Research group in Nonlinear Algebra

Head:
Bernd Sturmfels

Contact: Email
Phone:
+49 (0) 341 - 9959 - 750

Address:
Inselstr. 22
04103 Leipzig

Administrative Assistant:
Saskia Gutzschebauch
Email, Phone/Fax:
+49 (0) 341 - 9959
- 752
- 658

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Winter 2021/2022

Important information

  • Due to the pandemic, it is mandatory to register for a lecture series before attending one of the lectures. For this registration, we simply use the mailing list for the lecture series. So, if you plan to attend a lecture (series), please subscribe the mailing list which is linked below the respective lecture.
  • Obviously, this rule is only valid for in-person lectures. However, even for the online events it might be a good idea to subscribe. This way, you will receive the information on how to attend the online meetings.

Foundations of Algebraic Geometry

  • Lecturer: Daniele Agostini
  • Date: Monday, 11.15-12.45
  • Room: MPI MiS A3 01
  • Remarks: Please check the lecture webpage: https://personal-homepages.mis.mpg.de/agostini/Teaching/MPI-MiS/2021%20SS/Schemes/2021SS_schemes.html

Abstract

The idea of this reading group is to develop the basic notions of modern algebraic geometry, through the language of schemes. We will follow closely the wonderful lecture notes by Vakil, Foundations of Algebraic Geometry.

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Random Algebraic Geometry

  • Lecturer: Paul Breiding
  • Date: Tuesdays, 11:15-12:45
  • Room: Leipzig University, SG 2-14
  • Keywords: Algebraic Geometry, Probability, Expected Counting Problems, Average Topology
  • Prerequisites: Basic understanding of algebraic and differential geometry and probability
  • Remarks: Website of the course: https://pbrdng.github.io/rag.html

Abstract

This course will deal with the basic problem of understanding the structure (e.g. the geometry and topology) of the set of solutions of real polynomial equations with random coefficients. The simplest case of interest is the count of the number of real zeroes of a random univariate polynomial whose coefficients are Gaussian random variabl2s – this problem was pioneered by Kac in the 1940s. More generally algebraic geometers might be interested, for example, in the number of components of a random real plane curve, or in the expected number of real solutions of more advanced counting problems. I will present the basic techniques for attacking this type of questions, trying to emphasize the connections of classical algebraic geometry with random matrix theory and random fields.

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Introduction to Toric Geometry

  • Lecturer: Simon Telen
  • Date: Wednesday, 10.00-12.00
  • Room: MPI MiS G3 10
  • Keywords: toric varieties, toric geometry
  • Prerequisites: Basic algebraic geometry, at the level of introductory text books such as `Ideals, Varieties and Algorithms'.
  • Remarks: More info and a tentative schedule for the lectures can be found at https://simontelen.webnode.com/l/introduction-to-toric-geometry/.

Abstract

Toric varieties form a well-understood and intensively studied class of algebraic varieties. They provide a rich source of examples and test cases for theorems and conjectures. Moreover, they have direct applications in physics and in polynomial system solving. For instance, compact, projective toric varieties are the natural generalization of projective space considered in the study of discriminants and resultants for sparse polynomials. The theory consists of a nice interplay between algebra, geometry and combinatorics. In this course, we will start from embedded affine toric varieties via monomial maps to later discuss standard constructions of toric varieties from cones, fans and polytopes. We will motivate the theory by insights from sparse polynomial system solving, and (time permitting) present more advanced constructions such as the Cox ring and line bundles on toric varieties. Some important theorems and constructions that are featured include the orbit-cone correspondence, the Bernstein-Khovanskii-Kushnirenko theorem and the construction of a toric variety as a GIT (Geometric Invariant Theory) quotient.

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Spectra of tensors and their applications

  • Lecturer: André Uschmajew, Raffaella Mulas
  • Date: 6-8 lectures, on Wednesdays 2 pm, starting November 3, 2021
  • Room: MPI MiS G3 10
  • Prerequisites: basic knowledge in linear algebra, graph theory and optimization

Abstract

There exist several possibilities for defining eigen- and singular values for tensors, based on critical points of multilinear forms or homogeneous equations. In this mini course we introduce some of these concepts, discuss their basic properties as well as their relation to low-rank approximation and hypergraph theory. Geometric questions and computational aspects will also be considered.

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Algebraic operads, Koszul duality and Gröbner bases: an introduction

  • Lecturer: Pedro Tamaroff
  • Date: Friday, 09.00-10.30
  • Room: MPI MiS A3 01
  • Audience: The course is aimed at advanced MSc students and PhD students.
  • Keywords: algebraic operads, Gröbner bases, Koszul duality, higher structures
  • Prerequisites: We intend for the course to be as introductory as possible, so we will review any necessary material depending on the background of the participants, or provide necessary references. Nonetheless, some knowledge of basic homological algebra, algebraic topology, and representations of finite groups will be useful.

Abstract

This lecture series aim to offer a gentle introduction to the theory of algebraic operads and related topics, starting with the elements of the theory, and progressing slowly towards more advanced themes, including (inhomogeneous) Koszul duality theory, Gröbner bases, and higher structures. The course will consist of approximately 12 lectures, along with extra talks by willing participants, with the goal of introducing extra material to the course, and making them more familiar with the theory. In particular, participants will be encouraged to read (parts of) accessible research articles and present them during the last sessions of the lecture series. More information at https://ptamarov.github.io/operads2122.html.

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Other Lectures at MPI MIS

Please follow this link for other regular lectures at the Max-Planck-Institute.

29.11.2021, 02:30