A simple group is pseudofinite if and only if it is isomorphic to a (twisted) Chevalley group over a pseudofinite field. This celebrated result mostly follows from the work of Wilson in 1995 and heavily relies on the classification of finite simple groups (CFSG). It easily follows that any simple pseudofinite group $G$ is finite-dimensional. In particular, if $\operatorname{dim}(G)=3$ then $G$ is isomorphic to $\operatorname{PSL}(2, F)$ for some pseudofinite field $F$. In this talk, we describe the structures of finite-dimensional pseudofinite groups with dimension $<4$, without using CFSG. In the case $\operatorname{dim}(G)=3$ we show that either $G$ is soluble-by-finite or has a finite normal subgroup $Z$ so that $G / Z$ is a finite extension of $\operatorname{PSL}(2, F)$. This in particular implies that the classification $G \cong \operatorname{PSL}(2, F)$ from the above does not require CFSG. This is joint work with Frank Wagner.[-]

We give an arithmetic version of Tao's algebraic regularity lemma (which was itself an improved Szemerédi regularity lemma for graphs uniformly definable in finite fields). In the arithmetic regime the objects of study are pairs $(G, D)$ where $G$ is a group and $D$ an arbitrary subset, all uniformly definable in finite fields. We obtain optimal results, namely that the algebraic regularity lemma holds for the associated bipartite graph $(G, G, E)$ where $E(x, y)$ is $x y^{-1} \in D$, witnessed by a the decomposition of $G$ into cosets of a uniformly definable small index normal subgroup $H$ of $G$.[-]

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.[-]

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.[-]