Here, we provide a generic version of quantum Wielandt's inequality, which gives an optimal upper bound on the minimal length such that products of that length of n-dimensional matrices in a generating system span the whole matrix algebra with probability one. This length generically is of order $\Theta(\log n)$, as opposed to the general case, in which the best bound is $O(n^2)$.  This has implications for the primitivity index of random quantum channels, matrix product states and projected entangled pair states. Some results can be extended to Lie algebras. Joint work with Yifan Jia.[-]

This talk will focus on the fluctuations of a linear spectral statistic around its mean for $P\left(W_N, D_N\right)$ where $P$ is a polynomial, $W_N$ a Wigner matrix and $D_N$ a deterministic diagonal matrix. I will first consider the case when $P\left(W_N,D_N\right)=W_N+D_N$, based on a joint work with M. Février (U. Paris-Saclay). In the general case of $P$ a selfadjoint noncommutative polynomial, I will present results for the special case of the Stieltjes transform, based on a joint work with S. Belinschi (CNRS, U. Toulouse), M. Capitaine (CNRS,U. Toulouse) and M. Février (U. Paris-Saclay).[-]

We consider independent Hermitian heavy-tailed random matrices. Our model includes the Lévy matrices as well as sparse random matrices with O(1) non-zero entries per row. By representing these matrices as weighted graphs, we derive a large deviation principle for key macroscopic observables. Specifically, we focus on the empirical distribution of eigenvalues, the joint neighborhood distribution, and the joint traffic distribution. As an application, we define a notion of microstates entropy for traffic distribution which is additive under free traffic convolution.[-]

Quadratic enumerative geometry extends classical enumerative geometry. In this enriched setting, the answers to enumerative questions are classes of quadratic forms and live in the Grothendieck-Witt ring GW(k) of quadratic forms. In the talk, we will compute some quadratic enumerative invariants (this can be done, for example, using Marc Levine's localization methods), for example, the quadratic count of lines on a smooth cubic surface.
We will then study the geometric significance of this count: Each line on a smooth cubic surface contributes an element of GW(k) to the total quadratic count. We recall a geometric interpretation of this contribution by Kass-Wickelgren, which is intrinsic to the line and generalizes Segre's classification of real lines on a smooth cubic surface. Finally, we explain how to generalize this to lines of hypersurfaces of degree 2n − 1 in Pn+1. The latter is a joint work with Felipe Espreafico and Stephen McKean.[-]

In my talk, I will discuss an application of the theory of motives to transcendence theory, concentrating on the formal aspects. The Period Conjecture predicts that all relations between period numbers are induced by properties of the category of motives. It is a theorem for motives of points and curves, but wide open in general. The Period Conjecture also implies fullness of the Hodge-de Rham realization on Nori motives. If time permits, I will discuss how this generalizes (conjecturally) to triangulated motives and thus to motivic cohomology.[-]

The Grothendieck-Knudsen moduli space of stable rational curves n markings is arguably one of the simplest moduli spaces: it is a smooth projective variety that can be described explicitly as a blow-up of projective space, with strata corresponding to nodal curves similar to the torus invariant strata of a toric variety. Conjecturally, its Mori cone of curves is generated by strata, but this is known only for n up to 7. In contrast, the cones of effective divisors are not f initely generated, in all characteristics, when n is at least 10. After a general introduction to these topics, I will discuss what we call elliptic pairs and LangTrotter polygons, relating the question of finite generation of effective cones of blow-ups of certain toric surfaces to the arithmetic of elliptic curves. These lectures are based on joint work with Antonio Laface, Jenia Tevelev and Luca Ugaglia.[-]

The Grothendieck-Knudsen moduli space of stable rational curves n markings is arguably one of the simplest moduli spaces: it is a smooth projective variety that can be described explicitly as a blow-up of projective space, with strata corresponding to nodal curves similar to the torus invariant strata of a toric variety. Conjecturally, its Mori cone of curves is generated by strata, but this is known only for n up to 7. In contrast, the cones of effective divisors are not f initely generated, in all characteristics, when n is at least 10. After a general introduction to these topics, I will discuss what we call elliptic pairs and LangTrotter polygons, relating the question of finite generation of effective cones of blow-ups of certain toric surfaces to the arithmetic of elliptic curves. These lectures are based on joint work with Antonio Laface, Jenia Tevelev and Luca Ugaglia.[-]

An element g of an abstract group G is a distortion element if there exists a finite family S in G such that g belongs to the subgroup generated by S and the wordlength of gn (w.r.t. S) grows sublinearly in n. In this talk, we will be interested in the distortion elements of the group of Cr orientation-preserving diffeomorphisms of the closed interval, for different values of r. In particular, we will present some natural obstructions to distortion (such that the presence of hyperbolic fixed points in C1 regularity and the positivity of the so-called asymptotic distortion in C2 regularity (and higher)), and we will wonder whether they are the only ones.[-]

The KSBA moduli space of stable pairs ($\mathrm{X}, \mathrm{B}$), introduced by Kollár-Shepherd-Barron, and Alexeev, is a natural generalization of the moduli space of stable curves for higher dimensional varieties. This moduli space is described concretely only in a handful of situations. For instance, if $\mathrm{X}$ is a toric variety and $\mathrm{B}=\mathrm{D}+\varepsilon\mathrm{C}_{}^{}$, where D is the toric boundary divisor and $\mathrm{C}$ is an ample divisor, it is shown by Alexeev that the KSBA moduli space is a toric variety. More generally,for stable pairs of the form$\left( \mathrm{{X,D}+\varepsilon\mathrm{C}} \right)$ with $\left( \mathrm{X,D} \right)$ a log Calabi–Yau variety and C an ample divisor, it was conjectured by Hacking–Keel–Yu that the KSBA moduli space is still toric, up to passing to a finite cover. In joint work with Alexeev and Bousseau, we prove this conjecture for all log Calabi-Yau surfaces. This uses tools from the minimal model program, log geometry and mirror symmetry.[-]