Titoli e abstracts

Veronica Arena, The Tautological Ring and Non-tautological Cycles of H*(M_{g,2m})

When studying the cohomology ring of the moduli space of n-pointed curves H^*(M_{g,n}), one can identify a particular subring RH^*(M_{g,n}) called the Tautological ring. This ring has a simple construction via generators, however is very rich and the question of whether or not the equality RH^*=H^* is, in general, hard to answer. Examples of non-tautological classes are provided in work by Graber and Pandharipande and of van Zelm. In a generalisation of the technique used in the latter, in joint work together with S. Canning, E. Clader, R. Haburcak, A.Q. Li, S.C. Mok and C. Tamborini, we construct non-tautological classes for H^*(M_{g,2m}) for infinitely many g,m. 

Luca Battistella, Euler characteristics of double ramification loci in genus one

Fabio Bernasconi, Un controesempio alla decomposizione di Beauville--Bogomolov per varietà log canoniche

Negli ultimi anni, i lavori di Druel, Guenancia, Greb, Kebekus, Höring e Peternell hanno mostrato che un analogo della decomposizione di Beauville--Bogomolov (BB) vale per varietà $K$-triviali con singolarità klt sui numeri complessi. 

Motivati dal programma del Modello Minimale e dallo studio delle varietà aperte, è naturale chiedersi se una  decomposizione di tipo BB possa essere estesa al caso di singolarità log canoniche. 

In questo seminario darò una risposta negativa costruendo a partire dalla dimensione 3 una varietà log canonica $K$-triviale di dimensione $d$ che non ammette una decomposizione di tipo BB. 

Più precisamente, la mappa di Albanese di tale varietà non è birazionalmente isotriviale (in particolare, la componente abeliana non può essere scissa dopo un rivestimento étale), e il suo fascio tangente non è polistabile. Questo è un lavoro in collaborazione con S. Filipazzi, Zs. Patakfalvi, N. Tsakanikas e N. Müller.

Lucrezia Bottegoni, On infinitesimal braidings and pre-Cartier bialgebras

In the 1980s V.G. Drinfel’d introduced quasitriangular structures on bialgebras, the so- called universal R-matrices, as particular solutions of the quantum Yang-Baxter equation. It is known that a bialgebra is quasitriangular precisely if the monoidal category of modules over it is braided.

In this talk we present an infinitesimal version of a braided monoidal category, that we call braided pre-Cartier, whose corresponding infinitesimal braiding is a first-order deformation of the given braiding. This extends the (symmetric) Cartier categories, where an additional com- mutativity of the infinitesimal braiding with the involutive braiding is required. We consider the pre-Cartier structure for the categories of modules over a quasitriangular bialgebra and we characterize the resulting infinitesimal R-matrices on the bialgebra. Among their main properties, infinitesimal R-matrices are 2-cocycles in Hochschild cohomology and they satisfy an infinitesimal quantum Yang-Baxter equation. We discuss the dual picture for pre-Cartier coquasitriangular bialgebras. We provide explicit examples on some known Hopf algebras, such as on the Hopf algebras E(n) which are a generalization of the Sweedler’s Hopf algebra E (1).

This talk is based on a joint work with A. Ardizzoni, A. Sciandra, T. Weber, and on a joint work with F. Renda, A. Sciandra.

Carlo Collari, Uberhomology, combinatorics and double homology.

 In this talk we will introduce uberhomology, which is a combinatorially-defined homology theory for simplicial complexes. After proving some of its properties, we’ll show how this invariant is related to dominating sets and to the Mayer-Vietoris spectral sequence. Finally, we discuss the isomorphism between a specialisation of uberhomology and the double homology of the moment angle complex.  We will conclude with some open questions and problems.  All results are joint work with C.Caputi and D.Celoria.

Veronica Fantini, Invariants from the TS/ST correspondence

Building on the seminal paper by Candelas, De La Ossa, Green, and Parkes, topological string theory on Calabi-Yau threefolds has been a source of inspiration for enumerative geometers.

While mirror symmetry has been extensively studied in all its aspects, the so-called Topological String/Spectral Theory (TS/ST) correspondence of Grassi-Hatsuda-Mariño has been recently attracting more attention. Indeed, it predicts a new source of (resurgent) invariants, which should be conjecturally related to enumerative invariants.

In this talk, I will introduce these new resurgent invariants and discuss their expected connection with Gromov–Witten invariants, focusing on the examples of local P2.

Daniel Labardini Fragoso, Drawing and mutating infinite-dimensional modules.

Over 15 years ago, Derksen-Weyman-Zelevinsky defined mutations of finite-dimensional representations of quivers with potential. Also over 15 years ago, I associated a quiver with potential to each ideal triangulation of any surface with marked points, as well as a finite-dimensional representation of its Jacobian algebra to each arc \gamma, and proved that if one performs a flip and obtains another ideal triangulation, but keeps the arc \gamma fixed, then the quiver with potential and the representation that one reads off from the new ideal triangulation can be obtained by applying the corresponding DWZ-mutation to those read off from the old one.

In this talk, based on works in progress both alone and joint Lang Mou, I will show how one can read off other quivers with potential from triangulations of a surface with marked points, and how one can extend the aforementioned results, including DWZ's mutations, to infinite-dimensional representations.

Sergej Monavari, Partitions, motives and Hilbert schemes

Counting the number of higher dimensional partitions is a hard classical problem. Computing the motive of the Hilbert schemes of points is even harder. I will discuss some structural formulas for the generating series of both problems, their stabilisation properties when the dimension grows very large and how to apply all of this to obtain (infinite) new examples of motives of singular Hilbert schemes.

This is joint work with M. Graffeo, R. Moschetti and A. Ricolfi.

Alessio Savini, Actions, groupoids and their cohomology

Given a finitely generated group $G$ acting in a sufficiently nice way on a probability space $(X,m)$, measured group theory studies the interplay between the algebraic properties of $G$ and dynamical properties of the action on $(X,m)$. For instance, one can look at the macroscopic structure of $G$-orbits. For $G$ acting on $(X,m)$ and $H$ acting on $(Y,n)$, we say that the actions are orbit equivalent if there exists an isomorphism of measure spaces between X and Y which sends $G$-orbits to $H$-orbits. Orbit equivalence is an equivalence relation whose classes are quite difficult to understand. Actions of uncountable amenable groups are all orbit equivalent, whereas higher rank lattices in Lie groups have a far more rigid behaviour, since the existence of an orbit equivalence implies the existence of an isomorphism between the ambient Lie groups. 

As usual in Mathematics, we would like to introduce a simpler invariant to study orbit equivalence classes. This time we are lucky enough to translate the study of orbit equivalence into the more powerful language of measured groupoids. The latter offer us a unifying approach to equivalence relations, groups and groups actions, foliations and so on. For measured groupoids, we defined a cohomological theory which is precisely our desired invariant. In the case of group actions, we proved a Fubini-Tonelli like theorem that gives us new information about the orbit equivalence class. 

Joint work with Filippo Sarti.

Lisa Seccia, Singolarità di Frobenius e  decomposizione geometrica per vertici

 La teoria delle singolarità di Frobenius è diventata un tema centrale in algebra commutativa, poiché fornisce l’analogo in caratteristica p delle singolarità considerate in geometria birazionale (ad esempio, le singolarità log-canoniche).  In questo seminario, introdurrò i concetti fondamentali e le motivazioni alla base di questa teoria.  Successivamente,  presenterò una connessione tra le F-singolarità e una tecnica di degenerazione combinatoria nota come decomposizione geometrica per vertici (geometric vertex decomposition), che permette di scomporre ricorsivamente un ideale in pezzi più semplici preservandone alcune proprietà algebriche e geometriche. Questo seminario si basa su un lavoro in collaborazione con Emanuela De Negri, Elisa Gorla, Patricia Klein e Jenna Rajchgot.