Seminari periodici
DIPARTIMENTO DI MATEMATICA

Seminario di Algebra e Geometria

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Martedì
24 Maggio
Leone Slavich (Università di Pavia)
Totally geodesic immersions of hyperbolic manifolds
nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

seminario di algebra e geometria

 ore 14:15
  presso Seminario II
The study of totally geodesic immersions between (complete, finite-volume) hyperbolic manifolds is a classical problem in the field of hyperbolic geometry. There are two main approaches to this problem which often interplay with each other: 1) Given a hyperbolic manifold N, determine the hyperbolic manifolds in which N can be immersed geodesically; 2) Given a hyperbolic manifold, determine its totally geodesic immersed submanifolds. We will show how it is possible to build totally geodesic immersed submanifolds in a hyperbolic manifold M using finite subgroups in the commensurator of M. We will then focus on the class of arithmetic manifolds i.e. those whose fundamental groups is commensurable with the integral points of some k-form of Isom(H^n)=PO(n,1,R), for some real algebraic number field k. We will show how to characterise all totally geodesic immersions in this setting through the analysis of Vinberg's commensurability invariants: the adjoint trace field (which is an algebraic number field) and the ambient group (an algebraic group defined over the adjoint trace field). This is joint work with Mikhail Belolipetski, Nikolay Bogachev and Alexander Kolpakov.

Lunedì
30 Maggio
Alessandro D'Andrea
A differential geometric description of exceptional projective irreducible (discrete) representations of the Hamiltonian Lie algebra of Cartan type
nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

seminario di algebra e geometria

 ore 14:30
  presso Aula Seminario VIII piano
Some exact complexes of irreducible discrete representations of the unique irreducible central extension of H_n have only been constructed by hand. I will provide a differential geometric description by using base change in the pseudoalgebra language.

Martedì
31 Maggio
Margherita Lelli Chiesa
Irreducibility of Severi varieties on K3 surfaces
nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

seminario di algebra e geometria

 ore 11:00
  presso Seminario II
Let (S,L) be a general K3 surface of genus g. I will prove that the closure in |L| of the Severi variety parametrizing curves in |L| of geometric genus h is connected for h>=1 and irreducible for h>=4, as predicted by a well known conjecture. This is joint work with Andrea Bruno.

Martedì
31 Maggio
Stavroula Makri
Surface braid groups and the splitting problem of the mixed braid groups of the projective plane
nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

seminario di algebra e geometria

 ore 14:15
  presso Seminario II
The aim of this talk is to give an introduction to the surface braid groups and to present both the splitting problem of surface braid groups and certain results about this problem, concerning the mixed braid groups of the real projective plane. Surface braid groups are a generalisation, to any connected surface, of both the fundamental group of a surface and the braid groups of the plane, which are known as Artin braid groups and were defined by Artin in 1925. Surface braid groups were initially introduced by Zariski and then, during the 1960’s, Fox gave an equivalent definition from a topological point of view. In the first part of the talk, we will define the surface braid groups from both a geometric and a topological point of view and we will present their close relation to the symmetric groups. Moreover, we will present an important family of surface braid groups, the so-called mixed braid groups. Finally, we will describe the splitting problem of surface braid groups, which we will see in detail in the second part of the talk. In the second part of the talk, we will focus on the splitting problem, which, during the 1960’s, the period of the development of the theory of surface braid groups, was studied by many mathematicians; notably by Fadell, Neuwirth, Van Buskirk and Birman, and more recently by Gonçalves–Guaschi and Chen–Salter. In particular, we will focus on the case of the projective plane: we will present its braid groups as well as certain results that we obtained concerning the splitting problem of its mixed braid groups.

Venerdì
03 Giugno
Aline Zanardini
Moduli of rational elliptic surfaces of index two
nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

seminario di algebra e geometria

 ore 16:00
  presso - Aula Da Stabilire -
seminario on line • collegamento al meeting su zoom
Elliptic surfaces are ubiquitous in Mathematics. Examples include Enriques surfaces, Dolgachev surfaces, all surfaces of Kodaira dimension one, and many rational surfaces. In this talk we will focus on the latter. It is a classical result that all rational elliptic surfaces can be realized as a nine-fold blow-up of the plane, where the nine points (possibly infinitely near) are the base points of a pencil of plane curves of degree 3m, each of multiplicity m. The number m is called the index of the fibration. In joint work with Rick Miranda we have constructed a moduli space for rational elliptic surfaces of index two as a toric GIT quotient. The goal of my talk will be to explain our construction.

Martedì
07 Giugno
Marco Trevisiol (Sapienza Università di Roma)
Normality of Closures of Orthogonal Nilpotent Symmetric Orbits
nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

seminario di algebra e geometria

 ore 14:15
  presso Seminario I
Kraft and Procesi showed that the Zariski closure of the conjugacy classes of type A are all normal and, in type B, C and D, they have described which ones are normal. In their work the Lie group acts on its Lie algebra by the adjoint action. In types B, C, D, a similar question can be asked for the action of the Lie group on the odd part of the general linear Lie algebra; that is the orthogonal group acting on the symmetric matrices and the symplectic group acting on the symmetric-symplectic matrices. Ohta showed that in the latter case every orbit has normal closures while this conclusion is not valid in the former case. In this talk I will present the main result of my Ph.D. thesis which gives a combinatorial description of the orbit whose closures are normal in the orthogonal case.

Martedì
14 Giugno
Francesco Esposito (Università di Padova)
TBA
nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

seminario di algebra e geometria

 ore 14:15
  presso Seminario II

Martedì
21 Giugno
Francesco Sala (Università di Pisa)
Representations of cohomological Hall algebras of surfaces via stable pairs
nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

seminario di algebra e geometria

 ore 14:15
  presso Seminario II
During the first part of the talk, I will provide a gentle introduction to the theory of cohomological Hall algebras of surfaces and their categorification. Moreover, their “quantum” nature will be discussed. In addition, the Dolbeaut cohomological Hall algebras of curves will be introduced. The second part of the talk is devoted to the construction of a representation of the cohomological Hall algebra of a smooth projective surface S via moduli spaces of (rank r) stable pairs on S. I will describe a curve analog of this construction, which involves cyclic Higgs bundles (called also Bradlow-Higgs pairs).

Seminari passati


2022
10 Maggio
Simone Diverio
Sviluppi recenti sulla congettura di Lang: quozienti di domini limitati
nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

seminario di algebra e geometria

La congettura di Lang (1986) caratterizza le varietà complesse proiettive (o, più generalmente, Kähler compatte) iperboliche nel senso di Kobayashi come quelle di tipo generale assieme a tutte le loro sottovarietà. Lungi dall’essere dimostrata al momento, la congettura è però nota in una serie di casi paradigmatici ancorché particolari. Ci concentreremo in particolare su una direzione della congettura, spiegando come sia possibile verificare ad esempio che un quoziente libero e compatto di un dominio limitato dello spazio affine complesso abbia tutte sottovarietà di tipo generale (lavoro in collaborazione con S. Boucksom). Tempo permettendo, descriveremo alcune variazioni sul tema, considerando tipi di quozienti più generali: non più necessariamente lisci, né compatti (lavoro in collaborazione con B. Cadorel e H. Guenancia).

2022
09 Maggio
Alessandro D'Andrea
Moltiplicazione, numeri primi ed errori fortunati
nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

seminario di algebra e geometria

Parlerò di come affrontare due problemi familiari (moltiplicare numeri interi e stabilire se un numero sia primo) in modo insolito e di quali ricadute questo abbia al di fuori della matematica.

2022
06 Maggio
Aldo Conca
Resolution of Ideals Associated to Subspace Arrangements
nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

seminario di algebra e geometria

The ideal of definition of a linear subspace in a projective space has a very simple structure and hence a very simple free resolution, i.e. a Koszul complex. What can we say for the ideal that defines a finite collection of linear subspaces, subspace arrangements, in a projective space? Here we can take the intersection of the ideals defining the individual subspaces or their product. For the intersection, the structure of the resolution remains largely mysterious. For the product instead the resolution can be described and it turns out that it is supported on a polymatroid associated with the subspace arrangement. Joint work with Manolis Tsakiris (Chinese Academy of Sciences). arXiv:1910.01955v2

2022
03 Maggio
Florestan Martin-Baillon
Bifurcation currents for families of group representations in higher rank
nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

seminario di algebra e geometria

Finitely generated groups acting on projective spaces are examples of holomorphic dynamical systems which exhibit a variety of different behaviours. We introduce the notion of proximal stability which measures a form of dynamical stability for the action of a holomorphic family of group representations and we will explain how this property can be detected using a bifurcation current on the parameter space of the family. This bifurcation current measure the pluriharmonicity of the top Lyapunov exponent of the family of representation, defined using a random walk on the group.

2022
26 Aprile
Martina Lanini
Symmetric quivers and symmetric varieties
nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

seminario di algebra e geometria

In this talk I will report on ongoing joint work with Ryan Kinser and Jenna Rajchgot, on varieties of symmetric quiver representations. These varieties are acted upon by a reductive group via change of basis, and it is natural to ask for a parametrisation of the orbits, for the closure inclusion relation among them, for information about the singularities arising in orbit closures. Since the Eigthies, same (and further) questions about representation varieties for type A quivers have been attached by relating such varieties to Schubert varieties in type A flag varieties (Zelevinsky, Bobinski-Zwara, ...). I will explain that in the symmetric setting it is possible to interpret the above questions in terms of certain symmetric varieties. For example, we show that singularities of an orbit closure of a symmetric quiver representation variety are smoothly equivalent to singularities of an appropriate Borel orbit closure on a symmetric variety. As a consequence, we obtain an infinite class of symmetric quiver loci that are normal and Cohen-Macaulay.

2022
12 Aprile
Ludovico Battista
Hyperbolic 4-Manifolds with perfect circle-valued Morse functions
nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

seminario di algebra e geometria

In dimension 3, combining the study of the geometry and the topology of manifolds led to interesting and surprising results. The generalization of such a connection in dimension 4 seems to be a promising approach to better understand this complicated world. An intriguing 3-dimensional phenomenon is the existence of hyperbolic manifolds which fiber over the circle. Such manifolds cannot exist in dimension 4, due to a constraint given by Euler Characteristic and the Gauss - Bonnet formula. We will introduce the notion of "perfect circle-valued Morse function", which appears to be the natural generalization of "fibration over S^1", and we will introduce some tools to build a hyperbolic 4-manifold that admits such a function. To do this we will elaborate on a paper of Jankiewicz - Norin - Wise that makes use of Bestvina - Brady theory. Joint work with Bruno Martelli.

2022
05 Aprile
Michele Graffeo (SISSA)
On the Behrend function and the blowup of some fat points
nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

seminario di algebra e geometria

Not much is known about the geometric properties of the punctual Hilbert scheme of fat points of length n supported at the origin of the affine plane A^2. In order to investigate them, a huge number of invariants, for fat points, has been introduced (e.g. multiplicity, order, type, blowup tree...). I will focus on the Behrend number v_Z of a fat point Z in A^2. Such invariant can be defined in terms of the blowup of the affine plane with center the subscheme Z. I will discuss the problem of computing the Behrend number of a monomial fat point following a joint work with Andrea T. Ricolfi. In particular, I will explain, first in the normal setting, how toric geometry methods apply in the construction of the blowup and in the computation of v_Z. Then, I will move to the non-normal setting, and I will show some examples of computation. Finally, if time permits, I will show some difficulties that arise in higher dimension.

2022
05 Aprile
Allen Knutson (Cornell University)
Schubert calculus and quiver varieties
nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

seminario di algebra e geometria

Since the 1970s we have known that the structure constants for intersection theory on compact complex homogeneous spaces (such as Grassmannians) are nonnegative, but our only formulae for these constants (outside special cases) are essentially as alternating sums. The most effective tool to date for giving manifestly positive formulae are the "puzzles" that Terry Tao and I introduced, but the connection to quantum integrable systems observed by Paul Zinn-Justin made it clear that the puzzles should be solving a richer problem. This turns out to involve Nakajima quiver varieties, and has shed light even on the original problem of intersecting three cells in a Grassmannian. This work is joint with Paul Zinn-Justin.

2022
29 Marzo
Javier Aramayona (Instituto de Ciencias Matemáticas)
Asymptotic mapping class groups of Cantor manifolds and their finiteness properties
nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

seminario di algebra e geometria

A Cantor manifold C is a non-compact manifold obtained by gluing (holed) copies of a fixed compact manifold Y in a tree-like manner. Generalizing braided Thompson groups, we introduce the asymptotic mapping class group of C, whose elements are proper isotopy classes of self-diffeomorphisms of C that are ”eventually trivial.” This group B happens to be an extension of a Higman-Thompson group by a direct limit of mapping class groups of compact submanifolds of C. B acts on a contractible cube complex X of infinite dimension. We use the action to determine finiteness properties of B: in well-behaved cases, X is CAT(0) and B is of type F∞. More concretely, the methods apply when Y is a 2-dimensional torus, S2 × S1, or Sn × Sn for n at least 3. In these cases, the group B contains the mapping class groups of every compact surface with boundary, the automorphism groups of every finitely generated free group, or an infinite familiy of arithmetic symplectic or orthogonal groups. In particular, the cases where Y is a sphere or a torus in dimension 2 yields a positive answer to a question of Funar-Kapoudjian-Sergiescu. In addition, we find cases where the homology of B coincides with the stable homology of the relevant mapping class groups. (Joint work with Kai-Uwe Bux, Jonas Flechsig, Nansen Petrosyan, and Xiaolei Wu.)

2022
08 Marzo
Jerzy Weyman (Jagiellonian University)
On the structure of Gorenstein ideals of codimension 4
nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

seminario di algebra e geometria

I will discuss the new approach to Gorenstein ideals of codimension 4, with n generators. This allows us to construct (by a single calculation, starting from scratch) the examples of such ideals with 4<= n<= 8 generators. For n=4,5,6 they give generic models of resolutions of ideals of that type (in the sense that each such resolution comens from the generic model). We conjecture that for n=7,8 these resolutions are also generic models. The main idea is a construction of a certain generic ring which has unexpected symmetry of the type E_n. For n >= 9 such construction is not possible which indicates that the classification is much more difficult.

2022
01 Marzo
Andreas Knutsen
Severi Varieties on Enriques Surfaces
nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

seminario di algebra e geometria

Given a (smooth) projective (complex) surface S and a complete linear (or algebraic) system of curves on S, one defines the Severi varieties to be the (possibly empty) subvarieties parametrizing nodal curves in the linear system, for any prescribed number of nodes. These were originally studied by Severi in the case of the projective plane. Afterwards, Severi varieties on other surfaces have been studied, mostly rational surfaces, K3 surfaces and abelian surfaces, often in connection with enumerative formulas computing their degrees. Interesting questions are nonemptiness, dimension, smoothness and irreducibility of Severi varieties. In this talk I will first give a general overview and then present recent results about Severi varieties on Enriques surfaces, obtained with Ciliberto, Dedieu and Galati, and the connection to a conjecture of Pandharipande and Schmitt.

2022
22 Febbraio
Marco Moraschini (Università di Bologna)
Simplicial volume and aspherical manifolds
nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

seminario di algebra e geometria

Simplicial volume is a homotopy invariant for compact manifolds introduced by Gromov in the early 80s. It measures the complexity of a manifold in terms of singular simplices. Since simplicial volume behaves similarly to the Euler characteristic, a natural problem is to understand the relation between these two invariants. More precisely, a celebrated question by Gromov (~’90) asks whether all oriented closed connected aspherical manifolds with zero simplicial volume also have vanishing Euler characteristic. In this talk, we will introduce the notion of simplicial volume and then we will describe Gromov's question. Then, we will discuss some new possible strategies to approach the problem as well as the relation between Gromov’s question and other classical problems in topology. This is part of a joint work with Clara Löh and George Raptis.

2022
15 Febbraio
Stefano Riolo (Università di Bologna)
La segnatura delle 4-varietà iperboliche con cuspidi
nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

seminario di algebra e geometria

Le 4-varietà iperboliche (orientate) compatte hanno segnatura nulla per il Teorema della segnatura di Hirzebruch. Cosa si può dire, invece, per 4-varietà non compatte, complete e di volume finito? In questo caso, la segnatura dipende solo dalla topologia delle parti finali (le cuspidi) e si annulla sempre su qualche rivestimento finito della varietà. Inoltre, tutti gli esempi noti fino a poco tempo fa avevano segnatura nulla. Considerazioni naïve potrebbero quindi dare la sensazione che questo sia vero in generale. Vedremo invece che la segnatura può essere qualsiasi numero intero e, tempo permettendo, affronteremo di conseguenza qualche considerazione "geografica". In collaborazione con Sasha Kolpakov e Steve Tschantz.

2022
01 Febbraio
Giovanni Cerulli Irelli (Sapienza Università di Roma)
Symmetric representation theory of quivers and connections to Lie theory
nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

seminario di algebra e geometria

The representation theory of quivers and finite dimensional algebras deals with representations of products of general linear groups, and it is hence a ``type A situation''. There are many attempts to drag into the pictures groups of other nature. In this talk I will talk about a new attempt to get actions of groups of type B,C and D on the representation varieties associated to algebras with self-duality based on joint works with Magdalena Boos and partially with Francesco Esposito. For hereditary algebras this reduces to the approach due to Derksen and Weyman in 2002 when they introduced the so-called ``symmetric quivers''. In the first part I will mostly talk about quivers of type A and their symmetric representation theory, and state one of our main result with Lena which states that the symmetric orbit closures are induced by non-symmetric ones for symmetric quivers of finite type. Then I will talk about the connection with 2-nilpotent Borel orbits in classical Lie algebra worked out with Lena and Francesco and give an example that shows that in this context is not true the orbit closures of type D are induced by type A. I will close the talk by stating various conjectures and open problems concerning the problem of when symmetric orbit closures are induced by type A.

2021
07 Dicembre
Mattia Talpo
Level structures on logarithmic (and tropical) curves
nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

seminario di algebra e geometria

Level structures are extra data that can be added to some moduli problems in order to rigidify the situation. For example, in the case of curves, they yield smooth Galois covers of the moduli space M_g, and the problem of extending this picture to the boundary was studied by several authors, using in particular admissible covers and twisted curves. I will report on some work in progress with M. Ulirsch and D. Zakharov, in which we consider a tropical notion of level structure on a tropical curve. The moduli space of these is expected to be closely related to the boundary complex of the stack of G-admissible covers. As usual, logarithmic geometry stands in the middle and provides a convenient language to bridge the two worlds.

2021
23 Novembre
Amos Turchet
Special varieties and hyperbolicity
nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

seminario di algebra e geometria

Campana proposed a series of conjectures relating algebro-geometric and complex-analytic properties of algebraic varieties and their arithmetic. The main ingredient is the definition of the class of special varieties, which conjecturally identity the class of varieties with a potential dense set of rational points (when defined over a number field) and admitting a dense entire curve (when defined over the complex numbers). In the talk we will review the main conjectures and constructions, and we will discuss some recent results that give evidence for some of these conjectures. This is joint work with E. Rousseau and J. Wang

2021
16 Novembre
Andrea Petracci
From polytopes to singularities on moduli spaces of Fano varieties
nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

seminario di algebra e geometria

A projective algebraic variety is called Fano (after the Italian mathematician Gino Fano) if it has positive curvature. Fano varieties play a prominent rôle in algebraic geometry for many reasons. Recently there has been fundamental work on constructing moduli spaces of (certain) Fano varieties. The aim of my talk is to show how polytopes and combinatorics can help in proving that moduli spaces of Fano varieties are, in general, quite singular. My talk is based on joint work with Anne-Sophie Kaloghiros. The first 45 minutes of my talk do not require any knowledge in algebraic geometry.

2021
09 Novembre
Giovanna Carnovale
Chiusure di classi di Jordan in algebre di Lie, gruppi algebrici ed algebre di Lie Z_n-graduate
nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

seminario di algebra e geometria

Le classi di Jordan sono state introdotte da Borho e Kraft nel loro studio delle sheet per algebre di Lie semisemplici. Sono le classi di equivalenza di elementi in un'algebra di Lie che hanno stessa decomposizione di Jordan, o, equivalentemente di elementi che hanno stabilizzatori (per l'azione aggiunta) coniugati tra loro. Sono localmente chiuse, irriducibili, lisce, e le loro chiusure danno luogo ad una stratificazione finita. La stessa costruzione può essere adattata per definire le classi di Jordan in gruppi algebrici riduttivi: la stratificazione che ne risulta compare nello studio di Lusztig dei fasci carattere. In collaborazione con Ambrosio ed Esposito abbiamo osservato che localmente le chiusure di classi di Jordan nel gruppo si comportano come chiusure di classi di Jordan in un'opportuna algebra di Lie. Un analogo di classe di Jordan per algebre di Lie Z_2-graduate è stato introdotto da Tauvel e Yu e le chiusure sono state studiate da Bulois ed Hivert: si perdono alcune delle caratteristiche dei casi precedenti ma il quadro complessivo è ancora chiaro. Motivato dallo studio della modalità per azioni di gruppi, Popov ha recentemente introdotto le classi di Jordan anche per algebre di Lie ciclicamente graduate. In collaborazione con Esposito e Santi abbiamo fornito una descrizione geometrica locale delle loro chiusure, mostrando in particolare che anche in questo caso la chiusura delle classi di Jordan è un'unione di classi. Con una serie di esempi mostreremo affinità e divergenze tra i vari contesti e le situazioni nelle quali la partizione in classi di Jordan ha un ruolo importante.

2021
02 Novembre
Roberto Pagaria
Hodge Theory for Polymatroids
nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

seminario di algebra e geometria

In the first part, we present classical results about hyperplane arrangements and matroids: starting from the definitions, we present the construction by De Concini and Procesi of wonderful models. We discuss the cohomology of the wonderful model and of the complement of the arrangement. We also sketch the proof by Huh, Adiprasito, Katz of log-concavity of coefficients of the characteristic polynomial. In the second part we introduce subspace arrangements and polymatroids. We provide a generalization of the Goresky, MacPherson formula and we discuss the Hodge package (i.e. Poincaré duality, Hard Lefschetz and Hodge Riemann bilinear relations) for the Chow ring of a polymatroid. This is a joint work with Gian Marco Pezzoli.

2021
26 Ottobre
Florent Ygouf
Horocycle flows in moduli spaces of abelian differentials
nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

seminario di algebra e geometria

There is a fruitful analogy between Lie group actions and the dynamics of SL(2,R) on the moduli space of abelian differentials. Along the lines of this dictionary, the horocycle flow corresponds to a unipotent flow on a homogeneous space, for which Ratner’s theory is available. I will report on recent progress regarding the dynamics of the horocycle flow in the moduli space of abelian differentials.

2021
12 Ottobre
Andrea Ricolfi
Localisation in enumerative geometry
nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

seminario di algebra e geometria

Very few techniques are available for performing calculations in enumerative geometry. One of these is localisation. We will present different examples, flavours and refinements of the localisation formula, including applications to Donaldson-Thomas invariants.

2021
22 Giugno
Mauro Mantegazza
Functoriality of the c-map
nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

seminario di algebra e geometria

 
This talk will present some of the results obtained in the paper "The c-map as a functor on certain variations of hodge structure" by Mantegazza and Saha. In particular we will see how projective special Kähler manifolds can be interpreted in terms of variations of polarised Hodge structure, and how the latter can be used to give a description of the c-map that is manifestly functorial.

2021
30 Marzo
Paul-Konstantin Oehlmann (Uppsala University)
Non-flat elliptic fourfold and three-form cohomology (and strongly coupled theories in four dimensions)
nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

seminario di algebra e geometria

We consider compact elliptic four-folds whose fiber ceases to be flat over Riemann surfaces of genus g in the base. We show that those contributions generically lead to non-trivial threeform cohomology proportional to g and the number of non-flat fiber components. These non-flat components can be viewed as compactifications of non-flat three-folds where they correspond to superconformal matter theories. Moreover we show, that one can perform conifold transitions that remove those non-flat fibers, corresponding to non-flat fibers in codimension three and second to birational base changes. The former phase is interpreted as a non-perturbative gauge invariant fourpoint coupling and the second one is closer to a classical 4D Coulomb branch.

2021
16 Marzo
Santiago Estupinan Salamanca (Universidad de los Andes)
Schur and Power Sum Polytopes
nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

seminario di algebra e geometria

Aguiar and Ardila defined a Hopf monoidal structure on the collection of generalized permutahedra of all dimensions; and from it, constructed a Hopf algebra on the same polytopes, which is isomorphic to the Hopf algebra of symmetric functions, Sym. This endows each element of Sym with a formal sum of permutahedra, so that we can think of symmetric functions as members of McMullen’s polytope algebra. In this talk, we give geometric models for the Schur and power sum symmetric functions, when regarded as elements of the aforesaid polytope algebra. This is accomplished through a combinatorial rule for the former ones and in the way of an explicit description for the latter ones. We also characterize when the resulting geometric objects correspond to polytopes with missing faces. (Joint work with Carolina Benedetti and Mario Sanchez)

2021
16 Marzo
Luis Ferroni Rivetti (Università di Bologna)
EHRHART THEORY OF MATROID POLYTOPES
nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

seminario di algebra e geometria

Matroids are combinatorial structures that admit several "cryptomorphic" definitions. It is possible to define matroids as a certain kind of polytopes with vertices with 0/1-coordinates. Also, for every lattice polytope, the function counting the number of integer points inside of every integral dilation is known to be a polynomial, named after Eugene Ehrhart. In this seminar we will talk about the Ehrhart polynomials of matroid polytopes. We will discuss some open problems and recent results on the area, using just minimal prerequisites.

2021
09 Marzo
Aline Zanardini (University of Pennsylvania)
Stability of Halphen pencils of index two
nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

seminario di algebra e geometria

In this talk I will present some results about the GIT stability of Halphen pencils of index two under the action of SL(3). These are pencils of plane curves of degree six having nine (possibly infinitely near) base points of multiplicity two. Inspired by the work of Miranda on pencils of plane cubics, I will explain how we can explore the geometry of the associated rational elliptic surfaces.

2021
02 Marzo
Mariel Supina (University of California, Berkeley)
The universal valuation of Coxeter matroids
nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

seminario di algebra e geometria

Matroids are combinatorial objects that generalize the notion of independence, and their subdivisions have rich connections to geometry. Thus we are often interested in functions on matroids that behave nicely with respect to subdivisions, which are called valuations. Matroids are naturally linked to the symmetric group; generalizing to other finite reflection groups gives rise to Coxeter matroids. I will give an overview of these ideas and then present some recent work with Chris Eur and Mario Sanchez on constructing the universal valuative invariant of Coxeter matroids. Since matroids and their Coxeter analogues can be understood as families of polytopes with special combinatorial properties, I will present these results from a polytopal perspective.

2021
02 Marzo
José Bastidas (Cornell University)
The polytope algebra of generalized permutahedra
nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

seminario di algebra e geometria

McMullen used the fundamental operation of Minkowski sum to construct the polytope algebra of real vector space. In this talk, I will consider the subalgebra generated by deformations of a fixed zonotope and endow it with the structure of a module over the Tits algebra of the corresponding hyperplane arrangement. In the particular case of Coxeter arrangements of type A and B, we find striking relations between the corresponding module structure and certain statistics on permutations and signed permutations, respectively. I will explain how these statistics give information on families of polytopes that generate all (type B) generalized permutahedra as signed Minkowski sums.

2021
16 Febbraio
Thomas Lam (University of Michigan)
Positroid varieties
nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

seminario di algebra e geometria

Positroid varieties are intersections of cyclically rotated Schubert varieties in the Grassmannian, introduced in my work with Knutson and Speyer. I will discuss some aspects of these very nice spaces, including a recent computation of the cohomology of open positroid varieties in joint work with Galashin.

2021
16 Febbraio
Allen Knutson (Cornell University)
Kogan Schubert calculus from bumpless pipe dreams
nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

seminario di algebra e geometria

Kogan gave a combinatorial rule (rediscovered by Lenart and by Assaf) for computing the product of a Schur polynomial S_lambda(x_1..x_k) by a Schubert polynomial S_pi, subject to the condition that pi's last descent is at or before k. Yong and I gave a streamlined proof, using Lascoux' transition formula for Schubert polynomials. As the Lee-Lam-Shimozono bumpless pipe dream formula for Schuberts is tightly compatible with transition (an observation of Weigand's), using them we can give an even tighter formula for Kogan's coefficients.