An important problem in enumerative geometry is counting rational curves that interpolate a configuration of points on an algebraic surface. Over the complex numbers, the answer does not depend on the configuration of points and is called the Gromov-Witten invariant. In contrast, over the real numbers, this invariance fails. To recover it, Welschinger invented an “sign” rule that gives rise to Welschinger invariants. Recently, Kass, Levine, Solomon, and Wickelgren constructed an invariant over an (almost) arbitrary field. The small “inconvenience” is that these latter invariants are no longer numbers, but quadratic forms. In a current work with Erwan Brugallé and Kirsten Wickelgren, we establish direct relationships between these different types of invariants. In my talk, I want to give an introduction to this topic.

Cubic hypersurfaces over the rational numbers often contain infinitely many rational points. In this situation, the asymptotic behavior of the number of rational points of bounded height is predicted by conjectures of Manin and Peyre. After reviewing previous results, we discuss the chordal cubic fourfold, which is the secant variety of the Veronese surface. Since it is isomorphic to the symmetric square of the projective plane, a result of W. M. Schmidt for quadratic points on the projective plane can be applied. We prove that this is compatible with the conjectures of Manin and Peyre once a thin subset with exceptionally many rational points is excluded from the count.

We prove a conjecture of Cherednik describing the Betti numbers of compactified Jacobians of unibranch planar curves via superpolynomials of algebraic knots. The methods of the proof use the theory of orbital integrals and affine Springer theory. No prior knowledge about any of these will be assumed.

Tropical geometry studies a piecewise linear, combinatorial shadow of degenerations of algebraic varieties. In many cases, usual algebro-geometric objects such as divisors or line bundles on curves have tropical analogues that are closely tied to their classical counterparts. For instance, the theory of divisors and line bundles on metric graphs has been crucial in advances in Brill–Noether theory and the birational geometry of moduli spaces. In this talk, I will present an elementary theory of tropical principal bundles on metric graphs, generalizing the case of tropical line bundles to bundles with arbitrary reductive structure group. Our approach is based on tropical matrix groups arising from the root datum of the corresponding reductive group, and leads to an appealing geometric picture: tropical principal bundles can be presented as pushforwards of line bundles along covers equipped with symmetry data from the Weyl group. Building on Fratila's description of the moduli space of semistable principal bundles on an elliptic curve, we describe a tropicalization procedure for semistable principal bundles on a Tate curve. More precisely, the moduli space of semistable principal bundles on a Tate curve is isomorphic to a natural component of the tropical moduli space of principal bundles on its dual metric graph. This is based on ongoing work with Andreas Gross, Martin Ulirsch, and Dmitry Zakharov.

We will survey different approaches to the moduli theory of foliations before presenting a new KSB approach to constructing moduli spaces of foliations, which yields a proper moduli space for foliations on surfaces. We will also discuss different applications and open problems that arise from this construction. This is joint work with C. Spicer and R. Svaldi.

Odaka proposed the K-moduli conjecture in 2010, predicting the existence of a moduli space of K-polystable objects with an ample CM line bundle. While this conjecture has been solved in the Fano case, it remains open in general. Recent developments of Fine, Dervan-Sektnan and Ortu have highlighted the relevance of the existence of cscK metrics and K-stability for \((X,\epsilon A+L)\) for sufficiently small \(\epsilon\), where \(f\colon (X,A)\to (B,L)\) is a fibration. According to their works, such K-stability is closely related to some K-stability of fibers and the bases. Especially in the Calabi-Yau fibration over curve case, uniform K-stability in this context (uniform adiabatic K-stability) coincides with the log twisted K-stability on the base. In this talk, we will regard the base curve as a quasimap and introduce the notion of K-moduli of quasimaps. By using this framework, we address the K-moduli conjecture for Calabi-Yau fibrations over curves whose generic fibers are either Abelian varieties or HyperKahler manifolds. This is a joint work arXiv:2504.21519 with Kenta Hashizume.

For a real quadratic field K and positive integer r, we prove an asymptotic formula for the number of rational integers of bounded absolute value that can be written as a sum of r units of the ring of integers of K. This is joint work with M. Widmer and V. Ziegler and answers partially a question posed by M. Jarden and W. Narkiewicz.

I will report on work in progress with M. Groechenig and P. Ziegler where we aim to express certain enumerative invariants, so called refined BPS-invariants, in terms of integrals on p-adic analytic manifolds naturally associated with the enumerative problem. In the case of 1-dimensional sheaves on a K3 surface, our work implies that these invariants are independent of the Euler-characteristic, confirming a conjecture of Toda.

Let K be a number field. The Grunwald problem for a finite group (scheme) G/K asks what is the closure of the image of $H^1(K,G) → \prod_{v \in M_K} H^1(K_v,G)$. For a general G, there is a Brauer—Manin obstruction to the problem, and this is conjectured to be the only one. In 2017, Harpaz and Wittenberg introduced a technique that managed to give a positive answer (BMO is the only one) for supersolvable groups. I will present a new fibration theorem over quasi-trivial tori that, combined with the approach of Harpaz and Wittenberg, gives a positive answer for all solvable groups.

I will present a randomised algorithm to compute the local zeta function of a fixed smooth, projective surface over the rationals, at any large prime p of good reduction. The runtime of the algorithm is polynomial in log p, answering a question of Couveignes and Edixhoven. The main ingredient is to explicitly compute cocycles associated to a Lefschetz pencil on the surface. This is based on joint work with Nitin Saxena.