Winter School

Geometry, Algebra and Combinatorics of Moduli Spaces and Configurations II

Dobbiaco (Toblach), February 18-23, 2018

This is the second edition of the winter school held in Toblach in winter 2017. There will be three 5-hours minicourses, some talks and ample time for discussion.

Timetable

Lectures

Enrico Arbarello, Sapienza Università di Roma, K3 surfaces, their hyperplane sections and stability conditions;
Luca Moci, Université Paris 7 Diderot, Graded bialgebras and deletion-contraction invariants [lecture notes];

Talks:

Gilberto Bini, Università di Milano, Binational invariants of projective varieties.
Davide Bolognini, Université de Fribourg, Cohen-Macaulay binomial edge ideals and Hartshorne's connectdness theorem. [show abstract]
Abstract: For binomial edge ideals of bipartite graphs we prove the converse of Hartshorne’s Connectedness Theorem. In this way, we provide a complete classification of Cohen-Macaulay binomial edge ideals of bipartite graphs. This result represents a binomial analogue of the celebrated characterization of (monomial) edge ideals of bipartite graphs due to Herzog and Hibi. As application, we prove a Hirsch type conjecture of Benedetti-Varbaro on the diameter of dual graphs for this class of ideals. This is a joint work with Antonio Macchia and Francesco Strazzanti. [Hide abstract]
Nero Budur, KU Leuven, On some homotopy invariants of algebraic varieties. [show abstract]
Abstract: By work of Simpson, Kollár, Kapovich, every finitely generated group can be the fundamental group of an irreducible complex algebraic variety with only normal crossings and Whitney umbrellas as singularities. In contrast, we show that if an irreducible complex algebraic variety has no weight zero 1-cohomology classes, then the fundamental group is strongly restricted: the irreducible components of the cohomology jump loci of rank one local systems containing the constant sheaf are complex affine tori. Joint work with Marcel Rubió. [Hide abstract]
Alessandro Iraci, Università di Pisa, Parallelogram Polyominoes and the Delta Conjecture. [show abstract]
Abstract: The Delta Conjecture is an important open problem in algebraic combinatorics. It gives a combinatorial interpretation for certain instances of the Delta operator, which is a linear operator on the space of symmetric functions that acts diagonally on the basis of Macdonald polynomials. The Delta Conjecture is stated in terms of parking functions, which are labelled Dyck paths. We give a combinatorial interpretation for some particular instances of the Delta Conjecture in terms of decorated parallelogram polyominoes, showing some recursions and proving the “hh” case using a bijection with some subset of parking functions. [Hide abstract]
Roberto Pagaria, Università di Pisa, Orlik-Solomon Relations for Toric Arrangements. [show abstract]
Abstract: After a briefly recall on the theory of hyperplane arrangements, we introduce arrangements of hypertori. The aim of our research is to study the cohomology algebra of the complement of some hypertori in a torus. We discuss unimodular arrangements and coverings of tori in order to state our result: a presentation of the cohomology algebra by generators and relations. The relations generalize the ones found by Orlik and Solomon in the case of hyperplane arrangements. [Hide abstract]
Oscar Papini, Università di Pisa, Milnor Fibre and Characteristic Variety of Line Arrangements. [show abstract]
Abstract: In the theory of hyperplane arrangements one of the main research topics is to understand which properties of an arrangement depend only on its intersection poset. In this talk I focus on two of them: the monodromy of the Milnor fibre and the (first) characteristic variety for line arrangements in the real projective plane. For the first part I state a conjecture, due to Salvetti and Serventi, that links the a-monodromicity of an arrangement to a certain combinatorial graph. In the second part I describe the characteristic variety of an arrangement, showing some meaningful examples. I conclude with a construction, based on neighbourly partitions, that seems to detect the translated components of the characteristic variety. [Hide abstract]

[Click here for a temporary list of registered participants]

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 19, 2018.

Registration and support

Participants are required to register by contacting the organizers at the following email address: toblach.configurations2018@gmail.com.

The participants who are financially supported by the organization will be hosted by Casa per Ferie Europa that is located at 5 minutes walking distance from the Kulturzentrum. The other participants are encouraged to book in advance their room (the price of full board at the Casa per Ferie is 65 euro per person in double room).

This workshop is supported by:

INdAM - GNSAGA