9th June 2025

Rost nilpotence and direct sums

Stefan Gille (U. Alberta)

In the first half of the talk I will explain Rost nilpotence and its

applications, and then discuss whether this property holds for a finite direct sum of motives if it holds for all summands. 

It turns out that this is the case if one assumes a slightly stronger version of Rost nilpotence which -- as I will

explain -- actually holds for all cases where the 'original' form of Rost nilpotence has been verified.

The Mean Exit Time for Submanifolds

Leandro de Freitas Pessoa (UFPI)

We will discuss the mean exit time for Brownian motion on a submanifold. We will show that minimal submanifolds with infinite mean exit time cannot be isometrically immersed in cylinders (horocylinders), cones, and wedges of certain product spaces.

5th May 2025

On the birationality of the canonical map of some 3-folds of general type

Massimiliano Alessandro (U. Saarland)

The canonical map phi of a smooth complex algebraic variety X of general type is an important invariant in complex algebraic geometry. We focus on the case where phi is birational. In 2017 Catanese provided a representation-theory-based criterion to ensure that phi is birational under the assumption that X is a surface isogenous to a product of curves. In a joint work in progress with Christian Gleissner we adapt this criterion to 3-folds isogeneous to a product via an abelian group, providing an effective construction method which leads us to explicit examples of 3-folds where phi is birational.

Collapsed ancient solutions of the Ricci flow on compact homogeneous spaces

Francesco Pediconi (PoliTo)

Ricci flow solutions that exist for all negative times have special significance and are known as ancient solutions. In this talk, we describe the general structure of collapsed ancient solutions to the homogeneous Ricci flow on compact manifolds. Furthermore, we present a general existence theorem and show that, under certain algebraic assumptions, these solutions exhibit more symmetries than initially expected. This is joint work with S. Sbiti and A. M. Krishnan.

7th April 2025

Maximal branes of compact hyper-Kähler manifolds

Annalisa Grossi (UniBo)

On a hyper-Kähler manifold X, a real structure — or equivalently, an anti-holomorphic involution — is referred to as a brane involution.  Up to hyper-Kähler rotation, an anti-holomorphic involution can become an anti-symplectic involution. When the Smith–Thom inequality is realized as an equality, the brane involution is said to be maximal. While examples of non-compact hyper-Kähler manifolds admitting maximal branes are known, the compact case presents a more intriguing picture. In particular, although some K3 surfaces admit maximal brane involutions, the main result that I will show is the non-existence of maximal branes on compact hyper-Kähler manifolds of K3^[n]-type when n=2 or n is odd.

This talk is based on a joint work in progress with Simone Billi and Lie Fu.

Rigidity of an overdetermined heat equation and minimal helicoids in space-forms

Andrea Bisterzo (SNS)

In the seminal paper "Characterizations of the Mean Curvature and a Problem of G. Cimmino" of 1995, J. C. C. Nitsche proved that if a domain in R^3 is uniformly dense in its boundary, then the boundary must be either a plane or a right helicoid, thereby resolving an open problem proposed by G. Cimmino in 1932. This result, along with the techniques used in its proof, has inspired a significant line of research on rigidity phenomena related to overdetermined differential problems in possibly unbounded domains, with particular regard to those involving the heat equation. The aim of this talk is to present an ongoing work in collaboration with Professor Alessandro Savo, in which we characterize embedded minimal helicoids and totally geodesic hypersurfaces in three-dimensional space-forms through the concept of “constant boundary temperature”, an overdetermined condition involving the Cauchy problem for the heat equation. The result is obtained using a method that differs significantly from Nitsche's technique.

17th March 2025

Finite mapping class group orbits on character varieties

Arnaud Maret (U. Strasbourg)

A natural group to associate to an surface is its mapping class group which is the group of isotopy classes of diffeomorphisms of the surface. The mapping class group acts on the space of conjugacy classes of morphisms from the fundamental group of the surface into some Lie group - a so-called character variety. In this talk we'll investigate the rare phenomenon of finite orbits for mapping class group dynamics on character varieties. We'll see how to construct non-trivial examples of finite orbits and give some intuition on how to classify all finite orbits when the target Lie group is SL(2,C). Most of this work is a collaboration with Samuel Bronstein.

Arithmetic Zariski multiplets of irreducible plane curves

Matteo Penegini (UniGe)

(This is a joit project with M. Loenne.) 

Multicanonically embedded surfaces in projective space give rise to irreducible branch curves via projection from generic axes. Building on our previous work, we transfer results from the moduli space of surfaces to strata of plane curves. For instance, the faithful action of the Galois group on the connected components of the moduli spaces of surfaces isogenous to a product, as established by Bauer, Catanese, and Grunewald, gives rise to many arithmetic Zariski multiplets.

20th February 2025

Symmetry counts: an introduction to equivariant Hilbert and Ehrhart series

Alessio D'Alì (PoliMi)

The Ehrhart series of a lattice polytope P is a combinatorial gadget that counts the number of lattice points of P and of its dilations. The Hilbert series of a simplicial complex S counts how many monomials supported on faces of S exist in each possible degree. The aim of this talk is to introduce equivariant versions of such constructions, where we are not just interested in counting but we also want to record how the action of a finite group affects such collections of lattice points or monomials. Inspired by previous results by Betke-McMullen, Stembridge, Stapledon and Adams-Reiner, we will investigate which extra combinatorial features of the group action give rise to "nice" rational expressions of the equivariant Hilbert and Ehrhart series, and how the two are sometimes related. This is joint work with Emanuele Delucchi.

Riemannian twistor spaces in four-dimensional geometry: an overview and some rigidity results

Davide Dameno (UniMi)

In this talk, we will briefly present some unique features of four-dimensional Riemannian Geometry and their connections with twistor theory: indeed, it is well known that four-dimensional Riemannian manifolds carry many peculiar properties, which give rise to the existence of unique special metrics (e.g. half conformally flat metrics). In their study of self-dual solutions of Yang-Mills equations, Atiyah, Hitchin and Singer adapted the celebrated Penrose’s construction of twistor spaces to the Riemannian context, showing that a Riemannian four-manifold is half conformally flat if and only if its twistor space is a complex manifold: this paved the way for the study of many other characterizations of curvature properties for Riemannian four-manifolds. After giving an overview of the basic properties of twistor spaces in the four-dimensional case, we present some new rigidity results for Riemannian four-manifolds whose twistor spaces satisfy specific vanishing curvature conditions. This is based on joint works with Giovanni Catino and Paolo Mastrolia.

23rd January 2025

On the mass of initial data with positive cosmological constant

Stefano Borghini (UniTn)

The concept of mass for time-symmetric initial data has been extensively explored and is now a cornerstone in the study of contemporary Mathematical General Relativity, especially in relation to spacetimes with zero or negative cosmological constants. However, the case of a positive cosmological constant presents a distinct challenge: the renowned counterexamples to the Min-Oo conjecture by Brendle, Marques, and Neves highlight that even the rigidity statement in a potential positive mass theorem has not been correctly identified yet in this context. In this presentation, I will propose a novel approach to overcome this issue, leading to insights on a new notion of mass and to a characterization of the de Sitter spacetime. This is a joint work with Virginia Agostiniani and Lorenzo Mazzieri. 

Arithmetic is topology in disguise: Grothendieck's section conjecture

Giulio Bresciani (SNS Pisa)

After revolutionizing algebraic geometry, in his last years before retiring from mathematics, Grothendieck became very interested in arithmetic problems. As always in his career, he took a very naive approach to a subject that was relatively new to him. From his point of view, arithmetic is essentially topology, provided we take a fairly elastic definition of topology. This naive approach led him to define the étale fundamental group of an algebraic variety, which at once generalized the topological fundamental group and Galois theory, providing a bridge between the two. This new bridge inspired him to formulate a series of profound conjectures that predict how the entire geometry of certain varieties, so-called “anabelian,” is reconstructible from the étale fundamental group. Of these, the most profound is the so-called section conjecture, and it is to this day widely open. In the seminar, I will try to convince you that arithmetic is indeed topology, introduce the section conjecture, mention some results and future prospects.

13th December 2024

G2 structures on nilmanifolds and their moduli spaces

Giovanni Bazzoni (UnInsubria)

In this talk I will review non-integrable G2 structures on 7-dimensional nilmanifolds. I will dwell on purely coclosed G2 structures, constructing them from certain SU(3) structures in dimension 6. Also, I will illustrate some results on moduli spaces of (co)closed G2 structures on nilmanifolds. This is based on joint work with A. Garvín, A. Gil García and V. Muñoz.

Characteristic classes in motivic homotopy theory

Fabio Tanania (TU Darmstadt)

In this talk, I will give a brief introduction to the category of motivic spaces, highlighting some of its key features. I will then focus on classifying spaces of linear algebraic groups and discuss computations of their motivic cohomology rings, which yield various types of characteristic classes. Throughout the talk, the primary example will be provided by orthogonal groups (and also special orthogonal and spin groups), with their associated motivic Stiefel-Whitney classes, which provide invariants for quadratic forms.

14th Novemeber 2024

Quantization Commutes with Reduction on CR Manifolds

Andrea Galasso (UniMiB)

We consider a compact, torsion-free CR manifold that admits an action by a compact Lie group. Given an equivariant rigid CR line bundle, it is natural to study the space of invariant CR sections onto which a particular weighted invariant Fourier–Szegő operator projects. Under natural assumptions, we show that the group-invariant Fourier–Szegő projector admits a full asymptotic expansion. As an application, if the tensor power of the line bundle is sufficiently large, we prove that quantization commutes with reduction. This is joint work with Chin-Yu Hsiao.

Stable Minimal hypersurfaces: old and new

Alberto Roncoroni (PoliMi)

In this talk I will present an overview on the classification problem for stable minimal hypersurfaces in the Euclidean space, starting from the fundamental work of Bernstein about entire graphs, passing through the work of Schoen-Simon-Yau up to the recent  advancements.

17th October 2024

Conical linear series

Riccardo Moschetti (UniTo)

Cones over projective varieties are a very versatile and powerful tool in algebraic geometry. I will talk about a joint work with Pietro Pirola and Lidia Stoppino, in which we develop a theory of "conic linear series" , where we will use certain limits of cones to construct divisors on curves in P^3. I will talk about one possible application of this theory to the construction of certain pencils of planar curves.

Towards L^2 Hodge theory on non compact complex manifolds

Riccardo Piovani (UniTo)

L^2 Hodge theory has been mainly developed for elliptic complexes of first order differential operators such as the de Rham or the Dolbeault complexes. This excludes the complex associated to Aeppli and Bott-Chern cohomologies, which are useful invariants of non Kähler compact complex manifolds, since this complex has a differential of order two. In this seminar we will explore the path towards the establishment of a L^2 Hodge theory associated with this last complex on non compact complex manifolds.

Special structures in complex geometry

Tommaso Sferruzza (UnInsubria)

One of the main goals in complex geometry is to find a canonical notion of Hermitian metrics. For instance, Kähler metrics, i.e., Hermitian metrics whose fundamental form is closed, represent a very special class on compact complex manifolds: despite satisfying remarkable properties, their existence imposes several restrictions on the topology of the underlying manifold. As a result, starting from the 80's, much attention has been devoted to the study of candidate canonical structures which generalize the Kähler condition. In this seminar, I would like to give a brief introduction to some of such special structures and present some of their existence problems on compact complex manifolds.