# Saxonian Seminar Day on Algebra and Geometry

This one-day seminar features several talks by regional mathematicians on different topics from algebra and geometry. All talks take place at TU Dresden in room Z21 250 on **May 13**.

### Schedule

09:30-10:00 | Philip Dittmann | Determining solvability of equations over local fields of positive characteristic |

10:05-10:35 | Pierpaola Santarsiero |
A geometrical perspective on tensors and the identifiabilty problem |

Coffee break | ||

11:05-11:35 | Julian Vill | Gram spectrahedra of ternary quartics |

11:40-12:10 | Giles Gardam |
The Kaplansky conjectures |

Lunch break | ||

13:45-14:45 | Gennadiy Averkov | What matrix size can you afford? Semidefinite lifts with LMIs of bounded size |

Coffee break | ||

15:15-16:15 | Christian Lehn | Cubic hypersurfaces and Rational Curves on them |

After the talks Andreas will show us the touristic highlights of Dresden and we will have dinner together.

### Abstracts

**What matrix size can you afford? Semidefinite lifts with LMIs of bounded size** (Gennadiy Averkov)

Semidefinite optimization is a far-reaching generalization of linear programming dealing with optimization of a linear function subject to finitely many linear-matrix inequalities (LMIs), where an LMI is a positive semidefiniteness condition on a symmetric matrix whose entries are affine functions. Semidefinite optimization can be used to solve polynomial optimization problems in principle, but the main problem of the standard algebraic approach via the sum-of-squares certificates is that one has to solve semidefinite problems with LMIs of very large size. Computationaly, it is much easier to deal with many LMIs of small size. I will discuss up to what extent such a reduction of the size of the LMIs is possible.

**Cubic hypersurfaces and Rational Curves on them** (Christian Lehn)

Cubic hypersurfaces are the easiest example where many things in algebraic geometry become a lot more involved: Hodge theory, Chow rings, rationality questions. We will discuss constructions involving rational curves on cubic fourfolds that lead to surprisingly rich geometries.

**Gram spectrahedra of ternary quartics** (Julian Vill)

The Gram spectrahedron of a homogeneous polynomial is a compact, convex set that parametrizes all its sum of squares representations. As a compact, convex set its boundary is the union of all its faces. We give a description of the Gram spectrahedron when the polynomial is a quartic in three variables. Thereby building on earlier work of Plaumann, Sturmfels and Vinzant who mainly studied the extreme points of smallest rank on such Gram spectrahedra. We determine all possible dimensions of faces, as well as the dimensions of the semi-algebaic sets consisting of relative interior points of all faces of a fixed dimension.

**Determining solvability of equations over local fields of positive characteristic **(Philip Dittmann)

In both arithmetic geometry and model theory it is natural to wish to tell whether a given polynomial equation has any solution over a Laurent series field F_q((t)). More formally, we ask whether the existential theory of such a field in a suitable language is decidable. I shall discuss some results old and new (including joint work with Sylvy Anscombe and Arno Fehm) on this problem, assuming various forms of resolution of singularities in algebraic geometry.

**A geometrical perspective on tensors and the identifiabilty problem** (Pierpaola Santarsiero)

Over the last 60 years multilinear algebra made its way in the applied sciences. One of the main advantages of working with tensors instead of matrices is that tensors (very often) admit a unique rank decomposition, i.e. they are identifiable. After introducing some classical geometric tools to look at tensors, in this talk we investigate the identifiability problem in the case of specific tensors and we classify all identifiable tensors of rank either 2 or 3.

**The Kaplansky conjectures **(Giles Gardam)

There is a series of four long-standing conjectures on group rings that are attributed to Kaplansky. For example, the zero divisor conjecture states that the group ring of a torsion-free group over a field has no zero divisors. I will discuss what is known about these conjectures and my recent disproof of the unit conjecture.