I will discuss some classical and some recent results about group and algebra actions on abelian varieties and curves.

The relation between quantum cohomology and derived categories has been known since long ago. In 1998 Dubrovin formulated his celebrated conjecture relating semisimplicity of the quantum cohomology of Fano manifolds and the existence of exceptional collections in their derived categories of coherent sheaves. In this talk we will discuss recent joint work of the author with Anton Mellit, Nicolas Perrin and Maxim Smirnov studying the relation between quantum cohomology for certain isotropic Grassmannians and their quantum cohomology in the spirit of Dubrovin conjecture. In this case the small quantum cohomology turns out not to be semisimple but still an interesting decomposition of the derived category arises. If time permits new directions and some problems will be mentioned.

By a degeneration, we mean a process that transforms a geometric object \(X/F\) defined over a field \(F\) into a simpler object that retains many of the relevant properties of \(X\). Formally, any degeneration is realized by an integral model for \(X\); that is, a flat scheme \(X'/R\) defined over some integral domain \(R\) whose generic fiber is the original object \(X\).

In this talk, we endow the field \(F=K((t_1,\ldots,t_m))\) of quotients of multivariate formal power series with a (generalized) non-Archimedean absolute value \(|\cdot|\). We use this to establish the existence of (affine) integral models \(X'/R\) over the ring of integers \(R=\{|x|\leq 1\}\) for schemes \(X\) associated to solutions of systems of algebraic partial differential equations with coefficients on \(F\). We also concretely describe the specialization map of a model \(X'/R\) to the maximal ideals of \(R\), which are encoded in terms of usual (total) monomial orderings.

Recently, Oguiso addressed the following question, attributed to Gizatullin: ``Which automorphisms of a smooth quartic K3 surface \(D\subset \mathbb{P}^3\) are induced by Cremona transformations of the ambient space \(\mathbb{P}^3\)?'' When \(D\subset \mathbb{P}^3\) is a quartic surface, \((\mathbb{P}^3,D)\) is an example of a \emph{Calabi-Yau pair}, that is, a pair \((X,D)\) consisting of a normal projective variety \(X\) and an effective Weil divisor \(D\) on \(X\) such that \(K_X+D\sim 0\). In this talk, I will explain a general framework to study the birational geometry of mildly singular Calabi-Yau pairs. This is a joint work with Alessio Corti and Alex Massarenti.

Let S be a smooth complex surface fibered over a curve C. Under suitable assumptions, if E is a stable rank two bundle on C then its pullback is a H-stable bundle on S. In this sense we can relate moduli spaces of rank two stable bundles on the surface S with moduli spaces of rank two stable bundles on C. In this talk we aim to take advantage of this relation to provide examples of Brill–Noether loci on fibered surfaces.

We will give an overview of several aspects of the Langlands Program that interconnect different fields of mathematics, namely: Algebraic Geometry, Number Theory and Representation Theory. We study general results over global fields, and mention what is known on Langlands functoriality conjectures. In characteristic p, we have a rich interplay with algebraic geometry and we present results of the author in connection with the work of Laurent and Vincent Lafforgue. As a main application, we look at the Ramanujan conjecture, known for generic representations of the classical groups and GL(n) over function fields, for example.

Let \((A,\mathcal{L})\) be a complex abelian variety of dimension \(g\) with polarization of type \(D = \mbox{diag}(d_1,\dots,d_g)\). So \(A = V/\Lambda\) where \(V\) is a complex vector space of dimension \(g\) and \(\Lambda\) is a lattice of maximal rank in \(\mathbb{C}^g\) such that with respect to a basis of \(V\) and a symplectic basis of \(\Lambda\), \(A\) is given by a period matrix \((D\; Z)\) with \(Z\) in the Siegel upper half space of rank \(g\).

In this talk we will present part of the results in Auffarth, Lange, Rojas (2017), where we give a set of equations in the entries of the matrix \(Z\) which characterize the fact that \((A,\\mathcal{L})\) is non-simple.

We developed this criterion to apply it in the problem of finding completely decomposable Jacobian varieties. This research is motivated by Ekedahl and Serre's questions in Ekedahl, Serre (1993), which can be summarized into whether there are curves of arbitrary genus with completely decomposable Jacobian variety. Moreover, in Moonen, Oort (2004), the authors ask about the existence of positive dimensional special subvarieties \(Z\) of the Jacobian loci \(\mathcal{T}_g\) in the moduli space of principally polarized abelian varieties, such that the abelian variety corresponding with the geometric generic point of \(Z\) is isogenous to a product of elliptic curves.

In this direction, we use this criterion in combination with the so called Group Algebra Decomposition Lange, Recillas (2004) of a Jacobian variety \(JX\) with the action of a group \(G\). This is, the decomposition of \(JX\) induced by the decomposition of \(\mathbb{Q}[G]\) as a product of minimal (left) ideals which gives an isogeny \[B_1^{n_1} \times \cdots \times B_r^{n_r}\to JX\]

Although \(JX\) is principally polarized, the induced polarization on \(B_i\) is in general not principal. The question we address is to study the simplicity of \(B_i\), hence finding in some cases a further decomposition of \(JX\).

This is a joint work with R. Auffarth and H. Lange.

The classical representation theory of affine schemes over a field, has been widely considered and many of its main properties have been understood and put under contral--after the work of many mathematicians along the second half of the 20th century. In joint work with del Ángel and Rittatore, we proposed a representation theory for the more general situation where the group scheme is affine over a fixed abelian variety A (the classical case corresponds to the situation that A is the trivial abelian variety with only one point). In this talk we describe the main relevant definitions, where the representations are vector bundles over A with suitable actions of the group scheme G. Then, we show that in our context an adequate formulation of Tannaka--type of recognition and reconstruction results (due to Grothendieck, Saavedra, Deligne and others) remain valid.

We give a new practical method for computing subvarieties of projective hypersurfaces. By computing the periods of a given hypersurface X, we find algebraic cohomology cycles on X. On well picked algebraic cycles, we can then recover the equations of subvarieties of X that realize these cycles. In practice, a bulk of the computations involve transcendental numbers and have to be carried out with floating point numbers. As an illustration of the method, we compute generators of the Picard groups of some quartic surfaces. This is a joint work with Emre Sertoz.