Algebra and Geometry Seminar

Abstracts of talks 2010/2011

20 October 2010
Roberto Frigerio (Università di Pisa)
Rigidity of manifolds with(out) nonpositive curvature

In recent years, there has been an extensive amount of work done on proving rigidity results for various classes of non-positively curved spaces.
In this talk I describe some results obtained in collaboration with J.F. Lafont and A. Sisto, which concern analogous rigidity theorems for a class of manifolds which are ``mostly'' non-positively curved, but may not support any actual non-positively curved metric.
More precisely, we define a class of manifolds which contains non-positively curved examples. Building on techniques coming from geoemtric group theory, we show that smooth rigidity holds within our class of manifolds, and that our manifolds are topologically rigid (i.e. they satisfy the Borel conjecture).
We also discuss some results concerning the quasi-isometry type of the fundamental groups of mostly non-positively curved manifolds.

27 October 2010
Bruno Benedetti (Technische Universität Berlin)
Duality, discrete Morse theory, and constructions for manifolds

Reeb's Sphere Theorem says that the d-sphere is the only closed d-manifold that may admit a Morse function with exactly two critical points. A `triangulated version' of this result was obtained by Forman in 1999 using his discrete Morse theory. We present a new version of discrete Morse theory specific for manifolds with boundary and ``dual'' to the classical theory. We obtain a Ball Theorem, analogous to Forman's Sphere Theorem, and slightly stronger than a classical result by Whitehead. If time permits, we will discuss applications to combinatorics, differential topology, or enumerative combinatorics.

3 November 2010
Filippo Viviani (Università di Roma 3)
La mappa di Torelli: compattificazione e tropicalizzazione

La mappa di Torelli classica è la mappa dallo spazio dei moduli delle curve allo spazio dei moduli delle varietà abeliane, che manda una curva nella sua Jacobiana. Il teorema di Torelli asserisce che la mappa di Torelli è iniettiva sui punti geometrici.
In questo seminario, proporremo due estensioni del teorema di Torelli: una per la mappa di Torelli compattificata e una per la mappa di Torelli tropicale.
La mappa di Torelli compattificata (definita da Alexeev) è la mappa modulare dallo spazio dei moduli delle curve stabili allo spazio di moduli delle coppie semi-abeliche stabili, che manda una curva stabile di genere g nella Jacobiana compattificata di grado g-1 munita del divisore theta e dell'azione della Jacobiana generalizzata.
In un lavoro in collaborazione con Lucia Caporaso, descriviamo le fibre della mappa di Torelli compattificata. D'altro parte, in un lavoro in collaborazione con Silvia Brannetti e Margarida Melo, costruiamo lo spazio dei moduli delle curve tropicali e delle varietà abeliane tropicali e definiamo la mappa di Torelli tropicale.
In un altro lavoro in collaborazione con Lucia Caporaso, descriviamo le fibre della mappa di Torelli tropicale.
Nel corso del seminario, presenteremo un panoramica dei suddetti risultati cercando di evidenziare il legame tra il Torelli compattificato e il Torelli tropicale.

9 November 2010 - Special seminar! Dipartimento di Scienze di Base per l'Ingegneria - Aula 1E, 11:30-12:30
George Glauberman (University of Chicago)
A counterexample for Lie rings from a finite p-group

A beautiful theorem of Michel Lazard states that for certain "small" p-groups S, we may define operators denoted by x+y and [x,y] so that S becomes a Lie ring.
In this talk, we discuss the background of this theorem and describe how some, but not all, of its conclusions may be generalized to a wider family of p-groups.
Part of this work stems from discussions at the Ischia Group Theory 2010 Conference.

9 November 2010 - Unusual day and time: Aula di Consiglio 14-15
Marco Boggi (Universidad de los Andes)
Galois covers of moduli spaces of curves and loci of curves with symmetry

An idea which proved to be extremely useful in anabelian geometry is that the same amount of information contained in the algebraic fundamental group of a hyperbolic curve is contained in the first homology groups of all its finite coverings but then these are easier to handle. This idea was implicit in Looijenga construction of smooth Galois coverings of moduli spaces of stable curves.
My aim is to generalize Looijenga construction in order to implement the above philosophy to the study of arbitrary level structures over moduli of curves. Moreover, a simple geometric interpretation of these level structures is given in terms of loci of curves with symmetry. This allows to give a combinatorial description of the nerve of their Deligne-Mumford boundaries.

10 November 2010
Dmitri Panov (King’s College of London)
Polyhedral Kähler manifolds

Polyhedral Kähler metrics are piecewise flat Kähler metrics defined on complex manifolds that appear in various situations. For example, we can use it to prove that certain algebraic surfaces are of type K(π,1), and to provide new constructions of such surfaces. Or we can use it to find new "sphere type" conjectures in polyhedral geometry, such as the following one, that we will discuss in the talk:
Conjecture. Consider a manifold M with a polyhedral metric, i.e. a manifold obtained by gluing a collection of Euclidean simplexes. Call it positively curved if the conical angles along co-dimension 2 faces of M are at most 2π. Suppose that the holonomy of the metric on M is irreducible and b₂(M)>0. Then M has a natural holomorphic structure with respect to which it is biholomorphic to CPⁿ and the original polyhedral metric on M is a singular Kähler metric with respect to this complex structure.

17 November 2010
Alessandro Chiodo (Université de Grenoble)
La corrispondenza Landau-Ginzburg/Calabi-Yau

Nonostante molti tentativi in fisica e in matematica il problema del calcolo degli invarianti di Gromov-Witten delle curve di genere g tracciate su una ipersuperficie di Calabi-Yau (f=0) in uno spazio proiettivo CPⁿ resta irrisolto. Grazie alla teoria geometrica degli invarianti - attraverso un cambiamento della condizione di stabilità - si pu correlare la geometria di (f=0) in CPⁿ a quella della singolarità all'origine del cono corrispondente in Cⁿ⁺¹. Nel 1993, Witten ha enunciato l'idea che questi due modelli - l'ipersuperficie di Calabi-Yau e la singolarità - siano "due fasi della stessa teoria". Questa corrispondenza ammette una formulazione in termini di invarianti di Gromov-Witten. È stata dimostrata in genere zero in collaborazione con Yongbin Ruan ed è stata generalizzata e messa in relazione con l'equivalenza di Orlov in collaborazione con Hiroshi Iritani e Yongbin Ruan.

23 November 2010 - Unusual day and time: Aula di Consiglio, 13-15
Corrado De Concini (Università "Sapienza" di Roma)
Introduzione alla fibrazione di Hitchin

(abstract)

24 November 2010
Riccardo Salvati Manni (Università "Sapienza" di Roma)
Siegel threefolds with a Calabi-Yau model

Recentemente in collaborazione con Freitag abbiamo descritto delle varietà modulari di Siegel che ammettono un modello di weak Calabi-Yau (non necessariamente proiettive). I punti di partenza della nostra investigazione sono: una compattificazione toroidale introdotta da Igusa e/o una varietà modulare introdotta da Van Geemen e Nygaard. In questo modo abbiamo trovato circa 4000 varietà modulari di Siegel che ammettono un modello di weak Calabi-Yau. Daremo un criterio di proiettività. Per alcuni casi significativi abbiamo calcolato i numeri di Hodge.

1 December 2010
Maria Gorelik (Weizmann Institute of Science)
On Kac-Wakimoto Denominator Identities for Lie superalgebras

In 1972 I. G. Macdonald generalized a classical formula of H. Weyl, obtaining, in particular, a formula for certain powers of η-function which include some classical identities of Jacobi. In 1994 V. Kac and M. Wakimoto conjectured a super-analogue of Macdonald identities and proved it for some special cases. Specializations of these identities give, in particular, Jacobi and Legendre formulas for representing an integer as a sum of squares or a sum of triangular numbers, respectively. In this talk I will review recent results in this area.

7 December 2010 - Unusual day and time: Aula di Consiglio, 14-15
Pierre Albin (Institut de Mathématiques de Jussieu)
Equivariant cohomology and resolution

We extend to general group actions the simple statement: the equivariant cohomology of a space is the cohomology of the space of orbits. This is literally true only for free actions; we show that otherwise the equivariant cohomology can be computed by a de Rham-like complex on a compactification of the regular part of the orbit space.
We also extend the `delocalized' cohomology of Baum, Brylinski, and MacPherson from Abelian group actions to arbitrary compact group actions. This is joint work with Richard Melrose.

Four lectures on Hitchin fibration, endoscopy and mirror symmetry
Luca Migliorini (Università di Bologna)

Lecture I: 9 December 2010, Aula Picone, 13-15
Lecture II: 10 December 2010, Aula II, 14-16
Lecture III: 15 December 2010, Aula B, 13:30-15:30
Lecture IV: 17 December 2010, Aula B, 13:30-15:30

Abstract

Bibliography on Ngô's support theorem: survey papers of De Cataldo (section 3) and Ngô.

12 January 2011
Leandro Arosio (Università "Sapienza" di Roma)
Resonances and direct limits in Loewner equations

Classical Loewner theory in the unit disc of the complex plane was introduced by C. Loewner in 1923 and has been since then used to prove several deep results in geometric function theory. Loewner theory is one of the main ingredients in the proof of the Bieberbach conjecture given by de Branges in 1985, and gives the basis for the Schramm-Loewner evolution introduced by Schramm in 2000 to study the scaling limits of two-dimensional lattice models in statistical physics. Recently Bracci, Contreras and Daz-Madrigal proposed a generalization of the theory on the unit disk. I present the theory in several complex variables, proving by a direct limit procedure the existence of an abstract solution for any Loewner PDE. I also show, by solving a parametric Schroeder equation, that any Loewner PDE of dilation type admits a solution in the classical sense, that is with range in Cⁿ.

19 January 2011
Christophe Soulé (IHES)
Secant varieties and arithmetic surfaces

Let Σ be a secant variety associated to a smooth projective curve. We give an upper bound for the dimension of linear subspaces in Σ. We then use this result to get lower bounds for the successive minima of the first cohomology group of an arithmetic surface, with coefficients in some hermitian line bundle.

26 January 2011 - Unusual time - Aula di Consiglio 13:30-14:30
Ben Moonen (Universiteit van Amsterdam)
The Torelli locus and special subvarieties

I shall discuss what is known about special subvarieties ('Shimura subvarieties') in the moduli space A_g that are contained in the Torelli locus. The study of such subvarieties is motivated by a conjecture of Coleman, via a conjecture of Andre-Oort. (I will explain what these conjectures are about.) In my talk I shall discuss the non-trivial examples that are presently known, leading to counterexamples to Coleman's conjecture for small genera. I shall also discuss restrictions coming from work of Hain and de Jong-Zhang, among others.

2 February 2011 - Unsual time: Aula di Consiglio 15:30-16:30
Gérard Laumon (Université Paris-Sud 11)
An extension of Ngo Bao Chau's decomposition theorem

The proof of the Langlands-Shelstad fundamental lemma by Ngo Bao Chau, and its extension to the Arthur weighted fundamental lemma by Pierre-Henri Chaudouard and myself, are based on an important cohomological property of the Hitchin fibration.
In the talk I would like to present this cohomological property in the particular case of the Hitchin fibration for GL_n in characteristic 0, and to sketch its proof.

16 February 2011
Stefano Trapani (Università di Roma "Tor Vergata")
Classification of taut Stein surfaces with proper R-actions

In this talk I will describe describe a complete classification of two dimensional Stein taut manifolds having a non trivial connected subgroup of the group of biholomorphisms.
The first step in the classification comes from a previous result by C. Miebach and K. Oeljeklaus which states the possibility of embedding such surfaces into a principal C-bundle over a Riemann surface.
This is joint work with Andrea Iannuzzi.

23 February 2011
Winfried Kohnen (Universität Heidelberg)
Generalized modular functions

Generalized modular functions are holomorphic functions on the complex upper half-plane, meromorphic at the cusps, which satisfy the usual transformation formula of a classical modular function of weight zero, however with the important exception that the character need not be unitary. The theory of such functions has been partly motivated from CFT in physics.
In this talk I will report on recent joint work with G. Mason (2010) on arithmetic properties of their Fourier coefficients and their characters.

2 March 2011
Eduard Looijenga (Universiteit Utrecht)
Homotopical properties pertaining to the moduli spaces of curves and of principally polarized Abelian varieties

In the moduli space of g-dimensional principal Abelian varieties, the decomposible ones make up a closed subvariety. We describe some homotopy properties of this pair (joint work with Wilberd van der Kallen). We do similarly for the pair consisting of the moduli space of stable curves of genus g with compact Jacobian and the locus therein parameterizing singular curves.

9 March 2011
Carlos Simpson (Université de Nice)
Structures on nonabelian cohomology

The first nonabelian cohomology of a variety is the moduli space of representations of its fundamental group. There are several different algebraic varieties corresponding to this space, and these have various interesting structures. We'll discuss these structures, their relationships, and how some of them might be generalized to higher nonabelian cohomology.

16 March 2011
Vladimir Lazić (Imperial College of London)
MMP revisited, I

I will talk about joint work with P. Cascini which gives a self-contained proof of the finite generation of the canonical ring by induction on the dimension, while avoiding standard techniques of Mori theory.

23 March 2011
Urs Schreiber (Universiteit Utrecht)
Cocycles for differential characteristic classes

Differential cohomology is a means to speak of connections on higher fiber bundles, hence a way to speak of differential refinements of ordinary cohomology classes. I will talk about a transparent general abstract formulation of the theory in terms of simplicial presheaves and then indicate how this allows the construction of concrete Cech-representative of cocycles for differential characteristic classes, such as fractional Pontryagin classes.

30 March 2011
Christian Pauly (Université de Montpellier II)
On the monodromy of the Hitchin connection

In this talk I will show that the monodromy representation of the projective Hitchin connection on the sheaf of generalized theta functions on the moduli space of vector bundles over a curve has an element of infinite order in its image. I will explain the link with conformal blocks.

6 April 2011
Max Nazarov (University of York)
Generalized Harish-Chandra isomorphism

This is a joint work with S.Khoroshkin and E.Vinberg. For any complex reductive Lie algebra g and any locally finite g-module V, we extended to the tensor product of U(g) with V the Harish-Chandra description of g-invariants in the universal enveloping algebra U(g). In our subsequent work with S.Khoroshkin, this result was used to give explicit realizations of all simple finite-dimensional modules of Yangians and their twisted analogues.

13 April 2011
Umberto Zannier (Scuola Normale Superiore di Pisa)
Punti di torsione simultanei su superfici ellittiche

Il seminario illustrerà la soluzione, sviluppata con David Masser, della seguente congettura da lui a suo tempo formulata.
Consideriamo la famiglia di Legendre di curve ellittiche E_λ, definita da y^2=x(x-1)(x-λ), (λ≠0,1), e siano P_λ, Q_λ due punti su E_λ, con ascisse risp. 2,3. Allora ci sono solo un numero finito di valori complessi di λ per cui entrambi P_λ, Q_λ sono di torsione.
Discuterò inoltre alcune variazioni di questo problema, che si può anche vedere come versione "relativa" della congettura di Manin-Mumford e come caso speciale di congetture assai generali di Pink.

20 April 2011
Yuly Billig (Carleton University)
Irreducible representations for the Lie algebra of vector fields on a torus

The goal of this talk is to construct irreducible bounded weight modules for the Lie algebra of vector fields on a torus. These modules have a weight decomposition with finite-dimensional weight spaces and possess the property that the energy operator has spectrum bounded from below. We use vertex algebra technique to give an explicit free-field realization of a family of such representations. The modules in this family are irreducible unless they belong to the chiral de Rham complex, introduced by Malikov, Schechtman and Vaintrob.
This is a joint work with V.Futorny.

27 April 2011
SPEAKER (University)
TITLE

(abstract)

4 May 2011
Stefano Francaviglia (Università di Bologna)
Il teorema di Royden per l'Outer Space

L'outer space di un gruppo libero è l'equivalente dello spazio di Teichmüller di una superficie e si può descrivere come lo spazio dei grafi metrici marcati con lo stesso gruppo fondamentale. Come il mapping class group agisce sul Teichmüller, il gruppo degli automorfismi esterni di un gruppo libero agisce sul suo outer space. Il teorema di Royden per lo spazio di Teichmüller afferma che il gruppo delle isometrie dello spazio di Teichmüller è il mapping class group. Nel seminario si darà una introduzione alla teoria dell'outer space, si discuteranno alcune possibili metriche invarianti e si discuterà l'equivalente del teorema di Royden: il gruppo di isometrie dell'outer space di un gruppo libero di rango n è Out(F_n).

11 May 2011
David Hernandez (Université de Paris 7)
Asymptotic representations and Drinfeld rational fractions

We introduce and study a category of representations of the Borel algebra, associated with a quantum loop algebra of non-twisted type. We construct fundamental representations for this category as a limit of the Kirillov-Reshetikhin modules over thequantum loop algebra and we establish explicit formulas for their characters. We prove that general simple modules in this category are classified by n-tuples of rational functions in one variable, which are regular and non-zero at the origin but may have a zero or a pole at infinity.
This is joint work with M. Jimbo.

18 May 2011
Paolo Salvatore (Università di Roma "Tor Vergata")
Cyclic formality of the operad of genus zero stable curves with tangent rays

Kontsevich and Tamarkin proved that the little 2-discs operad is formal, i.e. its chain and homology operads are isomorphic in the homotopy category. The framed little 2-discs operad by Getzler is homotopy equivalent to the Kimura-Stasheff-Voronov cyclic operad of genus zero stable curves with tangent rays at punctures and nodes. We show that this cyclic operad is formal (joint with J. Giansiracusa).

25 May 2011
Francesco Bonsante (Università di Pavia)
Diffeomorfismi lagrangiani minimali del piano iperbolico

Gli omeomorfismi quasi-simmetrici del bordo del piano iperbolico sono le tracce all'infinito dei diffeomorfismi quasi-conformi del piano iperbolico.
Il problema trattato nel seminario è il seguente: trovare un'estensione "canonica" di un omeomorfismo quasi-simmetrico del bordo ad un diffeomorfismo quasi-conforme del piano. Tale problema è correlato alla possibilità di trovare marking canonici degli elementi dello spazio di Teichmüller.
Schoen ha congetturato che esiste un'unica estensione armonica.
In un recente lavoro con Schlenker abbiamo dimostrato che esiste un'unica estensione minimale lagrangiana.
Nel seminario esporrò la problematica in generale e spiegherò la tecnica della dimostrazione del nostro risultato.

1 June 2011 - Unusual room: Aula E, 14:30-15:30
Marco Boggi (Universidad de los Andes)
Characterizing closed curves on Riemann surfaces via homology groups of coverings

Let S_g,n, for 2g-2+n>0, be a closed oriented Riemann surface of genus g from which n points have been removed. The purpose of the talk is to show that closed curves on S_g,n$ are characterized by the submodules they determine in the homology groups of finite unramified coverings of S_g,n. More precisely, for a given finite unramified covering π: S'→ S_g,n, let us denote by S' the closed Riemann surface obtained filling in the punctures of S'. Then, for a given closed curve γ on S_g,n, the irreducible components of π^-1(γ) in S' span a submodule V_γ of the homology group H₁ (S',Z). A non-power closed curve γ on S_g,n is simple if and only if, for a fixed prime p, every finite unramified p-covering π:S' → S_g,n is such that the associated submodule V_γ of H₁(S' ,Z) is isotropic for the standard intersection pairing on S'. If γ and γ' are two non homotopic simple closed curves on S_g,n, then there is a finite unramified p-covering π:S'→ S_g,n such that V_γ≠ V_γ' in the homology group H₁ (S',Z). As an application, we give a new geometric proof of conjugacy p-separability for oriented surface groups.

7 June 2011 - Unusual day and time: Aula Consiglio, 14-15
Lawrence Ein (University of Illinois)
Invariants of singularities of pairs

We consider pairs of the form (X, Z), where X is a normal algebraic variety and Z is a subvariety of X. We study various invariants attached to such a pair. In particular, we investigate the log-canonical threshold of the pair (X,Z). In this talk we'll describe some of the properties and applications of this invariant.

8 June 2011
Claudio Procesi (Università di Roma "La Sapienza")
Some geometric and algebraic problems arising in the study of the completely resonant non-linear Schrödinger equation on a torus

(Joint work with Michela Procesi)
We discuss a class of normal forms of the completely resonant non-linear Schrödinger equation on a torus.
iu_t - Δu = κ |u|^2qu + ∂ _u G(|u|²) ,   1≤q∈N
where u:=u(t,φ), φ∈Tⁿ and G(a) is a real-analytic function whose Taylor series starts from degree q+2. The case q=1 is of particular interest and is usually referred to as the cubic NLS. We stress the geometric and combinatorial constructions arising from this study. We aim at applications to quasi-periodic solutions. These require a careful study of the first 3 Melnikov non-degeneracy conditions in order to apply a KAM algorithm. Of particular relevance is the fact that the infinite-dimensional quadratic form appearing in the normal form is described by a finite number of combinatorially defined graphs which produce interesting polynomials and problems on their nature.

15 June 2011 - Unusual room: Aula Picone, 14:30-15:30
Linda Chen (Swarthmore College)
Equivariant Quantum Cohomology and Flag Varieties

Enumerative geometry problems have been studied since the nineteenth century: Schubert calculus describes the cohomology rings of Grassmannians and other flag varieties. Inspired by ideas in physics, surprising answers to numbers of rational curves were established through quantum cohomology. I will describe the equivariant quantum cohomology of partial flag varieties. Some tools and consequences include degeneracy locus formulas, equivariant transversality in spaces of maps, and Graham-type positivity. This is joint work with Dave Anderson.

15 June 2011 - Unusual room and time: Aula Picone, 15:45-16:45
Siegfried Böcherer (Universiteit Mannheim)
Congruences for Siegel modular forms and applications to Selmer groups

By a construction due to Yoshida, we can associate to a pair (f,g) of elliptic modular forms of squarefree level a Siegel modular form Y(f,g) of degree 2. If a certain L-value is divisible by a prime ideal λ of the field generated by the Hecke-eigenvalues of f and g, then we can show that there is another Hecke eigenform G of degree 2, whose eigenvalues are congruent to those of Y(f,g) modulo λ. Under some technical conditions, G is not an endoscopic lift. Such a result is interesting in its own right, but it also allows to construct a nontrivial element of a Selmer-group attached to the tensor product of motives of f and g; this fits well to the predictions made by the Bloch-Kato conjecture for the near center value of the L-function L(f⊗g,s) (joint work with N.Dummigan and R.Schulze-Pillot).

22 June 2011 - Aula Consiglio, 14:30
Alessio Fiorentino (Università di Roma "La Sapienza")
Su un problema relativo alla mappa dei gradienti Theta in genere 2 ed argomenti correlati

Nel caso di genere 2 è noto che la mappa dei gradienti delle funzioni Theta non è iniettiva. A tal riguardo verrà, dunque, presentata una descrizione del sottogruppo di congruenza del gruppo modulare di Siegel sul cui rispettivo spazio quoziente la mappa è ancora ben definita ed iniettiva; verrà, inoltre, fornita una presentazione in termini di generatori e relazioni dell'anello delle forme modulari rispetto a tale sottogruppo, ed una nuova descrizione per una nota forma modulare introdotta da Igusa, che coinvolge costruzioni pertinenti al problema.

22 June 2011 - Unusual time: Aula Consiglio, 15:30
Stefano Pascolutti (Università di Roma "La Sapienza")
Annullamento delle thetanull sul luogo iperellittico e sulla sua chiusura

È naturale chiedersi se si possa risolvere il problema di Schottky per il bordo dello spazio di moduli di curve iperellittiche, che consiste di curve di tipo compatto. Tsuyumine risolve il problema per i divisori, ma lo stesso ragionamento si può generalizzare a qualunque componente al bordo, con metodi combinatorici.
Tempo permettendo, esporrò anche un risultato che ottenuto in collaborazione con Claudio Fontanari, che esibisce una soluzione esplicita, per genere g=2,3,4,5, del fatto che M_g è unione di g-1 aperti affini. Il problema per genere g≥6 è tutt'ora irrisolto.