2024/2025 Seminars

Abstract: In this talk I will present a result due to Stein concerning the sharp summability condition on the weak gradient of a Sobolev function to ensure continuity and almost everywhere differentiability. We will follow a recent approach by Kauhanen, Koskela and Malý, which also provides a possible generalization of the Fundamental Theorem of Calculus in high dimensions. We will then extend this result to the non-abelian setting of Carnot groups, discussing the main ideas behind this generalization.

Based on a joint work with A. Pinamonti and F. Serra Cassano.

Abstract: We consider independent Bernoulli bond percolation with parameter p on the d-dimensional cubic lattice. In the subcritical regime - that is, for sufficiently small values of p - the size of any cluster is known to be almost surely finite, and the probability that a cluster reaches a given size decays exponentially fast. In this talk, I will present a recent and concise proof of this exponential decay, due to Hugo Vanneuville (2024). The proof relies on stochastic comparison techniques, originally developed by Russo in the 1980s, using the notion of pivotality. I aim to provide a general and accessible overview of these techniques.

Abstract: In this talk, we start providing a friendly introduction to Knot Theory and Cryptography. From the intersection of these fields, we propose an innovative key exchange protocol whose security relies on two hard problems: the Decomposition Problem (finding the prime factorization of a knot) and the Recognition Problem (deciding if two diagrams represent the same knot). Joint work with Arno Wildi.

Abstract: Topological K-theory was developed by Atiyah and Hirzebruch in 1961 based on Grothendieck’s work on algebraic varieties. The idea is to capture the topology of a space X by studying vector bundles over it. It is possible to define K-theory directly from X, or in terms of matrices of continuous complex-valued functions on X. This provides a more algebraic approach which extends to matrices over general Banach algebras. In fact, since the early 1970s Ktheory has been a powerful tool in C∗-algebras theory. I will give an introduction to this topic and present some instances of the interplay between operator algebras and homotopy theory.

Abstract:  A systematic bridge between the enumerative geometry of stable curves and integrable hierarchies of PDEs was build by Buryak; based on the geometry of the so-called double ramification cycles, one associates the integrable DR hierarchy to a collection of cohomology classes that respect i.a. the gluing of two curves along a node. This talk aims to discuss moduli spaces of stable curves and differentials, the double ramification cycles, and how these form the geometric input for two striking examples of this DR construction. Specifically, Buryak- Rossi-Zvonkine proved that the hierarchy associated to the Poincaré dual classes of the closures of the loci of curves that admit residueless meromorphic differentials with specified orders of zeroes and poles reduces to the paradigmal Kadomtsev-Petviashvili (KP) hierarchy, describing 2D shallow water waves, up to a coordinate transform. The proof demonstrates how the reconstruction of the integrals that count higher genus curves with certain conditions from only a couple geometric computations is understood as the reconstruction of the KP hierarchy from a few initial equations and the commutativity of the flows. Taking into account the spin-parity of the differentials, this method produces the BKP hierarchy, as demonstrated together with Klompenhouwer.

Abstract:  For an analytic function g in Hol(D), we discuss some properties of the integral operator Tg. In particular, we discuss about the largest space of analytic functions, which is mapped by Tg into the Hardy space Hp. We call this space the optimal domain of integration operator and we investigate its structural properties. Motivation for this comes from the work of G. Curbera and W. Ricker who studied the optimal domain of the classical Cesáro operator, while also investigating properties of non-radial weighted Bergman spaces. The talk is based on articles, which were in joint work with C. Bellavita, A. Belli, V. Daskalogiannis, G. Styllogiannis.

Abstract: Generalising Frobenius manifolds, the notion of F-manifold sits at the crossroad of a variety of areas of mathematics. In this talk, we first present progressive instances of F-manifolds and then introduce the class of F-manifolds with compatible connection, in relation with integrable systems of hydrodynamic type. Finally, we disclose how such a relation proves useful in extending the generalised hodograph method, originally developed to solve diagonalisable integrable systems of hydrodynamic type, to the regular non-diagonalisable setting.

Abstract: In 1977, D. C. Robinson developed a method for proving static vacuum Black Hole uniqueness in General Relativity. This method has recently been generalized to higher dimensions by C. Cederbaum, A. Cogo, B.Leandro, and J. Paolo dos Santos. It turns out that the same philosophy can also be used to prove geometric inequalities, such as the Willmore Inequality in Eucledian space by C. Cederbaum and A. Miehe.

In my talk, I will show how to adjust this philosophy to prove the Agostiniani-Fogagnolo-Mazzieri Willmore-type inequality for Riemannian manifolds with non-negative Ricci curvature as well as its application to the proof of Hamilton’s pinching conjecture. This is joint work with C. Cederbaum.

Abstract: Classically, mathematical theories are built on a strict notion of equality, and category theory provides a successful common language for those. Richer structures in modern mathematics (homotopy theory, representation theory, algebraic geometry) lead to more flexible notions of equivalence. Infinity-category theory capture these ideas by incorporating higher morphisms that witness identifications between identifications. 

In this talk, I will introduce the key ideas behind infinity categories and conclude by sketching their role in my research in representation theory.

Abstract: In 1927, Artin proposed a conjectural density for the set of primes p for which a given integer g is a primitive root mod p. The problem studied by Artin has an inherently geometric nature as it concerns the multiplicative group scheme over the integers, and thus admits various generalizations to other algebraic groups. Already in the case, first studied by Lang and Trotter in 1977, where the multiplicative group is replaced by an elliptic curve over the rationals, one can observe new phenomena that do not occur in Artin’s original problem.

 In this talk, I will trace the history of Artin’s conjecture and explain the heuristic behind it. I will then compare what is known for Artin’s original problem to more recent developments in the context of elliptic curves.

Abstract: Let M be a 3D contact sub-Riemannian manifold and S a surface embedded in M. We study the Schrödinger evolution of a particle constrained on the characteristic foliation F of S. Specifically, we define the Schrödinger operator Δℓ on each leaf ℓ as the classical "divergence of gradient", where the gradient is the Euclidean gradient along the leaf and the divergence is taken with respect to the surface measure inherited from the Popp volume, using the sub-Riemannian normal to the surface. We then study the self-adjointness of the operator Δℓ on each leaf by defining a notion of “essential self-adjointness at a point”, in such a way that Δℓ will be essentially self-adjoint on the whole leaf if and only if it is essentially self-adjoint at both its endpoints. We see how this local property at a characteristic point depends on a curvature-like invariant at that point. Additionally, we study self-adjoint extensions for the particular case of an infinite leaf.

Abstract: The study of boolean functions arises naturally in the context of computer science and combinatorics, but over the last 30 years connections with statistical physics started to be recognised. In this talk, we investigate the concept of noise sensitivity for functionals of i.i.d. random variables, which refers to the property that a small perturbation in the underlying randomness leads to an asymptotically independent functional and present some extensions of main classical results.

Abstract: This seminar aims to be a naive introduction to the theory of minimal surfaces in Riemannian and sub-Riemannian contexts. We begin with a review of some fundamental tools, such as affine connections, the second fundamental form and the mean curvature. The Gauss-Codazzi equations are then presented, along with Simons' formula in the special case of the Euclidean space. We show how these ideas can be extended to the sub-Riemannian setting, focusing on the Heisenberg group as a leading example. As an application of the Riemannian theory, the Schoen-Simon-Yau curvature estimates are discussed in the Euclidean setting. We conclude by briefly addressing analogous results in the Heisenberg group.

Abstract: Simple periodic motions for the n-body problem, in which each body moves on a Keplerian elliptical orbit while the shape of the system remains constant (up to rotations and scalings), arise from specific mass arrangements. In dimension d ≤ 3, such motions are produced only by central configurations. However, in dimension d ≥ 4, the higher complexity of the orthogonal group allows to define a broader class of balanced configurations, which includes central configurations as a special case.

 In this talk we study the existence of bifurcation points along trivial branches of planar balanced configurations in dimension 4 and give a lower bound on their number. This follows from an abstract bifurcation result which shows that, for a continuous family of C2 functionals on a finite dimensional manifold, the non vanishing of the spectral flow of the associated family of Hessians along a trivial branch of critical points implies the existence of bifurcation points. 

Abstract: Mumford’s Geometric Invariant Theory (GIT) is a powerful tool in algebraic geometry, used to form quotients of actions of reductive groups on schemes. Many examples of coarse/good moduli spaces of semistable objects can be realised as a reductive GIT quotient, including moduli spaces of Gieseker semistable sheaves and semistable quiver representations. However, if one is interested in constructing moduli spaces of unstable objects, often one needs to be able to form quotients of actions of parabolic groups.

After first giving a review of classical GIT, this talk will explain how in certain cases one can form quotients by non-reductive groups, using the Non-Reductive GIT of Kirwan et al. Time permitting, I will begin to explain how this can be applied in certain examples, such as with moduli of Gieseker unstable sheaves.

Abstract: This talk will be an introduction to representations of algebraic groups, with examples, focusing on a gadget called the Hecke algebra. After a short tour of the classical representation theory of finite groups, we will study the theory for p-adic groups which occur in number theory and in geometry. These groups have rich internal structure, making their theory analogous to that of finite groups, and our talk will focus on these similarities. We will demonstrate, with examples, how the Hecke algebra turns complicated problems into elementary linear algebra.

Abstract: In this talk I will give a crash course in the representation theory of p-adic groups. Using this knowledge, I will then explain why I'm interested in an invariant of complex representations of p-adic groups known as the canonical dimension. With the help of a combinatorial tool called the Bruhat-Tits building we can give a lower bound for the canonical dimension of a certain class of representations known as depth-zero supercuspidals. I will give a big picture sketch of the proof of this, focussing on concepts and ideas rather than technical details.