Geometry, Algebra and Combinatorics of Moduli Spaces and Configurations VI

This is the sixth edition of the winter school held in Toblach in winter 2017, winter 2018, winter 2019, winter 2020 and winter 2023. There will be three 5 hours minicourses, some talks and ample time for discussion.

Mini-courses

• Concettina Galati, Università della Calabria, Severi varieties of curves and surfaces (part 1, part 2, part 3). [show abstract]

  Abstract: This course will be a brief introduction to Severi varieties of singular curves on a smooth projective surface S and Severi varieties of surfaces with nodes and triple points in the complex projective space. In the first lesson we will consider the case S is the complex projective plane: we will define Severi varieties, describe their tangent spaces by using the classical approach of Severi and we will introduce the Zariski's dimensional characterization theorem. We will then briefly introduce the "Severi problem" and Harris' theorem which proves the irreducibility of Severi varieties of nodal plane curves, thus showing the necessity of describing Severi varieties in the language of modern deformation theory. In the second lesson we will focus on the "existence problem" of Severi varieties of nodal curves in the case S is an Enriques surface. We will in particular introduce several deformation methods of curves, surfaces and singularities of curves. The last lesson will be devoted to Severi varieties of surfaces with nodes and triple points in the complex projective space. We will in particular show how to apply limit linear system techniques to the study of deformations of curve and surface singularities. Part of these results are a joint work in progress with C. Ciliberto. [Hide abstract]

• Anton Mellit, University of Vienna, Symmetric functions in geometry. [show abstract]

  Abstract: I will explain in explicit and hands-on way how various symmetric functions, such as Schur, Hall-Littlewood and Macdonald, encode geometric information. Examples will be Schubert calculus, finite and affine Springer fibers, quiver varieties and moduli spaces of bundles and Higgs bundles, Hilbert scheme of C^2. [Hide abstract]

• Roberto Pagaria, Università di Bologna, Hitchin fibrations and related combinatorics (notes). [show abstract]

  Abstract: The mini course is divided into two parts. In the first one, we gently introduce vector bundles on algebraic curves, Higgs bundles, and their moduli space. We present the Hitchin fibration and spectral curves, and we describe line bundles over reduced curves. The goal is to tackle the geometry of the moduli space by understanding its intersection cohomology. In the second part we translate the problem in combinatorics and present the tools to solve it. In particular, we consider integral points in polytopes, posets of certain partitions, and their symmetries. The techniques consist in shellability, Ehrhart polynomial, Möbius inversion, and fixed point methods. [Hide abstract]

Talks:

• Christian Krattenthaler, University of Vienna. Identities for cylindric Schur functions. [show abstract]

  Abstract: I shall tell the story of a very small seed (a specific enumeration problem) that in the end gives rise to an unexpected general result (in the theory of symmetric functions) that at first sight does not seem to have anything to do with the original problem. The original problem is about finding a bijection between two specific sets of lattice paths. After reformulation, this problem leads us in the end to discover a far-reaching generalisation of a classical, but difficult-to-prove result in the theory of symmetric functions due to Basil Gordon, as realised by Bender and Knuth. The latter result provides a determinantal formula for a specific sum of Schur functions that plays and important role in other contexts, most notably in the enumeration of plane partitions and in the representation theory of classical groups. In my talk I shall explain all the above terms and connections, and describe the above sketched path of discovery of an unexpected new result, while the original problem of finding a bijection was left unresolved ... This is joint work with JiSun Huh, Jang Soo Kim and Soichi Okada. [Hide abstract]

• Viola Siconolfi, Politecnico di Bari. Zeta functions for class two nilpotent groups. [show abstract]

  Abstract: The notion of Zeta function for groups was introduced in a seminal paper from Grunewald, Segal and Smith and proved to be a powerful tool to study the subgroup growth in some classes of groups. In this seminar I will introduce this Zeta function presenting some general properties for this object. I will then focus on some results obtained for class two nilpotent groups. In particular, I will describe some combinatorial tecniques used to tackle this problem, namely the study of series associated to polyhedral integer cones. This is a joint work with Christopher Voll and Marlies Vantomme. [Hide abstract]

Schedule


You can download a pdf with the schedule of the winter school here.

Patercipants:

• Houcine Ben Dali, Université de Lorraine/Université de Paris
• Riccardo Biagioli, Università di Bologna
• Sabrina Botticchio, Università di Pisa
• Giusi Capobianco, Università di Roma Tor Vergata
• Lorenzo Campioni, Università dell'Aquila
• Nicoletta Cantarini, Università di Bologna
• Fabrizio Caselli, Università di Bologna
• Nirvana Coppola, Unaffiliated
• Michele D'Adderio, Università di Pisa
• Luca Dal Molin, Università di Trento
• Davide Dal Martello, University of Birmingham
• Francesco Esposito, Università di Padova
• Dario Faro, Università di Pavia
• Lorenzo Fassina, Università di Pavia
• Marco Fava, University of Liverpool
• Marco Franciosi, Università di Pisa
• Giovanni Gaiffi, Università di Pisa
• Concettina Galati, Università della Calabria
• Lorenzo Giordani, Università di Bologna/Ruhr-University Bochum
• Franco Giovenzana, Université Paris-Saclay
• Luca Giovenzana, University of Sheffield
• Giorgio Grisolia, Università di Roma 'La Sapienza'
• Álvaro Gutièrrez Cáceres, University of Bristol
• Giovanni Interdonato, Scuola normale
• Alessandro Iraci, Università di Pisa
• Christian Krattenthaler, University of Vienna
• Vadym Kurylenko, SISSA
• Antonio Lacopo, Università di Pavia
• Pietro Leonardini, Università di Pisa
• Hsin-Chieh Liao, University of Miami
• Valerio Melani, Università di Firenze
• Anton Mellit, University of Vienna
• Luca Migliorini, Università di Bologna
• Martina Miseri, Università di Roma Tre
• Luca Morstabilini, Università di Pisa
• Lorenzo Noè, Università di Pisa
• Clemens Nollau, Universität Tübingen
• Roberto Pagaria, Università di Bologna
• Giovanni Paolini, Università di Bologna
• Margherita Paolini, Università di Roma 'La Sapienza'
• Paolo Papi, Università di Roma 'La Sapienza'
• Lorenza Parisi, Università del Salento
• Simone Pesatori, Università di Roma Tre
• Maddalena Pismataro, Università di Bologna
• Alejandro Polo Molina, Comillas Pontifical University (ICAI)
• Roberto Riccardi, Scuola normale
• Marino Romero, University of Vienna
• Maria Teresa Ruggiero, Università di Pavia
• Nikola Sadovek, FU Berlin
• Mario Salvetti, Università di Pisa
• Elena Sammarco, Università di Roma Tre
• Alessio Sgubin, Università di Pisa
• Viola Siconolfi, Politecnico di Bari
• Lorenzo Venturello, Università di Siena
• Eduardo Venturini, Università di Pisa/Scuola normale

Venue

The school will be held in Dobbiaco (Toblach) at the Kulturzentrum Grand Hotel Centro Culturale, in "La Sala degli Specchi". Kulturzentrum Grand Hotel Centro Culturale is located in front of Toblach train station (see the map).

The first talk will start on Monday February 12, 2024.

Registration and support

Participants are required to register by contacting the organizers at the following email address: toblachconfigurations2024@gmail.com.

Limited funding will be available to cover local expenses (lodging, breakfast and dinner) of some PhD students. Participants who want to apply for financial support should send an email with their CV to toblachconfigurations2024@gmail.com. The extended deadline is December 17, 2023.

The participants who are financially supported by the organization will be hosted by Casa per ferie Europa or by Hotel Union. The other participants are encouraged to book in advance their room.

This workshop is supported by:



PRA_2022_10, Spazi di moduli, rappresentazioni e strutture combinatorie, University of Pisa.

PRIN 2022, ALgebraic and TOPological combinatorics (ALTOP), University of Pisa.

PRIN 2017, Moduli and Lie Theory, University of Pisa.

PRIN 2022, De Sole, University "La Sapienza" of Rome.

Progetto di Ateneo 2022, Teoria delle rappresentazioni e applicazioni, University "La Sapienza" of Rome.

GNSAGA, INDAM.

PRIN 2022, GEOMETRY OF ALGEBRAIC STRUCTURES: MODULI, INVARIANTS, DEFORMATIONS, Università di Bologna.

Organizers: Michele D'Adderio , Marco Franciosi , Giovanni Gaiffi , Alessandro Iraci , Paolo Papi and Lorenzo Venturello .