I will present some algorithms for computation of arithmetic subgroups of groups related to algebraic geometry developed in series of papers by A.S.Detinko, D.L.Flannery and E.A. O’Brien, and A.S. Detinko,D.L. Flannery, A. Hulpke. I will start with some motivation: arithmetic groups in $\operatorname{SL}_n(\mathbb Z)$ related to the level structures for elliptic curves and monodromy groups of the Picard-Fuchs differential equations for the period integrals. I will briefly recall the basic definition and then discuss the example of fourteen $CY(3)$ operators: seven with arithmetic and seven with thin monodromy groups.

In the second part of a talk based on a series of papers by A.S.Detinko, D.L.Flannery and E.A. O’Brien, and A.S. Detinko,D.L. Flannery, A. Hulpke I will present an algorithm to compute the level and the maximal Proncipal Congruence Subgroup for an arithmetic subgroup of the group $\operatorname{Sp}(n,\mathbb Z)$. Principal congruence subgroup of level $m$ in $\operatorname{Sp}(n,\mathbb Z)$ is the kernel of the reduction homomorphism $\phi_m:\operatorname{Sp}(n,\mathbb Z)\longrightarrow \operatorname{Sp}(n,\mathbb Z/m\mathbb Z)$, a subgroup in $H\subset \operatorname{Sp}(n,\mathbb Z)$ is arithmetic (i.e. has finite index in $\operatorname{Sp}(n,\mathbb Z)$) iff it containce a Principal Congruence Subgroup.

A $\mu$-stable vector bundle $\mathcal{E}$ of rank 2 with $c_1 (\mathcal{E})=0$ on $\mathbb{P}_{\mathbb{C}}^{3}$ is called a mathematical instanton bundle if $\mathrm{H}^1 (\mathbb{P}^{3}, \mathcal{E}(-2))=0$. This type of bundle has been generalized to other varieties in various ways. First, it has been generalized to odd-dimensional projective spaces by M. M. Capria and S. M. Salamon, then to non-locally free sheaves of any rank on arbitrary projective spaces by M. Jardim. Then, D. Faenzi and A. Kuznetsov extended the definition to other Fano threefolds, and later, V. Antonelli and F. Malaspina modified the definition to apply to any polarization of Fano threefolds, introducing the concept of an $h$-instanton bundle. Finally, V. Antonelli and G. Casnati further broadened the definition to cover any polarized variety $(X,h)$. In this talk, we will focus on rank 2 $h$-instanton sheaves on ruled Fano threefolds with Picard rank 2 and index 1. This is a joint work with Marcos Jardim.

In my talk I will define Kac-Moody Lie algebras and Koszul modules. Then I will introduce Koszul modules associated with (graded) Kac-Moody Lie algebras. I will give a precise criterion for when these modules are of finite length, as well as an exact description of all nilpotent Kac-Moody Koszul modules. The talk is based on a part of my PhD thesis

Kontsevich suggested that the Landau—Ginzburg models provide a good formalism for investigations around the mathematical mirror symmetry problem. A different perspective on Landau-Ginzburg models is discussed in light of this claim. As a result, certain results of Abouzaid-Auroux-Katzarkov can be recovered differently. By using this different angle, we can make new advances regarding a conjecture of Kontsevich--Soibelman on a version of the Strominger-Yau-Zaslow mirror problem