# Algebra colloquium

It takes place on Thursdays 15:40 at K3.

## Information

Algebra colloquia are designed for master's and doctoral students, and the faculty. The aim is to offer a platform for researchers to present their own results or current trends in their area of interest in a way that is easy to understand for non-experts.

Colloquia talks will have 45 minutes and will be followed by questions and informal discussions (and cookies, in non-covid times :) ). Everyone is most welcome to attend, including people from other departments. The colloquium is co-organized by Zuzka and Víťa. If you want to give a talk, please, let one of us know.

## Schedule

• 7.10. Víťa Kala: Integers represented by ternary quadratic forms
Abstract: The study of integers represented by quadratic forms (such as sums of squares $$x^2+y^2, x^2+y^2+z^2, \ldots$$ ) has rich history that goes back at least to ancient Babylonia and that features impressive works by giants such as Gauss, or Bhargava (2014 Fields medalist). In the talk, I'll focus primarily on quadratic forms in 3 variables - I'll review some elementary results, as well as connections to modular forms. If time permits, I hope to finish with a surprising conjecture that came up in my recent paper with Tomas Hejda.
• 14.10. Francesco Genovese: Linearized scaffolds of spaces: an invitation to homological algebra
Abstract: It is common to treat complicated mathematical objects by reducing them to something which is "linear", so that one can use the powerful tools of linear algebra - think of differentials of functions, for example. In this talk I will show you that even topological spaces can be in a certain way "linearized" and reduced to nice algebraic gadgets called *chain complexes*, which we can conveniently manipulate to extract a useful invariant - namely, homology. Chain complexes (and homology) can be manipulated abstractly, and this essentially what *homological algebra* is about. I will show you a few basics of the subject and then give a glance at some more advanced tools I use in my daily life as a researcher.
• 21.10. Michael Kompatscher: Solving equations: a computational perspective
Abstract: In this talk I would like to discuss the computational complexity of solving polynomial equations $$t(x_1,\ldots,x_n)=s(x_1,\ldots,x_n)$$ over a fixed algebra $$\mathbf{A}$$. For infinite algebras this equation solvability problem can be arbitrarily hard and even undecidable, as the MRDP theorem famously showed for $$(\mathbb Z,+,0,-,\cdot)$$ (settling Hilbert's tenth problem). However, over finite algebras, equations can always be decided by „guessing" solutions; in other words, solving equations is in NP. It is then of major interest to distinguish algebras for which the equation solvability problem has an efficient algorithm (P), is hard (NP-complete), or possibly of some NP-intermediate complexity. I am going to discuss some known results (and pitfalls) when classifying the equation solvability problem for finite rings and groups. If the time permits, I will also present some generalization to algebras with a Maltsev term.
• 28.10. no seminar -- Independent Czechoslovak State Day
• 4.11. no seminar -- Fall school
• 11.11. Kristóf Huszár (INRIA): Topology from the Computational Viewpoint
Abstract: Ever since its early developments, topology has been interspersed with combinatorial ideas, and thus it is not surprising that many of its fundamental problems call for an algorithmic solution. Indeed, for over a century a great deal of research in topology has been driven by the Homeomorphism Problem (the problem of deciding whether two triangulated manifolds are homeomorphic) and special cases thereof, such as the problems of Sphere Recognition or Unknot Recognition. This talk aims to give a glimpse into some of the computational aspects of topology, in particular in low dimensions. After discussing Markov's classical result on the undecidability of the Homeomorphism Problem in dimensions four or above, we shift our focus on 3-dimensional manifolds, where, even though problems are often decidable, there are many challenges to be resolved.
Slides
• 18.11. Eric Nathan Stucky: Algebra of Parking Functions
Abstract: Originally considered in the 1960s by computer scientists, parking functions have risen to be a central object of study in combinatorial theory. In this talk we will define what these objects are and discuss some of their algebraic aspects, with an eye toward the parking space conjectures of Armstrong, Reiner, and Rhodes.
• 25.11. Liran Shaul: The Cohen-Macaulay property, its applications and generalizations
Abstract: In linear algebra, we learn that the maximal number of independent linear equations we can impose on $$\mathbb R^n$$ is equal to $$n$$. Generalizing this simple fact to commutative algebra and algebraic geometry leads to the notions of Cohen-Macaulay rings and Cohen-Macaulay varieties. In this talk we explain this notion, discuss its algebraic and geometric meaning, and demonstrate its various applications in algebra, geometry and combinatorics. Finally, if time permits, we discuss a recent generalization of this notion.
• 2.12. no seminar
• 9.12. Jordan Williamson: Local cohomology and classification theorems
Abstract: Local cohomology was introduced by Grothendieck in the 1960s and has since become an indispensable tool in commutative algebra. In this talk I will focus on introducing some more modern applications of local cohomology to the field of tensor-triangulated geometry where it can be used to define support theories and classify certain kind of classes.
• 16.12.Gábina Těthalová: From Polyphony to Painting via Möbius stripe.. and back
Abstract: We will introduce selected principles that can serve as inspiration in art work and illustrate them on concrete examples. A joint feature of these points of inspiration - both counterpoint in polyphony music and the Möbius strip - is their existence in the language of mathematics. Also repeated picture patterns and work with visual language and its structure can act similarly. To what extent one actually uses these rules or, conversely, in artistic rendering breaks them, or if it is simply a subjective interpretation - each listener will have to form their own opinion on this :-)