Algebra

Elliptic curves and higher-dimensional abelian varieties are fundamental objects of study in number theory. Our investigations focus mainly on their Galois representations, especially over number fields, in the spirit of the Mumford-Tate and Sato-Tate conjectures. We consider in particular the possible existence of nontrivial endomorphisms on a given abelian variety (with special attention given to the case of Jacobians), with implications for the arithmetic of low-genus curves. We also work on related problems, including Kummer theory (the study of division points of non-torsion points on commutative algebraic groups) and the determination of rational points on curves.

Hopf-Galois module theory is a generalisation of Galois module theory, where the action of a Galois group is replaced by that of a $K$-Hopf algebra. This generalisation is fruitful both for enlarging the set of extensions under consideration and for better describing wild extensions. We work on questions in both these settings.
The Hopf-Galois structures on a $G$-Galois field extension are in one-to-one correspondence with suitable subgroups of the permutation group on $G$, and are closely connected with skew braces. The focus of our research is both on the classification of Hopf-Galois structures and on the study of the properties of the skew braces as well as their equivalent from the Hopf-Galois point of view.
We are also employing the language and the properties of the braces in the study of some problems of commutative algebra.

Iwasawa theory studies modules (like class groups and Selmer groups) associated to global fields and abelian varieties in all characteristics, in an attempt to link algebraic invariants (like orders of class groups and ranks of elliptic curves) to analytic ones (i.e. special values of $L$-functions).
The study of Drinfeld modular forms is currently focused on their structure in an attempt to define old forms and new forms, and on their slopes with respect to the action of the Hecke algebra. The main goal is to establish an analog of Hida theory for function fields.

The theories of continuum quantum groups and continuum Kac-Moody algebras associated with one-dimensional manifolds (‘real curves’) can be thought of as continuum colimits of quantized enveloping algebras and Kac-Moody Lie algebras, respectively. These algebraic structures have rather peculiar properties, e.g. they are governed by a continuum root system with infinitely many elements and no simple roots.
At the moment, we are investigating ‘continuum’ analogs of representations, integrable systems, and affinizations of quantum groups.

Frenkel and Gaitsgory proposed a version of the Langlands correspondence in the case of the field $\mathbb{C}((t))$. Given a quasi-simple group $G$ such a proposal relates the geometry of the space of $G$ connections on the disk with a singularity in the origin and representations of the Langlands dual group of $G$. The tool that allows one to build a bridge between these objects is the Feigin-Frenkel theorem which describes the center of the enveloping algebra of an affine Lie algebra at the critical level as the functions on a space of connections for the dual algebra, called the space of Opers, introduced by Beilinson and Drinfeld. In this theory, vertex algebras play a key role in both the proof of the Feigin-Frenkel theorem and the development of the theory of Frenkel and Gaitsgory.

The local geometric correspondence is largely conjectural, but in the case of “spherical” representations, which corresponds to the case of monodromy-free connections, Frenkel and Gaitsgory gave complete proof.

We are studying a version of this correspondence in the case of families of connections on the disk with two singular points $0$ and $a$. In this case, we consider an analogue of the affine algebra, which we denote by $\mathfrak{g}(2)$, on the ring $\mathbb{C}[[a]]$. In the case $\mathfrak{g}=\mathfrak{sl}_2$ we were able to construct a family of elements that generalize Sugawara operators and generate the center of the enveloping algebra of $\mathfrak{g}(2)$ at the critical level. We have also provided a description of endomorphisms of a Weyl module similar to the one given by Frenkel and Gaitsgory in the usual case. We intend to continue the study started with this work by extending the results in various directions.

If $G$ is a reductive group and $H$ is the subgroup of points fixed under an involution then $G/H$ is called a symmetric variety. They have been extensively studied in relation to problems coming from representation theory or algebraic geometry and in particular enumerative geometry. Symmetric varieties are a particular case of spherical varieties: normal $G$ variety with a finite number of $B$ orbits. The classification of spherical varieties has been obtained recently. We study different aspects of symmetric and spherical varieties: compactifications, description of the coordinate ring, applications to the study of normality of particular classes of $G$ variety important in representation theory.
An important theme is a study of the classification of $B$ orbits in these varieties and their Bruhat order.