The Chow ring CH(G) of a split semi-simple linear algebraic group G is one of the key geometric invariants in the theory of linear algebraic groups, torsors, motives of twisted flag varieties. Starting from pioneering works by Grothendieck and Borel, it has been studied for decades and computed for all simple groups (see e.g. Kac 1985, Duan 2015's). In the present talk we explain how to describe (and, hence, to compute) an oriented cohomology (Borel-Moore homology) functor A(G) using the localization techniques of Kostant-Kumar and the techniques of 2-monoidal categories: we show that the natural Hopf-algebra structure on A(G) can be lifted to a 'bi-Hopf' structure on the T-equivariant cohomology A_T(G/B) of the complete flag variety. More generally, we prove that the structure algebra of a Bruhat moment graph of a root system is a Hopf algebroid with respect to the right Hecke and left Brion-Knutson-Tymoczko actions. As an application, we obtain an effective combinatorial way to compute the coproduct on A(G).
  This is a joint work with Martina Lanini and Rui Xiong.

The optimal matching of blue and red points is prima facie a combinatorial problem. It turns out that when the position of the points is random, namely distributed according to two independent Poisson point processes in d-dimensional space, the problem depends crucially on dimension, with the two-dimensional case being critical [Ajtai-Komlos-Tusnady]. Optimal matching is a discrete version of optimal transportation between the two empirical measures. While the matching problem was first formulated in its Monge version (p=1), the Wasserstein version (p=2) connects to a powerful continuum theory. This connection to a partial differential equation, the Monge-Ampere equation as the Euler-Lagrange equation of optimal transportation, enabled [Parisi~et.~al.] to give a finer characterization, made rigorous by [Ambrosio~et.~al.]. The idea of [Parisi~et.~al.] was to (formally) linearize the Monge-Ampere equation by the Poisson equation. I present an approach that quantifies this linearization on the level of the optimization problem, locally approximating the Wasserstein distance by an electrostatic energy. This approach (initiated with M.~Goldman) amounts to the approximation of the optimal displacement by a harmonic gradient. Incidentally, such a harmonic approximation is analogous to de Giorgi's approach to the regularity theory for minimal surfaces. Because this regularity theory is robust --- measures don't need to have Lebesgue densities --- it allows for sharper statements on the matching problem (work with M.~Huesmann and F.~Mattesini).

In this talk I will report recent progress on the Yamabe equation defined either on a punctured disk of a smooth manifold or outside a compact subset of $R^n$ with an asymptotically flat metric. What we are interested in is the behavior of solutions near the singularity. It is well known that the study of the Yamabe equation is sensitive to the dimension of the manifold and is closely related to the Positive Mass Theorem. In my recent joint works with Jingang Xiong (Beijing Normal University) and Zhengchao Han (Rutgers) we proved dimension-sensitive results and our work showed connection to other problems.

  A new algorithmic approach for computation of S_n Kazhdan-Lusztig polynomials, through their restriction to lower rank Bruhat intervals, was recently presented by Geordie Williamson and DeepMind collaborators.
  In a joint work with Chuijia Wang we fit this hypercube decomposition into a general framework of a parabolic recursion for Weyl group Kazhdan-Lusztig polynomials. We also show how the positivity phenomena of Dyer-Lehrer and Grojnowski-Haiman come into play in such decompositions.
  Staying in type A, I will explain how the new approach naturally manifests through the KLR categorification of (dual) PBW and canonical bases.

  Kazhdan-Lusztig (KL) polynomials play a central role in several areas of mathematics. In the 80's, Dyer and Lusztig, independently, formulated the Combinatorial Invariance Conjecture (CIC), which states that the KL polynomial associated with two elements u and v only depends on the poset of elements between u and v in Bruhat order. With the help of certain machine learning models, recently Blundell, Buesing, Davies, Velickovic, and Williamson discovered a formula for the KL polynomials of a Coxeter group W of type A, and stated a conjecture that implies the CIC for W (see [Towards combinatorial invariance for Kazhdan-Lusztig polynomials, Representation Theory (2022)] and [Advancing mathematics by guiding human intuition with AI, Nature 600 (2021)]. In this talk, I will present a formula and a conjecture about R-polynomials of W. The advantage in considering R-polynomials rather than KL polynomials is that the corresponding formula and conjecture are less intricate and have a dual counterpart. Our conjecture also implies the CIC.
  This is based on joint work with F. Brenti.

In this talk we present some recent results regarding compactly supported solutions of the 2D steady Euler equations. Under some assumptions on the support of the solution, we prove that the streamlines of the flow are circular. The proof uses that the corresponding stream function solves an elliptic semilinear problem -Delta phi = f(phi) with abla phi=0 at the boundary. One of the main difficulties in our study is that f can fail to be Lipschitz continuous near the boundary values. If f(phi) vanishes at the boundary values we can apply a local symmetry result of F. Brock to conclude. Otherwise, we are able to use the moving plane scheme to show symmetry, despite the possible lack of regularity of f. We think that such result is interesting in its own right and will be stated and proved also for higher dimensions. The proof requires the study of maximum principles, Hopf lemma and Serrin corner lemma for elliptic linear operators with singular coefficients.

  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.)