Topics and Main lecturers

Title: Tensor spaces with methods from algebraic geometry and combinatorics

Abstract: In recent years, modern algebraic geometry has been very successfully applied to combinatorics and tensors, resolving old conjectures, giving easier proofs of classical theorems, and opening new research paths. Our series of lectures will provide an introduction to tensors, algebraic geometry, and some representation theory. We will cover some of the more advanced topics, including interactions between metric geometry, matroid theory, and intersection theory as in the recent work of June Huh and beyond, using varieties of complete linear maps and complete quadrics.

Mateusz Michalek (University of Konstanz)

Keywords: graphs and permutohedra, log-concavity of coefficients of chromatic polynomials, matroidal Schubert varieties, generalized Schubert varieties for tensors and characteristic numbers of tensors, Chow quotients, asymptotic rank, and some explicit tensors.

• Lecture Note (updated 08. 01)

• Exercises

Giorgio Ottaviani (University of Florence)

Keywords: Quadratic and Hermitian forms on Sym^d V, Bombieri-Weyl form, Critical points and Euclidean Distance degree (EDdegree) - basics and duality properties, Spectral theory of tensors compared with the matrix case, More advanced results on EDdegree, the EDdiscriminant, the EDpolynomial.

• Lecture Note 1, 2 (Introduction to Tensors, Tensor Rank, Basic on classical algebraic varieties and representation theory)

• Lecture Note 3, 4, 5 (Topics) 

• Exercise 1

• Exercise 2

Each lecturer will give five 90-minutes lectures.

Contributed Talks

1. Maciej Galazka (BRL AGSTA)

Title: Computations of different versions of rank in algebraic geometry
Abstract: We present different notions of rank in algebraic geometry. We  include examples of polynomials whose rank is known. We state some open  problems connected with the computation of rank. 

2. Hyunsuk Moon (Sungshin Women's University)

Title: Real Waring rank of polynomials and related topics
Abstract: We will introduce the real Waring rank of polynomials : where the linear forms have only real coefficients. It is known that it is bigger than or equal to the complex Waring rank, but there are many unsolved problems related to it. In particular, we will introduce the typical Waring rank and real Waring rank of monomials. Also, we will consider the problem of determining the coefficients of the Waring decomposition.

3. Chia-Yu Chang (Toulouse Mathematics Institute)

Title: Generic Border Subrank
Abstract:  The subrank and border subrank play central roles in several  areas including algebraic complexity theory. In this talk, I will define  subrank and border subrank of tensors, which are generalizations of  matrix rank and first introduced by Strassen. The rank and border rank  of a generic tensor are the same and equal to the maximal border rank.  However, we know less about the behavior of the subrank and border  subrank. I will state the main result that the growth rate of the  generic subrank is the same as the growth rate of the generic border  subrank. Then I will give an idea of the proof via a generalized version  of Hilbert-Mumford criterion. This is joint work with Benjamin Biaggi,  Jan Draisma, and Filip Rupniewski. 

Each lecturer is planned to give a 45-minutes lecture.

Registration

Registration Form: forms.gle/FcZTA21Z8BAbndMQ9 

Registration Deadline: June 15th, 2025

There is no registration fee and we can provide accommodation for undergraduate/graduate students who are interested in this possibility (2 people per room).

Registrations are now closed. If you would like to register, please contact us via the Contact email above.