## Applied Algebraic Geometry Group

## Research Seminars

## 2022

## Ghent University

Welcome to the **Ghent Algebra and Geometry Weekly Seminars**.

Organizer: **Fatemeh Mohammadi**

We will meet at the main lecture room on the ground floor in the S12 building.

**Schedule (April 15, Friday, 2022):**

**April 15, 2022 (13:00-1****4:00****):**Leandro Vendramin (Vrije Universiteit Brussel)

**April 15, 2022 (14:00-14:45):****April 15, 2022 (15:00-15:10):**Xian Wu**April 15****, 2022 (15:20-15:30):**Francesca Zaffalon**April 22****, 2022 (13:00-14:00):**Yairon Cid-Ruiz**May 17****, 2022 (13:00-14:00):**Oleg Karpenkov

**Titles and Abstracts****:**

**May 17****, 2022 (13:00-14:00):**Oleg Karpenkov

**Configuration spaces of tensegrities**

Abstract: In this talk we consider a natural stratification of the configuration space of tensegrities. We discuss several results about the structure of the configuration space of (mostly two-dimensional) tensegrities with a small number of points. In particular we briefly describe the technique of surgeries that is used to find geometric conditions for tensegrities. We conclude the talk with several open problems related to the stratification space of tensegrities.

**April 22****, 2022 (13:00-14:00):**Yairon Cid-Ruiz

**The fiber-full scheme**

Abstract: We introduce the fiber-full scheme which can be seen as the parameter space that generalizes the Hilbert and Quot schemes by controlling the entire cohomological data. In other words, the fiber-full scheme controls the dimension of all cohomologies of all possible twistings, instead of just the Hilbert polynomial. We also present some applications that derive from the existence of the fiber-full scheme. This talk is based on joint work with Ritvik Ramkumar.

**April 15, 2022 (Friday, 13:00-1****4****:****00****):**Leandro Vendramin

**Groups, rings and the Yang-Baxter equation**

Abstract: Braces are ring-theoretical algebraic structures to study non-degenerate set-theoretic solutions of the Yang-Baxter equation. A typical example of a brace is a Jacobson radical ring. In this talk, we will discuss the basic properties of braces and how these structures are related to a discrete version of the celebrated Yang-Baxter equation. We will also discuss intriguing connections between the Yang-Baxter equation and braces, groups and non-commutative rings.

**April 15, 2022 (Friday, 14:00-14:45):**

**Continuous Clusters **

Abstract: We will look at how the cluster combinatorics from Fomin and Zelevinksy’s cluster algebras have evolved to the continuous versions that exist today. In particular, we will highlight the viewpoint from representations of quivers. Along the way, we’ll mention connections to other areas of math and physics.

**April 15, 2022 (Friday, 15:00-15:10):**Xian Wu

**(Graded) Nilpotent Orbits vs. Schubert Varieties **

Abstract: In a simple Lie algebra, the graded nilpotent orbits are classified by Kraskiewicz-Weyman. Weyman conjectures that for the closure of each orbit, there exists a Schubert variety, whose intersection with the unipotent radical coincides with the normalization of the closure of the orbit. I will explain the conjecture and some ongoing work with S.A.Filippini and J.Weyman.

**April 15****, 2020 (Friday, 15:20-15:30):**Francesca Zaffalon

**Computing positroid cells of rank two **

Abstract: Positroids are families of matroids introduced by Postnikov in the study of non-negative Grassmannians. We will provide a combinatorial characterization of positroids for Gr(2,n) in terms of certain graphs. This can be used to compute the dimension, the boundary and intersections of positroid cells. Our techniques rely on determining, given some dependent sets, different ways to enlarge this family so that it represents the dependencies among the columns of a matrix with non-negative maximal minors.