Let $(M,g)$ be a closed Riemannian manifold of dimension $n ge 3$ and $P_g$ be a conformally-covariant operator on $(M,g)$. We consider in this talk two problem at the crossroads of conformal geometry and spectral theory: 1) determining the extremal value that the renormalized eigenvalues of $P_g$ take as $g$ runs through a fixed conformal class and 2) determining whether these extremal values are attained at an extremal metric. Examples of such operators $P_g$ include the famous conformal Laplacian of the Yamabe problem, $P_g = Delta_g + c_n S_g$, but also its higher-order generalisations such as the GJMS operators of order $2k$ for any positive integer $k$.
NB:This talk is part of the activity of the MUR Excellence Department Project MATH@TOV CUP E83C23000330006

We prove a conjecture of Cherednik describing the Betti numbers of compactified Jacobians of unibranch planar curves via superpolynomials of algebraic knots. The methods of the proof use the theory of orbital integrals and affine Springer theory. No prior knowledge about any of these will be assumed. Note: This talk is part of the activity of the MIUR Department of Excellence Project MatMod@TOV (2023-2027) and the PRIN 2022 Moduli Spaces and Birational Geometry and Prin PNRR 2022 Mathematical Primitives for Post Quantum Digital Signatures

&nbsp; Given a simple finite Lie algebra over the complex numbers, we can consider two other Lie algebras attached to it: its Langlands dual Lie algebra and the affine algebra at the critical level. It is a theorem of the nineties, by Feigin and Frenkel, that the center of the completed enveloping algebra of the affine algebra at the critical level is canonically isomorphic to the algebra of functions on the space of Opers on the pointed disk for the Langlands dual Lie algebra. These objects are actually pointwise instances of a more general picture: the space of opers for example enhances to a space which lives over an arbitrary smooth curve that is equipped with a natural factorization structure. This structure is fundamental for the Geometric Langlands community: factorization patterns allow for local to global arguments. In this talk I will explain the construction of the objects mentioned above and elaborate on a joint work with Andrea Maffei in which we prove the factorizable version of the Feigin-Frenkel theorem. <br> &nbsp; <em> <strong><u>N.B.</u>:</strong> this talk is part of the activity of the MIUR Excellence Department Project MatMod@TOV (CUP E83C23000330006) </em>

  In this talk, I will introduce restricted Poisson algebras in characteristic 2 and explore their connection with restricted Lie–Rinehart algebras. For the latter, a cohomology theory is deve-loped and abelian extensions are investigated. I will also construct a cohomology complex for restricted Poisson algebras in characteristic 2 that controls formal deformations. This complex is shown to be isomorphic to the cohomology complex of a suitable restricted Lie–Rinehart algebra. Several examples are provided to illustrate the constructions.
  This is a joint work with Sofiane Bouarroudj and Quentin Ehret.
  N.B.: this talk is part of the activity of the MIUR Excellence Department Project MatMod@TOV (CUP E83C23000330006)

Cubic hypersurfaces over the rational numbers often contain infinitely many rational points. In this situation, the asymptotic behavior of the number of rational points of bounded height is predicted by conjectures of Manin and Peyre. After reviewing previous results, we discuss the chordal cubic fourfold, which is the secant variety of the Veronese surface. Since it is isomorphic to the symmetric square of the projective plane, a result of W. M. Schmidt for quadratic points on the projective plane can be applied. We prove that this is compatible with the conjectures of Manin and Peyre once a thin subset with exceptionally many rational points is excluded from the count. Note: This talk is part of the activity of the MIUR Department of Excellence Project MatMod@TOV (2023-2027) and the PRIN 2022 Moduli Spaces and Birational Geometry and Prin PNRR 2022 Mathematical Primitives for Post Quantum Digital Signatures

In this talk, we introduce a family of functionals approximating the conformally invariant Dirichlet n-energy of maps between two Riemannian manifolds (M^n,g) and (N,h), which admit critical points. Along the approximation process, these critical points may incur a bubbling phenomenon, due to the conformal invariance of the limit Dirichlet n-energy. We prove an energy identity result for this approximation, ensuring that no energy gets lost along the formation of bubbles, under a Struwe type entropy bound assumption. We then show that min-max problems for the n-energy are always solved by a "bubble tree" of n-harmonic maps. This is a joint work with T. Lamm.<br> <b>NB</b>:<i>This talk is part of the activity of the MUR Excellence Department Project MATH@TOV CUP E83C23000330006</i>

  Singular cohomology of a space carries a natural CDGA structure induced by cup product at the level of cochains. While cup product is a well-defined operation on singular cochains, commutativity only holds after passing to cohomology, due to the presence of non-trivial Steenrod operations. This fact will be the motivating example for introducing E_∞-algebras and to explain how the E_∞-structure on cochains encodes this phenomenon in a precise way. Since the theory of E_∞-algebras needs the notion of operad, general recollections on the basics of operad theory will be provided.
  N.B.: this talk is part of the activity of the MIUR Excellence Department Project MatMod@TOV (CUP E83C23000330006)

  Shifted quantum affine algebras are quantum groups parameterized by a coweight of the underlying Lie algebra. In 2022, Hernandez introduced a category O of representations of these algebras, and in 2024, Geiss–Hernandez–Leclerc proved that the Grothendieck ring of this category O admits a cluster algebra structure. In this talk, after introducing the necessary background, I will explain how to construct a quantization of this cluster algebra, leading to a definition of the quantum Grothendieck ring for category O.
  If time permits, I will also discuss some applications and directions for future research.
  N.B.: this talk is part of the activity of the MIUR Excellence Department Project MatMod@TOV (CUP E83C23000330006)

batch sequential quadratic Hamiltonian (bSQH) algorithm for training convolutional neural networks with L2 and L0-based regularization is presented. This methodology is based on a discrete-time Pontryagin maximum principle. It uses forward and backward sweeps together with the layerwise approximate maximization of an augmented Hamiltonian function, where the augmentation parameter is chosen adaptively. The loss-reduction and convergence properties of the bSQH algorithm are analysed theoretically and validated numerically. Results of numerical experiments in the context of image classification with a sparsity enforcing L0-based regularizer demonstrate the effectiveness of the proposed method in full-batch and mini-batch modes. This is joint work with Sebastian Hofmann.

In this talk we illustrate the link between Deep Neural Networks and flows induced by control systems (Neural ODEs), and we relate the ''expressivity'' of a Residual Neural Network (ResNets) to the controllability properties of the corresponding Neural ODE in the space of diffeomorphisms. In case of control-linear Neural ODEs, a sub-Riemannian structure emerges. We show how the Lie Algebra Strong Approximating Property (see [Agrachev & Sarychev 2020,2022]) guarantees that, given two M-tuples of pairwise distinct points (M>1), we can steer one to the other. Moreover, this condition implies that we can approximate on compact sets any diffeomorphism isotopic to the identity using flows induced by the controlled dynamics. This ensures that ''sub-Riemannian'' ResNets are expressive.