Abstract: In this this talk I will give an introduction to the Topological Entropy Conjecture formulated by Shub, and highlight the most relevant related results from the litterature.

It is conjectured that the topological entropy of a diffeomorphism on a manifold can be controlled from below by the action induced by the diffeomorphism on the homology. The regularity of the diffeomorphism turns out to play a key role: the conjecture was disproved in the case of homeomorphisms, and conversely holds true if the diffeomorphism is smooth. For the finite regularity case, the problem is still open. We recently witnessed advancements, when some restrictions on the dynamics of the diffeomorphism were assumed.

Finally, we remark that this question led to the formulation of many other closely related problems, involving different global topological invariants.

Abstract: In this talk we will introduce Bressan's Fire Conjecture: it is concerned with the model of wild fire spreading in a region of the plane and the possibility to block it using barriers constructed in real time. The fire starts spreading at time t=0 from the unit ball centred at the origin in every direction with speed  1, while the length of the barrier constructed within the time  t has to be lower than σt, where  σ is a positive constant (construction speed). If σ≤1 Bressan proved that no barrier can block the spreading of the fire, while if σ>2 there exists always a strategy that confine the fire. In 2007 Bressan conjectured that if σ≤2 then no barrier can block the fire.

In this talk we will prove Bressan's Fire Conjecture in the case barriers are spirals. Spirals are thought to be the best strategies a firefighter can do in order to confine the fire for  σ≤2 . We will introduce the new concept of family of generalized barriers and we will prove that, if there exists such a family satisfying a diverging condition, then no spiral can confine the fire. This is a joint work with Stefano Bianchini.

Abstract: This talk is devoted to the prescribed mean curvature equation for t-graphs in a Riemannian Heisenberg group. Following  results due to Giusti, we will prove existence of $BV$ solutions in the so-called non-extremal domains without imposing Dirichlet conditions. Then the regularity will be improved in consecutive steps to obtain classical solutions in this domains. The main goal of the talk are the characterization of the existence of classical solutions of the equation and the study of several conditions that implies the uniqueness of them. Finally, by an approximation technique, we obtain solutions to the sub-Riemannian prescribed mean curvature equation. This result are obtained in collaboration with S. Verzellesi.

Abstract: Although the weight distribution doesn’t fully characterize a code, it provides significant information. Research in this area starts with MacWilliams’ work in 1963, where she proved the famous MacWilliams identities. These identities establish a relation between the Hamming weight enumerators of a linear code over a finite field and its dual. In the past few decades, the MacWilliams identities have been proven to hold for other metrics too, such as the rank metric or the sum-rank metric. When considering codes over f inite chain rings, the Lee metric or the Homogeneous metric are natural choices to endow the alphabet with. The classical MacWilliams identities do not hold for these metrics (apart from certain specific situations) due to the lack of knowledge about the zero positions of codewords for a fixed weight, which means that the Krawtchouk coefficients are not well-defined. To overcome this issue, we can choose a different approach to partition the ambient space. In this talk, we will partition the ambient space based on a natural decomposition that stems from the definition of the metric considered. This partition allows us to easily compute the weight of its elements and it gives rise to well-defined Krawtchouk coefficients in each case. Hence, MacWilliams-type identities based on the corresponding decomposition do exist..

Abstract: The Hochschild-Kostant-Rosenberg (HKR) isomorphism is one of the best features of Hochschild homology and cohomology, providing elegant decompositions of those. However, the resulting spaces are not always nice for varieties that are not compact. Compactification in algebraic geometry can be understood via logarithmic schemes. In this talk, I will present a joint work with Marci Hablicsek and Leo Herr devoted to adapting notions in this area and proving a logarithmic version of the HKR isomorphism.

Abstract: In this talk, through simple examples, I will explain the basic idea behind the Riemann-Hilbert correspondence. It is a correspondence between two different moduli spaces: the de Rham moduli space parametrizing meromorphic differential equations, and the Betti moduli space describing local systems of solutions and the representations of the fundamental group defined by them. We will see why such a correspondence breaks down for higher order poles.

Abstract: In this talk I will report on an ongoing project, jointly with G. Canneori and S. Terracini. We study a one dimensional model for the Helium atom in which the two electrons and the nucleus are collinear and subject to electric attraction/repulsion. The nucleus is fixed at the origin and the system is governed by a system of two non-linear singular differential equations.

Using a mountain pass type argument and a suitable smoothing of the system, we show the existence of a particular family of periodic solutions called frozen planet orbits for all negative values of the energy. For energy close to zero, one electron keeps collapsing into the nucleus whereas the outer one oscillates slowly and far from the other particles.

In the second part of the talk we investigate the case in which the repulsive force disappears and explore the connection with billiards dynamics. The ODE system decouples in two Kepler-like 1-dimensional equations and solutions converge to a period 2 billiard trajectory.

Abstract: Micro-swimmers like cilia and flagella propel through fluids using the axoneme, a structure composed of nine pairs of microtubules. Molecular motors, situated between these pairs, power motion through ATP. 

This talk explores microscopic modeling of the axoneme, including a '2-row model' extension. We discuss how this new system preserves axoneme symmetry, is well-posed and how it undergoes a supercritical Hopf bifurcation based on ATP levels. 

Moreover, we will present a further extension of the two-row model, introducing an arbitrary number $N$ of motor rows aimed at reproducing the real axoneme structure where $N=8$. The model is studied numerically through simulations, in which one can observe periodic oscillations past a certain ATP value, without any additional form of regulation in the system.

This is a joint work with Prof. Alouges F. (Centre Borelli, ENS-Paris Saclay, CNRS), Prof. De Simone A. (SISSA), Prof. Lefebvre-Lepot A. (CNRS, Fédération de Mathématiques de CentraleSupélec) and Levillain J. (CMAP,  CNRS, École polytechnique).

Abstract:  In applications, there are many systems whose dynamics can be influenced by discrete events. For instance, a power switch turned on and off, a thermostat turning the heat on and off, a car running on a street with some ice here and there. Such systems are called switched systems.

As for classical dynamical systems, stability for this class of systems is a very important issue. A natural way to measure the stability is to use Lyapunov exponents. 

In this talk, I will show how to compute exact Lyapunov exponents for a simple class of switched systems. This problem can be reduced to an Optimal Control Problem. Applying Pontryagin Maximum Principle, one can find all extremals for this problem and then choose among them the optimal one. This is a joint work with Prof. A. A. Agrachev

Abstract: We prove the global in time existence and the stability of the solutions of a class of quasi-periodically forced linear wave equations on the circle of a prescribed form.

This result is obtained by reducing the Klein-Gordon equation to constant coefficients, applying first a pseudo-differential normal form reduction and then a KAM diagonalization scheme.  A central point is that the equation is equivalent to a first order pseudo-differential system which, at the highest order, is the sum of two backward/forward transport equations, with non-constant coefficients, respectively on the subspaces of functions supported on positive/negative Fourier modes. The key idea is to straighten such operator through a novel quantitative Egorov analysis. A main point of interest, in view of applications to a nonlinear setting, is that the change of variables that reduce the equation satisfies tame bounds.

This is a joint work with M. Berti, R. Feola and M. Procesi.

Abstract: Almgren introduced for the first time the frequency function to prove that the singular set of so-called Dir-minimizing Q-valued functions on an n-dimensional real space has Hausdorff dimension at most n-2. This frequency function then played a fundamental role for estimating the Hausdorff dimension of mass-minimizing currents, and many variants of the frequency function have been developed by many authors to study regularity properties of some classes of varifolds and currents. In this talk I give a brief outline of Almgren's proof about the estimate of the Hausdorff dimension of the singular set of Dir-minimizing functions and how these techniques can be adapted to study rectifiability of the singular set of some varifolds with arbitrary codimension.

Abstract: A central problem in Kähler geometry, known as the Calabi problem and which was developed during the last half of the 20th century, is to understand the existence of canonical Kähler metrics representing a given deRham cohomology class (called a Kähler class) on a given compact complex manifold. The theory is now quite mature, and one of the most interesting directions in the field is its link to complex algebraic geometry through the notion of K-stability.

On this talk we will present briefly the classical Calabi problem for Kähler-Einstein metrics and its extension for constant scalar curvature and extremal metrics. For extremal Kähler metrics, the relation between extremal Kähler metrics and K-stability receives the name of Yau-Tian-Donaldson (YTD) conjecture. Our goal is to present this conjecture on the context of toric geometry. In this setting, explicit descriptions

of the objects appearing in the YTD conjecture can be given in terms of labelled polytopes in an n-dimensional real vector space . If time permits, we will also discuss modifications of this problem for toric generalized kähler geometry.

Abstract: In the context of continuous-time stochastic mean field games (MFGs), we introduce a generalization of MFG solution, called coarse correlated solution, which can be seen as the mean field game analogue of a coarse correlated equilibrium. The latter is a generalization of Nash equilibrium for stochastic games, which allows for correlation between the strategies of non-cooperative players. Our notion of solution can be justified by showing that approximate N-player correlated equilibria can be constructed starting from a correlated solution to the mean field game, and existence can be proved by means of a minimax theorem. If time allows, an application to an abatement game between greenhouse gas emitters will be presented, in which coarse correlated solutions lead both to greater abated quantities and higher payoffs than the usual MFG solution. The talk is based upon joint works with L. Campi (University of Milan "La Statale"), M. Fischer (University of Padua) and F. Cartellier (ENSAE Paris).

Abstract: We analyze an optimal transport problem with additional entropic cost evaluated along curves in the Wasserstein space which join two probability measures. The effect of the additional entropy functional results into an elliptic regularization for the (so-called) Kantorovich potentials of the dual

problem. Assuming the initial and terminal measures to have densities, we prove that the optimal curve remains positive and locally bounded in time. We focus on the case that the transport problem is set on a compact Riemannian manifold with  bounded Ricci curvature.

The approach follows ideas introduced by P.L. Lions in the theory of mean-field games about optimization problems with penalizing congestion terms. Crucial steps of our strategy include displacement convexity properties in the Eulerian approach and the analysis of distributional subsolutions to Hamilton-Jacobi

equations.

The result provides a smooth approximation of Wasserstein-2 geodesics.

Abstract: In this talk, we will present a local-global problem, arising from the Hasse principle for quadratic forms, called the Local-Global Divisibility Problem for commutative algebraic groups. It was first stated in 2001 by Dvornicich and Zannier, who also gave a cohomological interpretation of the problem. We will also see some results, in particular for algebraic tori: a generalization of the Grunwald-Wang Theorem to any algebraic tori with bounded dimension and a counterexample for the local-global divisibility by any power of an odd prime (joint work with Rocco Chirivì and Laura Paladino).

Abstract: In the last years it was observed that Residual Neural Networks (ResNets) can be interpreted as discretizations of control systems, bridging ResNets (and, more generally, Deep Learning) with Control Theory.     

In the first part of this seminar we formulate the task of a data-driven reconstruction of a diffeomorphism as an ensemble optimal control problem.

In the second part we adapt this machinery to address the problem of Normalizing Flows: after observing some samplings of an unknown probability measure, we want to (approximately) construct a transport map that brings a “simple” distribution (e.g., a Gaussian) onto the unknown target distribution. In both the problems we use tools from Γ-convergence to study the limiting case when the size of the data-set tends to infinity.

Abstract: Curved differential graded algebras are a generalization of the notion of differential graded algebras to a setting where the differential might fail to square to zero. I will explain how these emerge as deformations of differential graded algebras and the challenges that arise in trying to generalize the notion of quasi-isomorphism and derived categories to the curved setting.

Finally, I will give a tentative definition of a filtered derived category of a curved algebra and explain its main properties and possible applications.

Abstract: The aim of the talk is to give an introduction to linear degenerations of flag variety and to present some of their remarkable properties. 

In the first part of the talk, I am going to introduce these objects as a natural generalization of classical complete flag variety. 

In the second part, using suitable torus actions and tools from quiver representation theory, I will discuss geometric and combinatorial properties of linear degenerations. 

Finally, if there is enough time, I will present an overview of some recent results in this interesting research area.

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