The tensor functor called alpha-induction arises from a Q-system in a braided unitary fusion category. In the operator algebraic language, it gives extensions of endomorphism of N to M arising from a subfactor N subset M of finite index and finite depth giving a braided fusion category of endomorpshisms of N. We study this alpha-induction for bi-unitary connections, which give a characterization of finite-dimensional nondegenerate commuting squares. We show that the resulting alpha-induced bi-unitary connections are flat if we have a local Q-system.

alpha-induction is known to be a construction of a modular invariant full Q-system from a chiral Q-system of a conformal net. In this talk, I would like to introduce the equivariant version of this procedure. Indeed, we need a new construction of tensor categories, the neutral double construction, introduced by the speaker. Moreover, I would like to explain the relationship between the neutral double construction and an equivariant version of the center construction, which is also introduced by the speaker.

  The celebrated Cauchy identity rewrites a certain infinite product as a sum of products of Schur functions. The identity has a vast number of applications and interpretations; in particular, one can understand the infinite product as the character of polynomial functions on the space of square matrices and the products of Schur functions as characters of tensor products of irreducible gl(n) modules.
  The classical Cauchy identity has (at least) three natural generalizations: a nonsymmetric version, a q-version and their mixture. The representation theory of the nonsymmetric Cauchy is governed by the Borel subalgebra, the q-Cauchy identity is controlled by the representations of the current algebras and the nonsymmetric q-Cauchy identity has to do with the modules over the Iwahori algebra. We will discuss all these identities and explain the relevant representation theory.
  Based on joint works with Anton Khoroshkin, Ievgen Makedonskyi and Daniel Orr.

  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 Lie-theoretic 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.

In Stefan-type problems, free boundaries may not regularize instantaneously. In particular, there exist examples in which Lipschitz free boundaries preserve corners. Nevertheless, in the two-phase Stefan problem, I. Athanasopoulos, L. Caffarelli, and S. Salsa showed that Lipschitz free boundaries in space-time 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 one-phase 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.

  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.

  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.

  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.