10/03/23  Seminario  14:30  15:30  1201 Dal Passo  Alexander PÜTZ  RuhrUniversität  Bochum 
Algebra & Representation Theory Seminar (ARTS)
"Desingularizations of Quiver Grassmannians for the Equioriented Cycle Quiver"
Quiver Grassmannians are projective varieties parametrizing subrepresentations of quiver representations. Originating in the geometric study of quiver representations and in cluster algebra theory, they have been applied extensively in recent years in a Lietheoretic context, namely as a fruitful source for degenerations of (affine) flag varieties. This approach allows for an application of homological methods from the representation theory of quivers to the study of such degenerate structures. The resulting varieties being typically singular, a construction of natural desingularizations is very desirable.
We construct torus equivariant desingularizations of quiver Grassmannians for arbitrary nilpotent representations of an equioriented cycle quiver. This applies to the computation of their torus equivariant cohomology.

07/03/23  Seminario  14:30  15:30  1201 Dal Passo  Nicolò Forcillo  Università di Roma  Seminario di Equazioni Differenziali
The onephase Stefan problem: perturbative techniques for the free boundary regularity
In Stefantype problems, free boundaries may not regularize instantaneously. In particular, there exist examples in which Lipschitz free boundaries preserve corners. Nevertheless, in the twophase Stefan problem, I. Athanasopoulos, L. Caffarelli, and S. Salsa showed that Lipschitz free boundaries in spacetime become smooth under a nondegeneracy condition, as well as sufficiently "flat" ones. Their techniques are based on the original work of Caffarelli in the elliptic case.
In this talk, we present a more recent approach to investigate the regularity of flat free boundaries for the onephase Stefan problem. Specifically, it relies on perturbation arguments leading to a linearization of the problem, in the spirit of the elliptic counterpart already developed by D. De Silva. This talk is based on a joint work with D. De Silva and O. Savin. 
03/03/23  Seminario  14:30  15:30  1201 Dal Passo  Kirill ZAYNULLIN  University of Ottawa 
Algebra & Representation Theory Seminar (ARTS)
"Oriented cohomology of a linear algebraic group vs. localization in 2monoidal categories"
The Chow ring CH(G) of a split semisimple 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 (BorelMoore homology) functor A(G) using the localization techniques of KostantKumar and the techniques of 2monoidal categories: we show that the natural Hopfalgebra structure on A(G) can be lifted to a 'biHopf' structure on the Tequivariant 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 BrionKnutsonTymoczko 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. 
01/03/23  Colloquium  15:00  16:00  1201 Dal Passo  Felix Otto  MaxPlanckInstitut, Lipsia 
COLLOQUIUM
Optimal matching, optimal transportation, and its regularity theory
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 ddimensional space, the problem depends crucially on dimension, with the twodimensional case being critical [AjtaiKomlosTusnady].
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 MongeAmpere equation as the EulerLagrange 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 MongeAmpere 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).

14/02/23  Seminario  16:00  17:00  1201 Dal Passo  Lei Zhang  University of Florida  Seminario di Equazioni Differenziali
Asymptotic behavior of solutions to the Yamabe equation in low
dimensions
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
dimensionsensitive results and our work showed connection to other
problems.

10/02/23  Seminario  16:00  17:00  1201 Dal Passo  Mario MARIETTI  Università Politecnica delle Marche 
Algebra & Representation Theory Seminar (ARTS)
"Towards Combinatorial Invariance: KahdanLusztig Rpolynomials"
KazhdanLusztig (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 KazhdanLusztig 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 Rpolynomials of W. The advantage in considering Rpolynomials 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.

10/02/23  Seminario  14:30  15:30  1201 Dal Passo  Maxim GUREVICH  Technion  Israel Institute of Technology 
Algebra & Representation Theory Seminar (ARTS)
"Positive decompositions for KazhdanLusztig polynomials"
A new algorithmic approach for computation of S_{n} KazhdanLusztig 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 KazhdanLusztig polynomials. We also show how the positivity phenomena of DyerLehrer and GrojnowskiHaiman 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. 
07/02/23  Seminario  16:00  17:00  1201 Dal Passo  David Ruiz  Universidad de Granada  Seminario di Equazioni Differenziali
Symmetry results for compactly supported solutions of the 2D steady Euler equations
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.

03/02/23  Seminario  16:00  17:00  1201 Dal Passo  Francesco ESPOSITO  Università di Padova 
Algebra & Representation Theory Seminar (ARTS)
"Cohomology of quiver Grassmannians and Motzkin combinatorics"
Quiver Grassmannians are projective algebraic varieties generalizing ordinary Grassmannians and flag varieties. The cohomology of quiver Grassmannians of particular type has applications 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 LaniniStrickland and FangReineke.
In this talk, I relate on joint work with Cerulli IrelliFangFourier and Cerulli IrelliMarietti, 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 FangReineke.

03/02/23  Seminario  14:30  15:30  1201 Dal Passo  Lorenzo VECCHI  Università di Bologna 
Algebra & Representation Theory Seminar (ARTS)
"Categorical valuative invariants of matroids"
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. Information 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 nontrivial 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 subdivisions 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.
