Alpha-induction is a tensor functor producing a new fusion category from a modular tensor category and a Q-system. This can be formulated in terms of quantum 6j-symbols and braiding and gives alpha-induced bi-unitary connections. Last year, we showed that locality of the Q-system implies flatness of the alpha-induced connections. We now prove that the converse also holds.
The Operator Algebra Seminar schedule is here: https://sites.google.com/view/oastorvergata/home-page?authuser=0

Given the moduli space of hyperplanes in projective space, V. Alexeev constructed a family of compactifications parametrizing stable hyperplane arrangements with respect to given weights. In particular, there is a toric compactification that generalizes the Losev–Manin compactification for the moduli of points on the line. We study the first natural wall crossing that modifies this compactification into a non-toric one by varying the weights. In particular, we prove that in dimensions two the wall crossing corresponds to blowing up at the identity of the generalized Losev–Manin space. As an application, we show that any Q-factorialization of this blow-up is not a Mori dream space for a sufficiently high number of lines. This is joint work in progress with Patricio Gallardo.

Paul de Faget de Casteljau (1930-2022) was a highly gifted mathematician who worked in industry and made fundamental mathematical contributions. In this talk, I will focus on one central contribution that de Casteljau developed soon after he started working for Citroen in 1958. It is the algorithm of de Casteljau, a simple construction of polynomial curves from control points by iterated linear interpolation. This algorithm is not only very simple, very useful, and very well known, but it also has a great number of properties and generalizations that make it a fundamental and unifying theoretical tool for Geometric Design. As a tribute to an outstanding pioneer in CAGD, I will recall widely and little known generalizations and properties of this algorithm to remind us of its beauty, versatility and importance as THE algorithm and backbone of Computer Aided Geometric Design (CAGD).

Motivated by the problem of determining whether biexactness, the (AO)-property and von Neumann solidity are equivalent properties for a discrete countable group, I will discuss few recent results regarding analytic properties of SL(3,Z), related to hyperbolicity. I will focus on the role of measurable dynamics and proximality arguments in this context. Partly based on joint works with F. Radulescu and T. Amrutam.
Some references:
https://arxiv.org/abs/2305.16277
https://arxiv.org/abs/2111.13885
https://arxiv.org/abs/2403.05948

The log etale topology is a natural analogue of the etale topology for log schemes. Unfortunately, very few things satisfy log etale descent -- not even vector bundles or the structure sheaf. We introduce a new rhizomic topology that sits in between the usual and log etale topologies and show most things do satisfy rhizomic descent! As a case study, we look at tropical abelian varieties and give some exotic examples.

  Folded galleries, as introduced by Peter Littelmann in the 1990s, are combinatorial objects related to certain (subsets of) elements of Coxeter groups. They have shown to have versatile applications in algebra and geometry, making them an object of interest for current research. In this talk we will retrace the roots of their invention 192 years back in history, contemplate colorful illustrations of examples, and discover open questions for future applications.
  N.B.: the talk will be colloquium-style and aimed at a wide audience: no prerequisite of deep algebraic nor group theoretic knowledge is required.

  Given an arbitrary Coxeter system (W,S) and a nonnegative integer m, the m-Shi arrangement of (W,S) is a subarrangement of the Coxeter hyperplane arrangement of (W,S). The classical Shi arrangement (m=0) was introduced in the case of affine Weyl groups by Shi to study Kazhdan-Lusztig cells for W. As two key results, Shi showed that each region of the Shi arrangement contains exactly one element of minimal length in W and that the union of their inverses form a convex subset of the Coxeter complex. The set of m-low elements in W was introduced to study the word problem of the corresponding Artin-Tits (braid) group and they turn out to produce automata to study the combinatorics of reduced words in W.
  In this talk, I will discuss how to Shi's results extend to any Coxeter system and show that the minimal elements in each Shi region are in fact the m-low elements. This talk is based on joint work with Matthew Dyer, Susanna Fishel and Alice Mark.

According to physics literature, topologically ordered gapped ground states of 2-dimensional spin systems can be described by a topological quantum field theory. Many examples arise from microscopic models with local commuting projector Hamiltonians, namely Levin-Wen models.
In this talk, I will describe the general framework for classifying infinite volume gapped ground states (by Naaijkens, Ogata, et.al.) in the simple context of abelian Levin-Wen models. This framework is heavily inspired by the DHR analysis in relativistic quantum field theory. It applies to the doubled semion model whose anyon theory is a braided fusion category equivalent to the representation category of the twisted Drinfeld double of Z_2.
Based on joint work with Alex Bols and Alvin Moon, https://arxiv.org/abs/2306.13762.
Seminar schedule here: https://sites.google.com/view/oastorvergata/home-page.

In this presentation, we tackle the problem of reconstructing the optimal transport map $T$ between two absolutely continuous measures $mu, u in mathcal{P}(mathbb{R}^n)$, and for this approximation we employ flows generated by linear-control systems in $mathbb{R}^n$. We first show that, under suitable assumptions on the measures $mu, u$ and on the controlled vector fields, the optimal transport map is contained in the $C^0_c$-closure of the flows generable by the system. In the case that discrete approximations $mu_N, u_N$ of the measures $mu, u$ are available, we use a discrete optimal transport plan to set up an optimal control problem. With a $Gamma$-convergence argument, we prove that its solutions corresponds to flows that provide approximations of the optimal transport map $T$. Finally, in virtue of the Pontryagin Maximum Principle, we propose an iterative numerical scheme for the resolution of the optimal control problem, resulting in an algorithm for the practical computation of approximations of the optimal transport map. This approach can be interpreted as the construction of a ''Normalizing Flow'' by means of a Residual Neural Network (ResNet). Based on a joint work with Sara Farinelli. [1] A. Scagliotti, S. Farinelli. Normalizing flows as approximations of optimal transport maps via linear-control neural ODEs. arXiv preprint, 2023.
NB:This talk is part of the activity of the MIUR Excellence Department Project MATH@TOV CUP E83C23000330006

Blow-ups are fundamental tools in algebraic geometry, and there are several results (e.g the famous Castelnuovo's theorem) that can be used to determine when a variety is obtained as a blow-up of a smooth variety along a smooth center. Weighted blow-ups play a similar role for stacks. In this talk I will present a criterion for finding out if a smooth DM stack is a weighted blow-up. I will apply this result for showing that certain alternative compactifications of moduli of marked elliptic curves are obtained via weighted blow-ups (and blow-downs). This in turn will prove to be useful in order to compute certain invariants, like Chow rings or Brauer groups. First part of this talk is a joint work with Arena, Inchiostro, Mathur, Obinna and Pernice; the second part of this talk is a joint work with L. Battistella.