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.[-]

Over the last couple of years, it has become evident that matrix-valued semicircular elements establish strong links between free probability theory and noncommutative algebra. Another surprising connection of this kind was found in a recently finished project with Roland Speicher. We have shown that the Fuglede-Kadison determinant of an arbitrary matrix-valued semicircular element is essentially given by the capacity of its associated covariance map. In addition, we have improved a lower bound by Garg, Gurvits, Oliveira, and Widgerson on this capacity, by making it dimension-independent. Besides analytic tools from operator-valued free probability, these are the crucial ingredients in some novel algorithmic solution to the noncommutative Edmonds' problem which we described in collaboration with Johannes Hoffmann. In my talk, I will present our work and provide the background on free probability and noncommutative algebra required for this purpose.[-]

We report on the development of localization methods useful for quadratic enumerative invariants, replacing the classical Gm-action with an action by the normalizer of the diagonal torus in SL2.
We discuss applications to quadratic counts of twisted cubics in hypersurfaces and complete intersections (joint with Sabrina Pauli) as well as work by Anneloes Vierever, and our joint work with Viergever on quadratic DT invariants for Hilbert schemes of points on P3 and on (P1)3.[-]

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.[-]

Cellular A1-homology is a new homology theory for smooth algebraic varieties over a perfect field, which is often entirely computable and is expected to give the correct motivic analogue of Poincaré duality for smooth manifolds in classical topology. I will introduce cellular A1-homology, describe the precise conjectures about cellular A1-homology of smooth projective varieties and discuss how they can be verified for smooth projective rational surfaces. The talk is based on joint work with Fabien Morel.[-]

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.[-]

Given two algebraic ODEs, is there a differential-algebraic relation between generic tuples of their solutions? In recent work with Freitag and Moosa, we produce a bound on the length of tuples one must look at to f ind a relation. Our proof relies on two ingredients. The first is differential Galois theory, combined with the recent proof by Freitag and Moosa of the Borovik-Cherlin conjecture in algebraically closed fields. The second is some general model theory result which allows us to factor any relation through some minimal ODE. I will give a precise statement of our result and sketch the proof. I will also explain why our bound is tight.[-]

The mini-course is structured into three parts. In the first part, we provide a general overview of the tools available in the summation package Sigma, with a particular focus on parameterized telescoping (which includes Zeilberger's creative telescoping as a special case) and recurrence solving for the class of indefinite nested sums defined over nested products. The second part delves into the core concepts of the underlying difference ring theory, offering detailed insights into the algorithmic framework. Special attention is given to the representation of indefinite nested sums and products within the difference ring setting. As a bonus, we obtain a toolbox that facilitates the construction of summation objects whose sequences are algebraically independent of one another. In the third part, we demonstrate how this summation toolbox can be applied to tackle complex problems arising in enumerative combinatorics, number theory, and elementary particle physics.[-]