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.

The Bogomolov–Tian–Todorov (BTT) theorem states that smooth Calabi–Yau (CY) varieties over the complex numbers have unobstructed deformation spaces. While this fails over fields of positive characteristic, recent results of Brantner–Taelman and Petrov show that the BTT theorem does hold in positive and mixed characteristics for ordinary CY varieties without crystalline torsion, implying that such varieties are liftable. In this talk, we study the problem of liftability for singular CY surfaces over the ring of Witt vectors $W(k)$. In earlier joint work with Brivio–Kawakami–Witaszek, we proved that normal globally $F$-split CY surfaces admit an equisingular lifting over $W(k)$. Together with Q. Posva, building on the explicit birational geometry and the deformation theory of log CY surfaces, we further generalised this result to the case of globally $F$-split surfaces with semi-log canonical (in particular, not necessarily normal) singularities. In the first part, I will recall the setup and definitions in positive characteristic algebraic geometry, giving an overview of the proofs of some of the previous results. In the second part, I will focus on joint work with Posva and highlight the difficulties that arise in the non-normal case.

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.

Dynamical Galois groups are profinite groups that are constructed via iterations of rational functions. They are intimately connected with, and in a way they are a wild generalization of, Galois representations on Tate modules of elliptic curves. The problem of determining which rational functions yield abelian dynamical Galois groups has attracted quite some attention in recent years; in this talk I will survey on what is known about this problem, and explain how to solve certain instances of it over number fields and over function fields. This is based on joint works with P. Ingram, A. Ostafe, C. Pagano and U. Zannier.

In analytic number theory one needs to bound sums of oscillating functions, such as the Mobius function. Over function fields these are trace functions of sheaves, so their sums are controlled, in view of the Grothendieck--Lefschetz trace formula, by the cohomology groups of the sheaves. In joint work in progress with Will Sawin, building on arguments from works of Bombieri, Adolphson--Sperber, Deligne--Illusie, Katz, and a study of singularities, we obtain bounds on the cohomology of tame sheaves which leads to applications to the distribution of irreducible polynomials over finite fields in certain thin sets.

Exponential sums over finite fields are essential ingredients in the solution of many arithmetic problems. Their study often relies on algebraic geometry, and especially on Deligne's Riemann Hypothesis over finite fields, which reveals deep structural features of these sums. In turn, one can exploit these to construct some remarkable combinatorial objects. The talk will provide a survey of these various aspects. (Joint works with A. Forey, J. Fresán and Y. Wigderson)

È ben noto grazie a un lavoro di Ein del 1988 che le intersezioni complete molto generali di multigrado $(d_1,…,d_c)$ nello spazio proiettivo di dimensione n non contengono curve razionali non appena $d_1+…+d_c > 2n-c-1$. Questo risultato è stato reso ottimale ed esteso nel caso delle ipersuperfici grazie a un metodo introdotto da Voisin che ha ispirato lavori ulteriori di Clemens, Ran e miei. Nonostante lavori più recenti di Coskun, Riedl e Yang sempre nel caso delle ipersuperfici, il risultato di Ein è rimasto l’unico disponibile per intersezioni complete di codimensione arbitraria. Nel seminario parlerò di un lavoro in collaborazione con Francesco Bastianelli in cui estendiamo il lavoro di Ein, mescolando l’approccio di Voisin con quello di Coskun, Riedl e Yang.

Motivated by the Green-Griffiths and Lang-Vojta conjectures, it is expected that the algebraic exceptional set of a log-surface $(X,B)$ of log-general type - which parametrizes rational curves on $X$ meeting $B$ in at most two points - is finite. In this talk, I will discuss recent results on this set for the cases where $X$ is the projective plane or a Hirzebruch surface, and $B$ is a curve with three irreducible components. This talk is based on [arXiv.2507.13280] and work in progress.

A classical result due to Clebsch from the mid-nineteenth century confirms that every complex space sextic curve (given as an intersection of a quadric and a cubic surface in projective 3-space) has exactly 120 tritangent planes. In this talk we will show how to use combinatorial methods arising from tropical geometry to revisit this classical problem and perform the analogous count over the reals and extensions thereof. This is joint work with Yoav Len, Hannah Markwig and Yue Ren (arXiv:2512.24277).

I will study the Kodaira dimension of $A_g$, i.e., the moduli space of principally polarized Abelian $g$-folds, and of $X_g^n$, i.e., the space of Kuga $n$-fold varieties on these spaces. I will then use the results on the slope of Siegel modular forms to determine the Kodaira dimension for all Kuga varieties and $A_g$ $(g \neq 6)$. I will report the results for the case $g=6$. If I have time, I will report the results on the moving slope of $A_g$. These results were obtained in collaboration with: Dittmann, Scheithauer, Poon, Sankaran, Grushevsky, Ibukiyama, and Mondello.

We give necessary and sufficient conditions for the connectedness of some degeneracy loci. In the special case of Ulrich bundles, these degeneracy loci are called Ulrich subvarieties and we will see that they are always connected with a few exceptions. Joint work with V. Buttinelli and R. Vacca.

Cercherò di dare un’idea di alcuni dei risultati contenuti nel preprint ArXiv 2508.05105.

I will report on a joint work with S. Coughlan, R. Pardini and S. Rollenske. The investigation of (minimal) surfaces of general type with low invariants and their moduli spaces started with the work of Castelnuovo and Enriques and during the last decades of the 20th century many authors continued studying these surfaces. Nowadays Gieseker's moduli space of canonical models of surfaces of general type with K^2 and χ fixed is known to admit a modular compactification, namely the KSBA moduli space, obtained considering stable surfaces. The structure of such moduli space is not completely known and studying stable surfaces with low invariants is a starting point to see concrete examples and studying its properties. The aim of this talk is to give a description of the KSBA moduli space of stable surfaces with K^2=1 and χ=3, showing different ways to construct boundary components. After an overview of the know components, I will focus on the case of 2-Gorenstein surfaces, (with particular attention to the surfaces obtained by gluing two irreducible surfaces) and to the case of normal surfaces having rational singularities.

I will report on a joint work in progress with I. Biswas, E. Colombo and A. Ghigi in which we describe a canonical projective structure on every etale double cover of a curve $C$ of genus $g>6$. This projective structure is the restriction to the second infinitesimal neighborhood of the diagonal in $C\times C$ of the second fundamental form of the Prym map. It gives a section of the space of projective structures on $\mathbb{R}_g$ and the $(0,1)$-component of the differential of this section is proven to be the pullback via the Prym map of the Kaehler form on $A_{g-1}$. This generalises a previous result obtained in collaboration with Biswas, Colombo, and Pirola in the case of $M_g$, showing the existence of a canonical projective structure on every curve of genus $g>3$, obtained by the second fundamental form of the Torelli map.

The YTD conjecture predicts that the existence of "canonical" Kähler metrics on a polarized projective manifold is governed by a purely algebro-geometric notion known as K-stability. Recently, solutions to (variants of) this conjecture have been proposed, and the purpose of this talk is to review this recent progress.

In Kähler geometry, the Calabi-Yau theorem establishes the existence and uniqueness of Kähler metrics with prescribed volume form. In this talk, I will present a non-Archimedean version of this theorem for general Kähler manifolds, extending a theorem of Boucksom-Jonsson from the projective setting. This is joint work with David Witt Nyström.

I will give a gentle introduction with several examples to the following topic: Yau-Tian-Donaldson conjecture states that a polarised manifold $(X,L)$ admits a cscK metric in $c_1(L)$ if and only if $(X,L)$ is K-polystable. Matsushima proved in 1957 that existence of such cscK metric implies reductively of the automorphism group of $X$. In a positive direction on the YTD conjecture, we show that K-polystability implies reductively of the automorphism group of $X$, for smooth Fano threefolds. This is joint work with Paolo Cascini and Ivan Cheltsov.