A celebrated result in graph theory links the chromatic polynomial of a graph to the Tutte polynomial of the associated graphic matroid. In 2005, Helme-Guizon and Rong proved that the chromatic polynomial is categorified by a cohomological theory called chromatic cohomology.
  In this talk, I will describe how to associate a matroid to a directed graph G, called the multipath matroid of G, which encodes relevant combinatorial information about edge orientation. We also show that a specialization of the Tutte polynomial of the multipath matroid of G provides the number of certain "good" digraph colorings.
  Finally, analogously to the relationship between the chromatic polynomial and chromatic cohomology, I will show how the polynomial expressing the number of "good" digraph colorings is linked to multipath cohomology, introduced in a work with Caputi and Collari in 2021.

  The small quantum groups u_q(g) are finite-dimensional quotients of quantum universal enveloping algebras U_q(g) at a root of unity q for g a semisimple complex Lie algebra. After the work of Lusztig, the representation theory of these quantum objects was intensively studied because of its relation with the representation theory of semisimple algebraic groups in positive characteristic. In this talk, I will present some results on the representation theory of what we call "generalized" small quantum groups. A particular feature of these objects is that the role of the corresponding Cartan subalgebra is played by a finite non-abelian group. Nevertheless, they still admit a triangular decomposition and share similar properties with the standard quantum groups, like the existence of weights (that are no longer one-dimensional) and Verma modules.
  This talk is based on a joint work with Cristian Vay [Simple modules of small quantum groups at dihedral groups, Doc. Math. 29 (2024), 1-38].

Isogeometric Analysis (IGA) advocates for the direct utilization of spline-based geometric representations for performing spline-based numerical simulations. Over the last decade and a half, this philosophy has been applied to diverse applications with great success and has yielded theoretical developments that cement the status of smooth splines as excellent tools for approximation. In this talk I will focus on the use of Truncated Hierarchical B-spline (THB-spline) spaces and discuss the formulation of a local, L²-stable projector for them. The projector is an extension of the Bézier projection proposed by Thomas et al. (2015) and relies on a characterization of linear independence of THB-splines over certain macro-elements. This talk is part of the activity of the MUR Excellence Department Project MatMod@TOV (CUP E83C23000330006).

We study a geometric/combinatorial characterisation of supports for linear degenerations of flag varieties in an attempt to determine the coefficients of the decomposition theorem. We show that it behaves correctly when applying the operation suspension, and obtain positive results in the PBW-locus.

Energy density and its positivity properties represent a fundamental subject in classical and quantum physics. In this talk, we will investigate this topic in the thermal representation of a free massive quantum scalar field. After a brief review of the fundamental mathematical tools at the base of this work, we will construct the GNS representation of our QFT induced by a state at thermal equilibrium (KMS). Therein, we will identify the generator of the time evolution and its spatial density. The symmetry between the "particles" and "holes" makes evident the impossibility for a lower bound for the expectation value of the energy density in this representation. In order to tackle this problem, we will investigate and extend some results of modular theory and non-commutative Lp spaces. In this way, we will obtain a general result concerning the expectation value of operators affiliated to a von Neumann algebra. Finally, the proven results will be applied to our setting to derive an L4 state dependent non-trivial QEI.

The Operator Algebra Seminar schedule is here: https://sites.google.com/view/oastorvergata/home-page?authuser=0

In this talk we describe the influence of the initial data and the forcing terms on the regularity of the solutions to a class of evolution equations including the heat equation, linear and semilinear parabolic equations, together with the nonlinear p-Laplacian equation. We focus our study mainly on the regularity (in terms of belonging to appropriate Lebesgue spaces) of the gradient of the solutions.
NB:This talk is part of the activity of the MIUR Excellence Department Project MATH@TOV CUP E83C23000330006

One of the main motivations behind derived differential geometry is to deal with singularities arising from zero loci or intersections of submanifolds. Both cases can be considered as fiber products of manifolds which may not be smooth in classical differential geometry. Thus, we need to extend the category of differentiable manifolds to a larger category in which one can talk about "homotopy fiber products". In this talk, we will discuss a solution to this problem in terms of dg manifolds. The talk is mainly based on a joint work with Kai Behrend and Hsuan-Yi Liao.

I will give a brief survey of the formal geometry of groupoids.

Gibbs measures for nonlinear dispersive PDEs have been used as a fundamental tool in the study of low-regularity almost sure well-posedness of the associated Cauchy problem following the pioneering work of Bourgain in the 1990s. In the first part of the talk, we will discuss the connection of Gibbs measures with the Kubo-Martin-Schwinger (KMS) condition. The latter is a property characterizing equilibrium measures of the Liouville equation. In particular, we show that Gibbs measures are the unique KMS equilibrium states for a wide class of nonlinear Hamiltonian PDEs. Our proof is based on Malliavin calculus and Gross-Sobolev spaces. This is joint work with Zied Ammari.
In the second part of the talk, we will explain a general principle that allows us to obtain almost sure global solutions for Hamiltonian PDEs provided that one has a stationary probability measure. In this context, stationarity refers to a solution of the associated Liouville equation. This more general notion replaces the invariance from before. The second part of the talk is joint work with Zied Ammari and Shahnaz Farhat.

Note: This talk is part of the activity of the MUR Excellence Department Project MatMod@TOV (CUP E83C23000330006)

The Operator Algebra Seminar schedule is here: https://sites.google.com/view/oastorvergata/home-page?authuser=0

A classical result in approximation theory due to Korovkin asserts that a sequence of positive unital maps on C([0,1]) converges pointwise to the identity if they merely converge to the identity on the functions 1,x,x^2. This result was later generalized by Saskin, who showed that convergence to the identity on a generating function system implies convergence to the identity everywhere if and only if the system has full Choquet boundary. Arveson's last open conjecture in his seminal work on non-commutative boundary theory predicts that a non-commutative analogue of Saskin's result holds. We refute Arveson's conjecture with an elementary counterexample. All notions will be explained during the talk.

The Operator Algebra Seminar schedule is here: https://sites.google.com/view/oastorvergata/home-page?authuser=0