Matroids are combinatorial objects that abstract the notion of linear independence and can be used to describe several structures such as, for example, vector spaces and graphs. Informa-tion on matroids can be encoded in several polynomial invariants, the most famous one being the characteristic polynomial; some of these polynomials can also be upgraded to graded vector spaces via abelian categorification or, when the matroid has a non-trivial group of symmetries, to graded virtual representations.
  Moreover, to each matroid, one can associate a polytope that belongs to the more general class of generalized permutahedra; a matroid invariant is called valuative if it behaves well under subdivi-sions of matroid polytopes.
  After introducing matroids and their invariants, the goal of the talk is to formulate the new notion of categorical valuativity and give some examples.
  This is based on a joint ongoing project with Dane Miyata and Nicholas Proudfoot.

  Quiver Grassmannians are projective algebraic varieties generalizing ordinary Grass-mannians and flag varieties. The cohomology of quiver Grassmannians of particular type has appli-cations to the geometric interpretation of various algebraic objects such as quantized universal enveloping algebras and cluster algebras. The variation in the cohomology of families of quiver Grassmannians of equioriented type A has been studied by Lanini-Strickland and Fang-Reineke.
  In this talk, I relate on joint work with Cerulli Irelli-Fang-Fourier and Cerulli Irelli-Marietti, in which we prove an upper semicontinuity statement for the cohomology of quiver Grassmannians of type A and we study the relation with Motzkin combinatorics found in work of Fang-Reineke.

We consider random walks on groups of isometries of the hyperbolic plane, known as Fuchsian groups. It is well-known since Furstenberg that such random walks converge to the boundary at infinity, and the probability to reach a given subset of the boundary defines a hitting, or harmonic, measure on the circle. It has been a long-standing question whether this harmonic measure is absolutely continuous with respect to the Lebesgue measure. Conjecturally, this is never the case for random walks on cocompact, discrete groups. In the talk, based on joint work with Petr Kosenko, we settle the conjecture for nearest neighbour random walks on hyperelliptic groups. In fact, we show that the dimension of the harmonic measure for such walks is strictly less than one. This is also related to an inequality between entropy and drift.

Somewhat motivated by the original approach of J.-B. Fourier to solve the heat equation on a bounded domain, we formulate some new properties of countable discrete groups involving certain completely positive multipliers of the reduced group C*-algebra and norm-convergence of Fourier series. The stronger "heat property" implies the Haagerup property, while the "weak heat property" is satisfied by a much larger class of groups. Examples will be provided to illustrate the various aspects. In perspective, a challenging goal would be to obtain yet another characterization of groups with Kazhdan's property (T). (Based on joint work with E. Bédos.)

La retta tangente il grafico di una funzione sufficientemente regolare è la posizione limite di rette secanti il grafico. Ci si aspetterebbe che il piano tangente il grafico di una funzione a due variabili sufficientemente regolare sia la posizione limite di piani secanti il grafico in tre punti non collineari (a,f(a)), (b,f(b)) e (c,f(c)), ma così non è. Questo fenomeno paradossale è una versione locale del paradosso dell'area di una superficie curva (detto anche paradosso di Schwarz). In questo seminario visualizzerò il fenomeno paradossale, e mostrerò come il "coefficiente angolare vettoriale" di un piano secante possa esprimersi sia come combinazione vettoriale delle normali esterne al triangolo di vertici a, b e c, sia come rapporto vettoriale incrementale, quando il prodotto vettoriale è quello geometrico di Clifford. Se rimarrà tempo, accennerò anche a come modificare tale rapporto incrementale vettoriale per renderlo sempre convergente al gradiente di f.

In this talk, we will discuss the isoperimetric inequality and its high order version -- Alexandrov Fenchel inequality, which dates back to the Queen Dido in ancient Carthage era. We introduce the quermass integrals for compact hypersurfaces with capillary boundary. Then by using a constrained mean curvature type flow, one can obtain the Alexandrov-Fenchel inequality for compact hypersurfaces with capillary boundary.