Algebra and geometry seminar
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Abstracts
2018/2019
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3
October 2018 – Exceptionally at 14:30-15:30
Frédéric Patras (Nice)
Shuffle groups
There is a classical duality between algebraic groups
(and finite groups, continuous groups...)
and commutative Hopf algebras. Motivated by several theories and phenomena
(universal products in algebra, free probabilities, the combinatorics of
noncrossing partitions...), we investigate here a noncommutative extension of
the theory that gives rise to a family of groups and Lie algebras equipped with
several generalized logarithms (natural maps from the group to the Lie
algebra). The interaction between these various maps and their inverses has a
rich algebraic and combinatorial structure, related among others to the one of
noncrossing, boolean and monotone partitions. J.w. with K. Ebrahimi-Fard.
Catriona Maclean (Grenoble)
Classifying
approximable algebras
Approximable algebras, introduced by Huayi Chen, are
those for which a Fujita-type theorem holds.
Any full dimensional sub-algebra of the section
algebra of a big line bundle is approximable : we will see that the converse is
false, but becomes true if the notion of big line bundle is somewhat
generalised.
Simone Diverio (Roma
“Sapienza”)
La congettura di Lang per due classi di
esempi interessanti
La congettura di Lang
predice in particolare che una varietà proiettiva Kobayashi iperbolica
è di tipo generale, così come tutte le sue sottovarietà (lisce
o meno). Tale conseguenza della congettura di Lang non era nota fino a poche
settimane fa neppure per i seguenti due “esempi zero” della teoria:
varietà di Kähler compatte con curvatura sezionale olomorfa
negativa e quozienti liberi e compatti di domini limitati dello spazio affine
complesso. In questo seminario spiegherò il problema e darò
un’idea delle recenti dimostrazioni per queste due classi di esempi: la
dimostrazione per la prima classe è dovuta a H. Guenancia mentre per la
seconda a S. Boucksom e me medesimo.
Nicolas Bergeron (Jussieu)
On the cohomology
ring of the universal K3 surface
The Deligne decomposition theorem reduces the study of
the cohomology groups of the universal K3 surface, or more generally of
universal families of polarized hyperkähler varieties, to the study of
certain spaces of automorphic forms. This makes it possible to prove a
cohomological version of the generalized Franchetta conjecture due to O'Grady
but also to better understand the ring structure on the cohomology of these
universal families. This is a joint work with Zhiyuan Li.
Ivan Babenko (Montpellier)
Homology length
spectrum, systolic zeta-function and isoperimetric inequalities
In the talk we will define Dirichlet type series
associated with homology length spectra of Riemannian, or Finsler, manifolds,
or polyhedra, and will describe some of their analytical properties. As a
consequence we obtain an inequality analogous to Gromov's classical
intersystolic inequality, but taking the whole homology length spectrum into
account rather than just the systole.
Benoît Cadorel (Aix-Marseille)
Subvarieties of
quotients of bounded symmetric domains
The main conjecture of the field of complex
hyperbolicity, due to Green, Griffiths and Lang, states that a complex
projective manifold of general type should contain a proper algebraic
"exceptional" locus, containing all the entire curves, and all the
subvarieties which are not of general type.
This problem has recently attracted a lot of attention
for manifolds which are compactifications of quotients of bounded symmetric
domains by arithmetic lattices. Recent results of Brunebarbe and Rousseau show
that such a variety satisfies the Green-Griffiths-Lang conjecture, provided the
lattice is small enough. Our goal is to present a approach based on a metric
positivity criterion of Boucksom, which allows to state precise conditions for
the subvarieties to be of general type, or for the entire curves to be included
in the boundary. If we assume the Green-Griffiths-Lang conjecture, our criteria
permit in principle to get upper bounds for the dimension of the exceptional
locus, independently of the choice of lattice: we will illustrate this by
giving new effective examples for quotients of the ball, or of the generalized
upper half space of Siegel.
Jacopo Gandini (Pisa)
L'ordinamento di Bruhat sulle varietà
simmetriche hermitiane e sugli ideali abeliani
Sia G un gruppo algebrico
semplice, sia P un sottogruppo parabolico di G con radicale unipotente abeliano
e sia L un fattore di Levi di P, allora G/L è detta una varietà
simmetrica di tipo hermitiano per G. Se B è un sottogruppo di Borel di G
contenuto in P, allora B agisce con un numero finito di orbite sia su G/L che
sul radicale unipotente di P. Scopo del seminario sarà descrivere
l'ordinamento di inclusione tra le chiusure di tali orbite, parametrizzate
combinatoricamente mediante opportuni elementi del gruppo di Weyl di G. Tali
descrizioni, congetturate rispettivamente da Richardson e da Panyushev, sono
state ottenute in collaborazione con A. Maffei.
Generalizzando il caso dei
radicali unipotenti abeliani, considererò successivamente l'analogo
problema di decomporre in B-orbite un qualsiasi ideale abeliano dell'algebra di
Lie di B. Spiegherò come in questo caso le relazioni di inclusione tra
le chiusure delle B-orbite possono essere descritte mediante la combinatorica
del gruppo di Weyl affine associato a G. Questa seconda parte del seminario
è basata su una collaborazione con A. Maffei, P. Moseneder Frajria e P.
Papi.
Lidia Angeleri Hügel
(Verona)
Silting complexes
over hereditary rings
I will report on joint work with Michal Hrbek. Given a
hereditary ring, we use the lattice of homological ring epimorphisms to
construct compactly generated t-structures in its derived category. This leads
to a classification of all (not necessarily compact) silting complexes over the
Kronecker algebra.
Fabrizio Andreatta (Milano
UNIMI)
p-adic variations of
automorphic sheaves
Elliptic modular forms are sections of powers of the
Hodge bundle. Starting with
the works of J.P. Serre and N. Katz
more than 30 years ago, it was discovered that, given a prime number p, modular
forms live in p-adic families. This phenomenon is the geometric counterpart of
the theory fo p-adic deformations of Galois representations and has become a
basic tool in modern Number Theory. I will present joint work with A. Iovita
and V. Pilloni where we construct p-adic families of modular forms as sections
of p-adic powers of the Hodge bundle.
Angelo Vistoli (SNS Pisa)
Essential dimension
of finite groups
Essential dimension is a fundamental way of measuring
the complexity of a finite group G. It measures how many independent parameters
are needed to describe all Galois extensions E/K with group G, where K is an
extension of a fixed base field k. For example, if k contains a primitive n-th
root of 1 and G is a cyclic group of order n, all Galois extensions E/K with
Galois group G is obtained by taking an n-th root of an element of K; this
means that the essential dimension of a cyclic group of order n is 1.
I will review some of what is known about essential
dimension of finite groups and the main techniques that are used in the
subject, including some very recent applications of birational geometry.
Bernhard Krötz (Paderborn)
Plancherel theory for
real spherical spaces
The talk will be accessible to a general audience. We
wish to explain on how to relate the spectral theory of a real spherical space
to the orbit structure of a smooth toroidal compactification. When specialized
to real reductive groups we recover in an elementary way large parts of the
Plancherel theorem of Harish-Chandra.
Joint work with P. Delorme, F. Knop and H. Schlichtkrull. Paper link https://arxiv.org/pdf/1807.07541.pdf
Eduardo Esteves (IMPA)
Limit linear series:
an approach for all stable curves
I will present an approach to extend Eisenbud’s
and Harris’ notion of limit linear series to all stable curves. I will
describe the objects, how they vary in families and how a fine moduli space for
them can be constructed. The talk is based on ongoing work with Omid Amini (ENS
Paris), Margarida Melo and Filippo Viviani (U. Roma Tre).
Paolo Antonini (SISSA)
La congettura di Baum-Connes localizzata nell’elemento
unità di un gruppo discreto
In collaborazione con S.
Azzali e G. Skandalis abbiamo costruito per un gruppo discreto G, una versione della mappa di assembly di
Baum-Connes “localizzata nell’elemento identità di G”. Ne risulta una versione più
debole della congettura di Baum-Connes che ancora implica la congettura di
Novikov (forte) sull’invarianza omotopica delle segnature superiori delle
varietà.
Nella prima parte del
seminario verranno introdotte la congettura di Novikov e la congettura di Baum
Connes; nella seconda parte verrà illustrata la costruzione della mappa
localizzata.
In particolare il
collegamento con la congettura di Novikov è mostrato calcolando il
dominio della nuova mappa tramite un confronto preciso, in termini di K-teoria
a coefficienti reali, tra lo spazio classificante per azioni proprie di G con quello che classifica le azioni libere e
proprie.
Giovanna Carnovale (Padova)
Algebre di Nichols su gruppi finiti semplici
Le algebre di Nichols sono
una famiglia di algebre che si costruiscono per generatori e relazioni a
partire da un endomorfismo di V ÄV che soddisfa l'equazione delle trecce
(intrecciamento). Tra di esse appaiono l'algebra esterna, l'algebra simmetrica,
le algebre di Borel quantiche. Anche le algebre introdotte da Fomin e Kirillov
per lo studio del calcolo di Schubert nelle varietà delle bandiere per
GL(n) sono strettamente legate alle algebre di Nichols. È possibile
costruire un intrecciamento a partire da un G-modulo G-graduato con compatibilità
tra le due strutture ed è stato congetturato che tutte le algebre di
Nichols ottenute da un simile intrecciamento hanno dimensione infinita se G
è semplice finito non abeliano. La congettura è stata verificata
per il gruppo alterno e per quasi tutti i gruppi sporadici da Andruskiewitsch,
Fantino, Grana e Vendramin e parzialmente per PSL(2,q) da Freyre Grana e
Vendramin. In questo seminario descriverò risultati su questa congettura
ottenuti in collaborazione con N. Andruskiewitsch e G. Garcia per gruppi di
Chevalley e Steinberg e con Mauro Costantini per gruppi di Suzuki e Ree.
Giovanni Felder (ETH Zürich)
Representation
homology and supersymmetric gauge theory
Representation homology arises in the derived version of
the space parametrizing finite dimensional representations of an associative
algebra.
I will discuss some examples of this notion related to
quivers and its relation to supersymmetric gauge theory in 4 dimensions and
conformal field theory in 2 dimensions.
The talk is based on joint work with Y. Berest, M.
Müller-Lennert, S. Patotski, A. Ramadoss and T. Willwacher.
Thomas Willwacher (ETH Zürich)
Models for
configuration and embedding spaces via Feynman diagrams
Extending ideas of Kontsevich, we show that Feynman
diagrams of topological quantum field theories can be used to construct
combinatorial models for various spaces of interest in topology, including in
particular configuration spaces of points on manifolds and embedding spaces.
Tamás Hausel (IST Austria)
Representations of
quivers over Frobenius algebras
I will discuss an arithmetic Fourier transform
approach towards under-standing locally-free representations of quivers over
commutative Frobenius algebras. Joint work with Emmanuel Letellier and Fernando
R. Villegas.
Peter Teichner (MPI Bonn)
The group of two
2-spheres linked in 4-spaces
We’ll give an introduction to the mathematics of
4-dimensional knot theory, i.e. of embedded 2-spheres in 4-space, usually
called 2-knots. We’ll start with a simple construction that spins a
knotted arc in 3-space into such a 2-knot. Then we’ll discuss in how many
ways a second 2-knot can lie in the complement of the first. Finally,
we’ll report on recent joint work with Rob Schneiderman, in which we
proved the non-existence of a 4-dimensional Hopf link: Given any (possibly
singular) 2-sphere in the complement of a 2-knot, there is always a deformation
(link homotopy) to the trivial situation in such a way that both components
stay disjoint at all times.
Joel Kamnitzer (Toronto)
Mirkovic-Vilonen
cycles, preprojective algebra modules and Duistermaat-Heckman measures
Via the geometric Satake correspondence, the
Mirkovic-Vilonen cycles give bases for representations of semisimple Lie
algebras. Similarly, by work of
Lusztig, generic preprojective algebra modules give bases for these
representations as well. It is a
long-standing open problem to compare these bases.
We will explain a new geometric way to make this
comparison. As an application, we
will show that these bases do not agree in SL6.
Nima Moshayedi (Zürich)
Globalization of
Nonlinear AKSZ Sigma Models in the BV-BFV Formalism
We will give an introduction to the BV-BFV formalism
and discuss the setting of certain AKSZ theories. Moreover, we will describe a
globalization procedure using concepts of formal geometry, which extends the
Quantum Master Equation for manifolds with boundary. If time permits, we will also
talk about the special case of the Poisson Sigma Model and the relational
symplectic groupoid construction of the globalization of Kontsevich‘s
star product.
François Bergeron (UQAM)
Rectangular
multivariate modules of harmonic polynomials
Much of the work of the last 25 years on (modified)
Macdonald symmetric polynomials, and operators for which they are joint
eigenfunctions, has recently be nicely synthesized via an operator realization
of the elliptic Hall algebra (introduced by Burban-Vasserot-Schifmann). This
has opened the way to generalizations of the interaction between interesting
questions in many areas including: Algebraic Combinatorics (rectangular Catalan
combinatorics), Symmetric Functions (compositional shuffle conjecture/theorem,
nabla operator), Knot Theory (Khovanov-Rozansky homology of (m,n)-torus knots),
and Theoretical Physics (boson-fermion supersymmetry). We will present a new
link between all these subjects and representation theory, by describing modules
of bivariate diagonal polynomials that generalize to the (m,n)-rectangular
context the Garsia-Haiman module of diagonal harmonic polynomials (in 2 sets of
n variables). This makes it natural to extend most of the previous work to the
multivariate context (k sets of n variables). If time allows, we will finally
explain how such an extension unifies in a surprising manner many questions of
the domain.
Maria Gorelik (Weizmann)
Snowflake modules for
Lie superalgebras
Quasi-integrable modules over symmetrizable Kac-Moody
algebras form a semisimple subcategory
in the category O.
For affine Lie algebras these modules appear as modules over a simple vertex
algebra (Arakawa's Theorem).
In this talk I will speak about a joint project with V.
Serganova studying quasi-integrable (snowflake) modules for Lie superalgebras.
Olivier Benoist (ENS Paris)
On the
Clemens-Griffiths method over non-closed fields
The Lüroth problem asks whether every unirational
variety is rational. Over the field C of
complex numbers, it has a positive answer for curves and surfaces, but fails in
higher dimensions.
In this talk, I will consider the Lüroth problem
for geometrically rational varieties over a non-algebraically closed field k. Adapting in this context the strategy implemented by
Clemens and Griffiths over C, I
will describe new examples of k-varieties
that are geometrically rational, k-unirational,
but not k-rational.
This is joint work with Olivier Wittenberg.
Andrea Ricolfi (SISSA)
Una componente dello schema di Hilbert di una
jacobiana iperellittica
Data una curva liscia C di
genere g>2, immersa nella sua jacobiana J(C) tramite una mappa di
Abel-Jacobi, la componente H(C) dello schema di Hilbert di J(C) che contiene
tale immersione ha gli stessi punti chiusi di J(C). È noto che H(C) =
J(C) come schemi se e solo se C è non-iperellittica. Descriveremo la
struttura di schema di H(C) nel caso iperellittico. Come corollario, il
risultato determina la struttura di schema sugli spazi di moduli di fasci di
Picard su jacobiane.
Benoit Claudon (Rennes)
Fundamental groups of
compact Kähler threefolds
This talk will be concerned with the Kodaira problem
for the fundamental group which consists in asking whether the fundamental
group of a compact Kähler manifold can be also realized as the fundamental
group of a smooth projective variety. I will explain how to get a positive
answer to this question in dimension 3 (joint work with Hsueh-Yung Lin and
Andreas Höring).
Federico Binda (Regensburg)
Specialization
theorems for cycles of relative dimension 0
In this talk, we will present a relation between a
certain group of algebraic cycles, with finite coefficients, on a regular quasi-projective
scheme X, flat over an excellent Henselian DVR A with perfect residue field, and the motivic cohomology (with
compact support) of the special fiber, in the range classically corresponding
to the group of zero cycles. When X is projective over A, this relation was studied by Bloch and
Esnault–Kerz–Wittenberg, generalizing previous works by Sato and
Saito, and implies a finiteness result for the Chow group of zero cycles of a
smooth projective variety over a p-adic field. Our main result can be
interpreted as a proper base change theorem with compact support for relative
0-cycles.
This is a joint work (in progress) with Amalendu
Krishna.
Francesco Bonsante (Pavia)
Energia L1 di mappe tra superfici
iperboliche e teoria di Teichmüller
Nel seminario
parlerò di una energia L1 di mappe tra superfici iperboliche,
introdotta da Trapani e Valli.
I punti critici di tali
mappe sono caratterizzati da essere mappe che preservano l’area e il cui
grafico è una superficie minima.
Schoen ha mostrato che in
ogni classe di omotopia di mappe esiste sempre un unico rappresentante
minimizzante per questa energia L1.
Considerando lo spazio di
Teichmüller T(S) come lo spazio
delle metriche iperboliche su S a meno dell’azione dei diffeomorfismi
isotopi all’identità, considererò la funzione su T(S)×T(S) ottenuta minimizzando l’energia L1
dell’identità sulle orbite. In particolare discuterò come
tale funzione degeneri quando una delle variabili diverge ad un punto del bordo
di Thurston di T(S).
Nella seconda parte del
seminario proporrò una generalizzazione del funzionale nel caso di mappe
da una superficie iperbolica in una 3-varietà, ricavandone
l’equazione di Eulero-Lagrange e studiando lo spazio delle immersioni
minimizzanti.
Il lavoro presentato
è in collaborazione con Gabriele Mondello e Jean-Marc Schlenker.
Gordan Radobolja (Split)
Free field
realization of twisted Heisenberg-Virasoro algebra at level zero and its
applications
Twisted Heisenberg-Virasoro vertex algebra H is an important example of vertex
algebra with many applications in the representation theory and conformal field
theory. If the level of the corresponding Heisenberg vertex subalgebra is
non-zero, H is isomorphic to the
tensor product of Heisenberg vertex algebra and the (universal or simple)
Virasoro vertex algebra. The study of the twisted Heisenberg-Virasoro vertex
algebra at the level zero was initiated by Y. Billig motivated by applications
to the toroidal Lie algebras. It turns out that level zero has another interesting
application - in realization of a class of vertex algebras which appear in
physics literature under the name of Galilean algebra.
I shall present a joint work with D. Adamovic on
bosonic realization of this algebra and its highest weight representations.
Furthermore I will discuss on some of many applications of this realization.
Tueday 28 May 2019 – Aula di Consiglio,
14:30
Charles Favre (CNRS-CMLS)
Hybrid spaces
We shall explain how the construction of a hybrid
space by Berkovich enables one to understand the degeneration of natural
measures in various contexts in dynamics.
Boris Botvinnik (Oregon)
Spaces and moduli
spaces of metrics of positive scalar curvature
I will review several basic constructions related to
positive scalar curvature metrics: surgery constructions, Dirac operator, index
theory and some other relevant topological tools. Then we will discuss some
recent results on the spaces and moduli spaces of metrics of positive scalar
curvature.
Giulia Saccà
(MIT-Columbia)
I numeri di Hodge di O’Grady 10
In questo seminario
presenterò un lavoro svolto in collaborazione con A. Rapagnetta e M. de
Cataldo, in cui si calcolano i numeri di Hodge della varietà simplettica
olomorfa di dimensione 10 nota come O'Grady 10.
Il metodo usa un
raffinamento del teorema del supporto di Ngô.
Dylan Allegretti (Sheffield)
Quiver
representations, cluster varieties, and categorification of canonical bases
Associated to a compact oriented surface with marked
points on its boundary is an interesting class of finite-dimensional algebras.
These algebras are examples of gentle algebras, and their representation theory
has been studied by many authors in connection with the theory of cluster
algebras. An important fact about these algebras is that their indecomposable
modules come in two types: string modules, which correspond to arcs connecting
marked points on the surface, and band modules, which correspond to closed
loops on the surface. Thanks to the work of many mathematicians, the string
modules are known to categorify generators of a cluster algebra. In this talk,
I will explain how, by including band modules in this story, one can define a
family of graded vector spaces which categorify Fock and Goncharov's canonical
basis for the algebra of functions on an associated cluster variety. These
vector spaces are of interest in mathematical physics, where they are expected
to provide a mathematical definition of the space of framed BPS states from the
work of Gaiotto, Moore, and Neitzke.
Luca Schaffler (UMass – Amherst)
Compattificazioni dello spazio di moduli
delle superfici K3 con un automorfismo puramente non-simplettico di ordine 4
Nello studio delle
varietà algebriche di dimensione alta, gli spazi di moduli compatti
rivestono un ruolo centrale, e studiare compattificazioni diverse dà
informazioni sulla loro geometria birazionale.
In questo seminario,
consideriamo lo spazio di moduli delle superfici K3 con un automorfismo
puramente non-simplettico di ordine 4 ed una polarizzazione specifica. Kondo ha
costruito queste superfici a partire da otto punti in P1
come rivestimenti doppi di P1´P1. Segue dal lavoro di Deligne e Mostow che la
compattificazione GIT di questo spazio di moduli è isomorfa alla
compattificazione di Baily-Borel. Ma queste compattificazioni hanno un
significato geometrico debole.
Dimostriamo che la
desingolarizzazione parziale di Kirwan della compattificazione GIT ha un'interpretazione
modulare in termini di coppie stabili nel senso del Minimal Model Program, e
quindi ha un significato
geometrico molto ricco.
Descriviamo le degenerazioni parametrizzate dal bordo e le singolarità
che occorrono.
Questo lavoro è in
collaborazione con Han-Bom Moon.
3
July 2019 – Time 14:30 – Building CU022 – Aula Giacomini
Frances Kirwan (Oxford)
Moment maps and
non-reductive geometric invariant theory
When a complex reductive group acts linearly on a
projective variety the quotient in the sense of geometric invariant theory
(GIT) can be identified with an appropriate symplectic quotient. In general GIT
for non-reductive linear algebraic group actions is much less well behaved than
for reductive actions. However GIT for a linear algebraic group with internally
graded unipotent radical U (in the sense that a Levi subgroup has a central
one-parameter subgroup which acts by conjugation on U with all weights strictly
positive) is almost as well behaved as in the reductive setting, provided that
we are willing to multiply the linearisation by an appropriate rational
character. In this situation we can ask for moment map descriptions of the
quotient. This is related to the symplectic implosion construction (introduced
in a 2002 paper of Guillemin, Jeffrey and Sjamaar), work by Madsen and Swann on
multi-moment maps and recent work by Greb and Miebach on Hamiltonian actions of
unipotent groups on compact Kähler manifolds.