulas in
the case where all arrows of the quiver point the same direction, often having
positive structure constants in some particular basis. We generalize some of these formulas to
Type A quivers of arbitrary
orientation. The main ingredient is
the bipartite Zelevinsky map constructed in previous
work of Rajchgot and mine, which identifies orbit
closures with intersections of Schubert varieties and opposite Schubert cells
in a partial flag variety.
Fabio Gavarini (Roma « Tor Vergata »)
Affine supergroups and super
HarishChandra pairs
Together with any supergroup,
one can naturally associate the pair made of its classical (i.e. non super)
underlying group and its tangent Lie superalgebra,
two objects which obey some obvious mutual compatibility constraints; any similar
pair is called "super HarishChandra pair" (=sHCp). This construction leading from supergroups to sHCp's is functorial, and actually an equivalence, as an explicit
quasiinverse functor is known.
In this talk I present a new, totally different recipe
for such a quasiinverse: indeed, it extends to a much larger setup, with a
more geometrical method. I shall
mainly adopt the point of view of algebraic (super)geometry, but the bunch of
ideas and results we shall be dealing with actually applies to the real
differential, the real analytic and the complex analytic case as well.
Reference:
F. Gavarini, "Global splittings and super HarishChandra pairs for affine supergroups", Transactions
of the American Mathematical Society (to appear), 56 pages – see http://arxiv.org/abs/1308.0462
Kieran O’Grady (Roma « Sapienza »)
Sestiche EPW
Abstract
Paul Arne Østvær (Oslo)
Milnor's conjecture on quadratic forms
John Milnor proposed two influential
conjectures on Galois cohomology and quadratic forms in his 1969 landmark paper "Algebraic Ktheory and quadratic forms" (Inv. Math.). Both conjectures were solved in the affirmative by
Vladimir Voevodsky and collaborators
in papers published in 2003
(IHES journal) and 2007 (Annals of Math). We propose an alternate solution
to Milnor's conjecture on quadratic forms. The proof proceeds by an explicit spectral sequence computation. This is joint work with Oliver Rondigs (arXiv:1311.5833, to appear
in: Geometry and Topology).