Geometry, Algebra and Combinatorics of Moduli Spaces and Configurations VII

This is the 7th edition of the winter school held in Toblach in winter 2017, winter 2018, winter 2019, winter 2020, winter 2023 and winter 2024. There will be three 5 hours minicourses, some talks and ample time for discussion.

Mini-courses

• Giovanna Carnovale, Università di Padova, An introduction to Nichols algebras. [show abstract]

  Abstract: To any pair (V,c) consisting of a vector space V and a solution c ∈ GL(V ⊗ V) of the Yang-Baxter equation (c ⊗ id)(id ⊗ c)(c ⊗ id) =(id ⊗ c)(c ⊗ id)(id ⊗ c) one associates naturally an algebra, called Nichols (or shuffle) algebra. Nichols algebras are crucial for the classification program of Hopf algebras through the work of Andruskiewitsch, Angiono, Heckenberger, Schenider, Vendramin and their collaborators, and have had applications in the solution of Malle's conjecture in number theory by the work of Ellenberg, Tran and Westerland. Relevant but different algebras, such as the symmetric algebra, the exterior algebra, the positive part of quantized enveloping algebras, and Fomin-Kirillov algebras FK_3, FK_4 and FK_5 are examples of Nichols algebras. One of the basic questions is then how properties of (V,c) determine properties of the associated algebra, especially, for which pairs (V,c) is finite-dimensional, or finitely-generated. We will review some of the current and past approaches and solutions to this problem and mention two geometric interpretations of Nichols algebras, one, due to Meir, as elements in a close orbit for the action of a reductive group, and another, due to Kapranov and Schechtman, in terms of perverse sheaves on the space of complex monic polynomials.
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• Fabrizio Catanese, Universität Bayreuth, Finite automorphism groups of complex and algebraic manifolds. [show abstract]

  Abstract:
  I) Automorphism groups of projective varieties and compact complex manifolds. Examples and the case of algebraic curves: the classical results of Hurwitz, Klein, Lefschetz.
  II) Finiteness criteria for Aut(X). Upper bounds for the cardinality |G|, for G a finite subgroup of Aut(X): the case where G is Abelian, or X is a hypersurface.
  III) Automorphisms with some type of topological triviality, in particular Aut_Q(X) = the group of automorphisms acting trivially on the rational cohomology of X, respectively Aut_Z(X) = the group acting trivially on the integral cohomology of X. The benchmark case of surfaces, according to the Kodaira dimension.
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• Luis Ferroni, IAS Princeton, A gentle overview of Kazhdan-Lusztig-Stanley and Chow functions on posets. [show abstract]

  Abstract: In 1992 Stanley noticed that certain special functions on posets called "kernels" give place to special polynomial functions. These are now known as Kazhdan-Lusztig-Stanley (KLS) functions, and depending on the choice of the poset and the kernel, give place to important objects in algebraic combinatorics. Three foundational examples are the following: i) when the poset is the Bruhat order of a Coxeter group and the kernel are the R-polynomials, the KLS functions are the famous Kazhdan-Lusztig polynomials; ii) when the poset is the face lattice of a polytope and the kernel is the map [F_1, F_2] \mapsto (x-1)^{\dim F_2 - \dim F_2}, the KLS function is the so-called toric g-polynomial of the polytope; and iii) when the poset is the lattice of flats of a hyperplane arrangement (or matroid), the KLS function is the Kazhdan-Lusztig polynomial of the arrangement. We will recapitulate these central notions, and compare it with a new kind of functions called "Chow functions", that I introduced in an on-going collaboration with Jacob P. Matherne and Lorenzo Vecchi.
  No prior knowledge about matroids, polytopes, or Coxeter groups is necessary, but having at least one of these examples in mind will be helpful.
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Schedule

TBA


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 24, 2025.

Registration and support

Participants are required to register by filling in this registration form.
It would be very helpful for the organization if you fill in the form as soon as possible, thanks!

Limited funding will be available to cover local expenses (lodging, breakfast and dinner) of few Master or PhD students.
To apply for financial support you are required to fill in the same form.
The deadline to apply for funding is December 8, 2024.
Again, we encourage to apply as soon as possible in order to help the organization, thanks!

For any question, you can contact the organizers at the following email address: toblachconfigurations@gmail.com.

Organizers: Michele D'Adderio , Sabino Di Trani , Marco Franciosi , Alessandro Iraci and Paolo Papi.