# Algebra colloquium

It takes place on Tuesdays 15:40 at K1. We start on March 1st.

## 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.

## Spring semester

• 01. 03.   David Stanovský: A combinatorial approach to knot recognition
Abstract: The fundamental problem of knot theory asks the following: Given two knots (or knot diagrams), can one smoothly transform one into the other? Is it possible to unknot a given knot diagram? I will present the method of arc coloring, which provides a large class of invariants of various strengths. For example, SL(2,p)-coloring can be used to obtain a certificate of unknottedness which can be verified in polynomial time. In the end, I will mention the underlying algebraic structures, called quandles, and their relation to the theory of permutation groups.
• 08. 03.   Matteo Bordignon: Primes in arithmetic progressions and zeroes of Dirichlet L-functions
Abstract: Dirichlet theorem states that for any two positive coprime integers a and d, there are infinitely many primes of the form a  +  nd. To obtain this result Dirichlet introduced a set of specific complex functions, that are now called the Dirichlet L-functions, and linked the distribution of primes in arithmetic progressions to the position of the zeroes of these functions in the complex plane. The best possible result is that all the zeroes have real part equal to 1/2, this is the famous Generalized Riemann Hypothesis. In this talk we will give an overview on the properties and present the best known bound, obtained by Siegel, on these zeroes. At last, we will highlight the hurdles we encounter trying to make this bound explicit, and we will show what can be done to partially circumvent them.
• 15. 03.   Dmitriy Zhuk: The complexity of the Constraint Satisfaction Problem and its variations
Abstract: Many combinatorial problems, such as graph coloring and solving linear equations, can be expressed as the constraint satisfaction problem for some constraint language. In 2017 the complexity of the constraint satisfaction problem was described for any constraint language on a finite set, but there are many other variants of this problem whose complexity is still not known. For instance, we could allow to use both universal and existentialquantifiers, or require the solution to be surjective or balanced. Another variant is to require the input to satisfy an additional condition. As an example we could consider the problem of coloring a graph in 100 colors if we know that the graph is colorable in 3 colors. In the talk we discuss the complexity of these and other variants of the constraint satisfaction problem.
• 22. 03.   Marek Filakovský: Graph drawings and their higher-dimensional analogues: an overview
Abstract: A classical question in graph theory is whether a graph can be drawn into a plane without self-intersection. From an algorithmic perspective, we ask for an algorithm that decides existence of such drawing. Analogously, in higher dimensions, we consider a k-dimensional analogue of a graph called simplicial complex and ask whether it can be embedded in a d-dimensional Euclidean space. We will summarize (and try to explain) known results, discuss related problems such as extending embeddings (or extending drawings) and consider some open questions.
• 29. 03.   Karel Tůma: Mathematical description of shape memory alloys
Abstract: Shape memory alloys are very cool materials because of two interesting properties -- shape-memory effect and superelasticity -- that are caused by the changes on the microscopic level. Specifically, during the macroscopic deformation, the material undergoes a phase transformation that is accompanied with the formation and evolution of complex microstructure. We present a mathematical model capable of describing this process and show the results of numerical simulations that can capture the detailed microstructure, see the figure.
• 05. 04.   no seminar
• 12. 04.   no seminar
• 19. 04.   Olin Slávik: Monads: From Algebra to Topology (and many other places)
Abstract: As everything in category theory, monad is a concept encompassing a wide range of constructions in mathematics. We will go through many examples of monads and observe some typical situations in which one may meet them, journeying through parts of algebra, topology, set theory and more.Note: This talk should be mostly accessible to people with zero knowledge of category theory. It has also some overlap with my talk at the Fall School of the Department of Algebra, but the emphasis will be put elsewhere.
• 26. 04.   Sebastian Opper: Homological mirror symmetry: a bridge between two branches of geometry
Abstract: During his ICM address in 1994, Kontsevich proposed a highly influential conjecture which relates two vastly different areas of mathematics: symplectic geometry and algebraic geometry. The former is concerned with certain types of manifolds such as the surface of a donut or a cylinder. The latter studies zero sets of equations, for example the circle or the union of the two coordinate axes in the plane. The conjecture relates the two areas through the language of categories in a fascinating way making problems in one area approachable to the machinery (and big theorems) of the other area. I will give a basic overview about what the conjecture says and illustrate everything through accessible examples.
• 03. 05.   Pavlo Yatsyna: You crossed the line too many times...
Abstract: Unlike the lyrics of pop songs, we shall describe rigorously what it means to cross the line, how can we account for it and how many times is too many. It will be a gentle introduction to the topic utilising ideas and tools from combinatorics, discrete geometry, and number theory.
• 10. 05.   Piotr Miska (Kraków): Gosper's algorithm and multidimensional continued fractions
Abstract: Gosper's algorithm allows us to compute the coefficients of continued fraction expansion of e.g. linear combinations, products and quotients of two real numbers if we have their continued fraction expansions as the input. The aim of the talk is to present Gosper's algorithm and the proposition of its generalisation for multidimensional continued fractions.
• 17. 05.   no seminar

## Fall semester

• 07. 10.   Víťa Kala: Integers represented by ternary quadratic forms
Abstract: The study of integers represented by quadratic forms (such as sums of squares x2 + y2, x2 + y2 + z2, …) 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 is 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(x1,…,xn) = s(x1,…,xn) over a fixed algebra A. For infinite algebras this equation solvability problem can be arbitrarily hard and even undecidable, as the MRDP theorem famously showed for (Z, +, 0, –, •) (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
• 04. 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 Rn 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.
• 02. 12.   no seminar
• 09. 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 :-)

This is the old webpage.