Algebra and geometry seminar



Abstracts 2018/2019



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.




10 October 2018

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.




17 October 2018 – Aula L

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.



24 October 2018

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.



31 October 2018

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.



7 November 2018

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.




14 November 2018

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.




21 November 2018 – Aula L

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.




28 November 2018

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.




5 December 2018

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. 




19 December 2018

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




9 January 2019

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).




16 January 2019

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.




30 January 2019

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.




6 February 2019

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.




13 February 2019

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.



20 February 2019

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.



27 February 2019

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.




6 March 2019

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.




13 March 2019

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.




27 March 2019 – Aula G

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.




3 April 2019

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.




10 April 2019

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.




17 April 2019

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.




24 April 2019

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).




8 May 2019

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.



15 May 2019

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.



22 May 2019

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.




5 June 2019

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.




12 June 2019

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.




19 June 2019

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.




26 June 2019

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.