# Omsk Algebraic Online Webinar

Upcoming meeting:

### "On a symmetry group for some problems of quadratic programming"

Abstract. Solution and analysis of mathematical programming problems may be simplified when these problems are symmetric under appropriate linear transformations. In particular, a knowledge of the symmetries may help decrease the problem dimension, reduce the size of the search space by means of linear cuts. While the previous studies of symmetries in the mathematical programming usually dealt with permutations of coordinates of the solutions space, the present paper considers a larger group of invertible linear transformations. We study a special case of the quadratic programming problem, where the objective function and constraints are given by quadratic forms. We formulate conditions, which allow us to transform the original problem to a new system of coordinates, such that the symmetries may be sought only among orthogonal transformations. In particular, these conditions are satisfied if the sum of all matrices of quadratic forms, involved in the constraints, is a positive definite matrix. We describe the structure and some useful properties of the group of symmetries of the problem. Besides that, the methods of detection of such symmetries are outlined for different special cases as well as for the general case.

Latest announcements:

### "Orbits of actions of group superschemes"

Abstract. It is long known that any rational action of (not necessarily affine) algebraic group on a scheme of finite type has closed orbits. For example, if this group is reductive and affine, and the scheme on which the group acts is affine as well, then the closed orbits parametrize the points of the corresponding categorical quotient. In my talk I discuss the generalization of this fundamental result for algebraic group superschemes, rationally acting on superschemes of finite type.

### "Logical, universal, and classical algebraic geometry and algebraic groups"

Abstract. In this talk I will discuss two types of algebraic geometries: the universal one (see the book by Daniyarova, Miasnikov and Remeslennikov) and the logical one (proposed by B.Plotkin) – the roles they currently play in modern algebra and what kind of perspectives they provide. In particular, I will touch on a new notion of an “algebraic group” and how it appears in group theory.

### "Anderson modules and abelian extension of fields"

Abstract. The well known Kronecker-Weber theorem affirms that every finite abelian extension of the field $$Q$$ of rational numbers belongs to some cyclotomic extension $$Q(t|t^n=1)$$. In his 12th problem D.Hilbert asked how to generalize this theorem for other global fields.
In this talk, we give the exposition of atual state of this problem together with the connection with Carlitz-Drinfeld-Anderson modules.

Recall that Anderson module $$M$$ is a (left)module over non-commutative ring $$R=C_p[T,\tau]$$, $$T\tau=\tau T$$, $$\tau a=a^p \tau$$, where $$C_p$$ is a some field of characteristic $$p>0$$, such that $$M$$ is free finite generated over subrings $$C_p[T]$$ and $$C_p\{\tau\}$$.