Schedule 

In the first lecture, we will present a Hodge theoretic motivation for the introduction of questions related to "anchored" curves. The main theorem can be seen as a logarithmic version of the classical Clemens conjectures.

In the second lecture, we study a specialization argument that allows us to count rational curves on del Pezzo 3-folds. We then explain how the theorem from the first lecture follows mainly from this argument.

In the third lecture, as an application of the main theorem, we will introduce log conifolds transitions and study its Hodge theoretic properties. This is a mechanism of construction non-Kähler threefolds with lots of rational curves but complicated topology. We will finish with some open questions.

Over three talks, we plan to discuss the following topics (not necessarily in this order):

1. The classic counts of lines on a smooth cubic surface and on a general quintic threefold, respectively, via intersection theory; and their enriched Grothendieck--Witt-valued analogous to to Kass--Wickelgren and Sabrina Pauli.

2. Sheldon Katz' count of 2875 lines on a general quintic threefold via specialization.

3. Clemens' finiteness conjecture for rational curves, and its proof in small curve degrees.

We aim to discuss many topics, such as 27 lines of cubic surfaces, special components of Noether-Lefschetz loci and Clemens' conjecture on curves in quintic threefolds. The lectures are  based on the articles: 

On a counterexample to a conjecture of J. Harris for octic surfaces, arXiv:2606.26944 [math.AG]

Computing the lines of a smooth cubic surface Arkiv for Matematik, Vol. 63, Issue 2 (2025), pp. 369-374.

Periods of families of curves in threefolds, Manuscripta Mathematica, Vol. 177, No. 6, (2026). 

Special Components of Noether-Lefschetz loci , Rendiconti del Circolo Matematico di Palermo Series 2 volume 70, pages 861-874 (2021).

Frobenius geometry appears naturally in several areas of mathematics, including singularity theory, Hodge theory, quantum cohomology, and noncommutative geometry. From a categorical point of view, many of these examples can be placed within the broader framework of noncommutative Calabi–Yau geometry.

In this talk I will focus mainly on the case of isolated singularities and their deformations. I will review how Saito's theory of primitive forms brings together Gauss–Manin systems, Hodge-theoretic structures, and Frobenius manifolds, and discuss how chiral and elliptic structures naturally enter this picture. This provides a concrete setting in which one can explore possible chiral enhancements of classical Frobenius and Hodge-theoretic data.

Towards the end of the talk, I will describe a broader program aimed at extending this perspective to noncommutative Calabi–Yau spaces. Categories of matrix factorizations provide a natural bridge between the two settings, suggesting a common framework relating singularities, cyclic and Hodge-theoretic structures, Frobenius geometry, and chiral algebra.

In type IIB string compactifications on a Calabi–Yau threefold, the attractor equations for BPS black holes and the equations for Minkowski flux vacua are built from the same combinations of periods. We study to what extent this defines a map between charges and fluxes for one-parameter families. Fluxes along a single rank-one attractor charge never give a physical vacuum, whereas rank-two attractor points support two families of vacua, non-supersymmetric and supersymmetric, whose string coupling is fixed by the black hole entropy. Near the conifold and large complex structure points both attractors and vacua accumulate, with explicit but different expressions for the fixed complex structure. We then show that every Minkowski vacuum solves a one-parameter deformation of the attractor equation on its flux plane, DzZγ=μZγ, and that the orthogonal complement acts on μ as an antipodal map: flux vacua come in dual pairs at the same point in moduli space, with the same tadpole and inverted hierarchy between the (2,1) and (0,3) flux components.

We study the real bitangents of real algebraic plane curves from two perspectives. First, we show that there exists a signed count of such bitangents that depends only on the real (analytic) topological type of the curve. From this we get that a generic real algebraic curve of even degree $d$ has at least $d(d-2)/2$ real bitangents.

We also explain how to locate the (real) bitangents of a (real) perturbation of a (real) multiple conic in $CP^2$. As main applications 

1. we present a real sextic with 318 real and 6 complex bitangents, and 

2. we carry out asymptotic constructions that, to the best of our knowledge, provide the best lower bound of real bitangents of real algebraic plane curves of a given degree.

This is a joint work with Erwan Brugallé and Thomas Blomme.

The aim of this talk is to present a formulation of the Beilinson-Hodge conjecture for singular varieties in terms of Hanamura's motivic cohomology and Du Bois type singularities. As an application, we obtain several cases of the Beilinson-Hodge conjecture as a consequences of Lefschetz (1,1)-theorem for singular varieties. This is work in progress, joint with E. Y. Chen. 

How can we decide which geometric objects belong in a useful moduli space? Stability conditions offer a way to compare objects, break them into simpler pieces, and track how those pieces change.

Starting with vector bundles on curves, we explain the ingredients of Bridgeland stability and why Calabi-Yau threefolds pose a new challenge. We then discuss two ways forward: building stability from Brill--Noether bounds on curves, and transporting it through symmetries and crepant resolutions.

We conclude with how these constructions help organize moduli and compare Donaldson--Thomas counts, emphasizing what each result makes possible.

 Irreducible symplectic varieties are the natural higher-dimensional analogues of K3 surfaces, and they form one of the basic building blocks of manifolds with trivial first Chern class, alongside abelian and Calabi–Yau varieties. A central expectation, inspired by mirror symmetry and known as the SYZ conjecture (after Strominger–Yau–Zaslow), predicts that any such variety admitting a certain positivity condition on a line bundle should carry a special kind of fibration (a Lagrangian fibration) whose generic fiber is an abelian variety of half the total dimension.

In this talk I will introduce the SYZ conjecture and explain what is known about the geometry and moduli of Lagrangian fibrations, focusing on results of Kamenova–Verbitsky and Matsushita in the smooth setting. I will then discuss joint work with Claudio Onorati aimed at extending these results to the singular setting.

In this talk I will survey the state of the art of the Hodge conjecture for Fermat varieties, and new results obtained by introducing length reduction algorithms for Hodge characters. In the second part of the talk, I will present how similar methods apply in the context of Klein hypersurfaces, and relate the study of Hodge cycles on these varieties to Fermat varieties. In particular, we give sufficient conditions under which Klein varieties satisfy the Hodge conjecture. This is based on joint works with L. Martelotte, H. Movasati, M. Miranda and M. Pastrana.

In this talk, we explore the connections between Noether-Lefschetz theory, determinantal surfaces, and arithmetically Cohen-Macaulay (ACM) curves. We begin with a brief introduction to Noether-Lefschetz theory, which studies components in the space of surfaces in P^3 where the Picard group jumps. We then show how determinantal surfaces arise naturally as components of the Noether-Lefschetz locus, providing concrete geometric examples of this phenomenon. Next, we introduce ACM curves, which are a natural generalization of complete intersections that retain favorable cohomological properties. Finally, we generalize the notion of determinantal surfaces by introducing weak determinantal surfaces, and use this generalization to provide a classification of ACM curves contained in surfaces of arbitrary degree in P^3