The Einstein equations are the governing equations of general relativity and admit an initial value formulation as a system of nonlinear wave equations. The talk will present some stability and instability results for stationary (black hole) solutions to such evolution equations, enlightening interesting connections between the dynamics of solutions and the number of dimensions considered. No prior exposure to general relativity will be assumed.

Note: This talk is part of the activity of the MIUR Department of Excellence Project MatMod@TOV (2023-2027)

There are various notions of singular characteristics in the theory of propagation of singularities to viscosity solutions. In this talk, we will discuss a variational construction of generalized characteristics and strict singular characteristics. We proved the solution of generalized characteristics can be constructed in an intrinsic way using the idea of minimizing movements and certain process of homogenization. Moreover, the relation between the strict singular characteristics and the EDI-EVI frame of gradient flow was also discussed. These results bridge various different topics such as Hamiltonian dynamical systems, weak KAM theory, cut locus in geometry, homogenization and gradient flow theory.
This talk is based on our latest work with Piermarco Cannarsa, Jiahui Hong and Kaizhi Wang.
Note: This talk is part of the activity of the MIUR Department of Excellence Project MatMod@TOV (2023-2027)

See the attached pdf file

Note: This talk is part of the activity of the MIUR Department of Excellence Project MatMod@TOV (2023-2027)

Non-autonomous or random dynamical systems (RDS) provide useful and flexible models to investigate systems whose evolution depends on external factors, such as noise and seasonal forcing. Modern developments on multiplicative ergodic theory and transfer operators allow us to get useful insights into the long-term behavior of these systems. In this talk, we will present results in this direction, including (quenched) limit theorems and thermodynamic formalism for a class of RDS. Our results will be illustrated with examples, including random open and closed intermittent maps and non-transitive systems. This talk is based on collaborations with J. Atnip, D. Dragicevic, G. Froyland and S. Vaienti.

  In 2014 Baur and Dupont introduced the notion of an asymptotic triangulation. This is a combinatorial object arising naturally from the combinatorics of cluster algebras of type Ã. They showed that the set of all asymptotic triangulations has an interesting combinatorial structure: it is a poset and the edges of the Hasse graph can be obtained by flipping the asymptotic arcs. In this talk I will explain how this set parametrises certain 2-term complexes in derived category of a finite-dimensional algebra called a cluster-tilted algebra of type à and that the flip operation corresponds to a mutation operation.
  This is joint work with L. Angeleri Hügel, K. Baur and F. Sentieri.

  After a general introduction on differential calculi on noncommutative spaces, we equip a family of algebras whose noncommutativity is of Lie type with a derivation based differential calculus obtained, upon suitably using both inner and outer derivations, as a reduction of a redundant calculus over the Moyal four dimensional space.

We present a novel framework for the study of a large class of nonlinear stochastic partial differential equations, which is inspired by the algebraic approach to quantum field theory. The main merit is that, by realizing random fields within a suitable algebra of functional-valued distributions, we are able to use specific techniques proper of microlocal analysis.These allow us to deal with renormalization using an Epstein-Glaser perspective, hence without resorting to any specific regularization scheme. As a concrete example we shall use this method to discuss the stochastic Phi^3_d model and we shall comment on its applicability to the stochastic nonlinear Schrödinger equation.

In this talk, I will present a recent regularity result for isoperimetric sets with densities, under mild Holder regularity assumptions on the density functions. Our main Theorem improves some previous results and allows to reach the regularity class C^{1,frac{alpha}{2-alpha}} in any dimension. This is a joint work with L. Beck and C. Seis.
NB:This talk is part of the activity of the MIUR Excellence Department Project MATH@TOV CUP E83C23000330006

  In this talk I will present a new point of view on Schubert polynomials via bosonic operators. In particular, we extend the definition of bosonic operators from the case of Schur polynomials to Schubert polynomials. More precisely, we work with back-stable Schubert polynomials and our operators act on the left weak Bruhat order. Furthermore, these operators with an extra condition give sufficiently enough linear equations for the structure of the cohomology ring of flag varieties.

  In its simplest form, this correspondence is the map from symmetric functions to skew-symmetric functions given by multiplication by the Weyl denominator (the Vandermonde determinant). A generalization produces the motivating shadow of “geometric Satake”, a diagram which contains the Satake isomorphism, the center of the affine Hecke algebra and the Casselman-Shalika formula. In a miracle that I wish I understood better, the whole diagram generalizes to the case of Macdonald polynomials and sends the bosonic Macdonald polynomial to the fermionic Macdonald polynomial. Does this suggest an “elliptic version" of geometric Langlands?
  This talk is based upon arXiv2212.03312, joint with Laura Colmenarejo.