Guests and lectures
Minicourses
Xuding Zhu (Zhejiang Normal University)
Applications of Combinatorial Nullstellensatz to graph colouringsAbstrakt
The Combinatorial Nullstellensatz gives a sufficient condition for a polynomial P(x1,x2,...,xn) ∈ F[x1,x2,...,xn] to have a non-zero point in a given grid S1×S2×...×Sn, where Si ⊆ F. Many combinatorial problems can be stated as the existence of a non-zero point of a polynomial in a certain grid, and hence have the potential of applying Combinatorial Nullstellensatz. This series of lectures explains some applications of Combinatorial Nullstellensatz to graph coloring and related problems. We focus on methods that show certain monomials in the expansion of a polynomial are non-vanishing, i.e. having non-zero coefficients. These include Alon Tarsi orientations, interpolation formula and permanent method. These methods are illustrated with applications to problems in list colouring of graphs, vertex-edge weighting of graphs, list of forbidden out-degree orientations of graphs, etc.More information about our guest here
Andrzej Ruciński (Adam Mickiewicz Poznań University)
Random graphsAbstract
My course is intended to be a gentle introduction to the vast field of the theory of random graphs. We will begin with historical remarks and a presentation of basic models. Then, we will focus on the binomial model G(n,p) and present a number of results determining thresholds probabilities for various graph properties, like small subgraphs, perfect matchings, etc. If time permits some more recent results will also be presented.More information about our guest here
Sean Prendiville (Lancaster University)
Additive Combinatorics - polynomial progressions in finite fieldsAbstract
We introduce some techniques from additive combinatorics which allow one to locate patterns in dense subsets of finite fields. As an example application, we detail how to count certain polynomial progressions in such sets - an application which exhibits the essential features of the theory in an elementary manner.More information about our guest here
Michał Seweryn (Charles University in Prague)
Graph minorsAbstract
A minor of a graph is any graph obtained by deleting edges or vertices, and by contracting edges. A classical theorem of Wagner asserts that a graph is planar if and only if it contains neither $K_5$ nor $K_{3,3}$ as a minor. In a seminal series of papers, Robertson and Seymour showed that this is a special case of one of the deepest results in structural graph theory: every minor-closed class of graphs is characterized by a finite set of forbidden minors. Qualitative structure theorems describe the approximate shape of graphs that exclude a minor of particular type. Informally, they show that graphs excluding a fixed forest as a minor are ``path-like'', while graphs excluding a fixed planar graph as a minor are ``tree-like''. Of particular importance are large grid minors, which are closely tied to the notion of tangles. Tangles formalize the idea of highly connected regions in a graph, generalizing 2- and 3-connected components. The concept of tangles is central to understanding the structure of graphs excluding an arbitrary graph as a minor, culminating in the Graph Minor Structure Theorem. This series of lectures offers an introduction to graph minor theory, providing an overview of its most important concepts and results. We will also discuss some more recent developments in the field, such as the Clustered Hadwiger Conjecture and a new proof of the Graph Minor Structure Theorem.More information about our guest here
Lectures
Michał Pilipczuk (University of Warsaw)
Chi-boundednessAbstract
Consider the following two parameters of a graph G: the clique number omega(G), defined as the maximum size of a set of pairwise adjacent vertices in G, and the chromatic number chi(G), defined as the minimum number of colors needed to color the vertices of G so that no two adjacent vertices receive the same color. Clearly, chi(G) is never smaller than omega(G), but is there a relation in the other direction? That is, is there a function f such that chi(G) <= f(omega(G)) for every graph G? This turns out to be false in general --- there exist graphs with no triangles and an arbitrarily high chromatic number --- but still holds in many important classes of graphs; we call such classes chi-bounded. During the talk, we will give some non-trivial examples of chi-bounded classes, particularly Pt-free graphs and intersection graphs of rectangles in the plane, and discuss some questions around chi-boundedness.More information about our guest here
Jarosław Grytczuk (Jagiellonian University / Warsaw University of Technology)
The magic of rhythmAbstract
I will present a collection of problems and results concerning various rhythmical phenomena in combinatorial structures. Among the topics we will possibly cover are the issues of rhythmic canons, twins in words, non-repetitive sequences, and the mystery of the duplication constant.More information about our guest here
Jonathan Chapman (Warwick University)
Colouring patterns in the integersAbstract
A fundamental problem in arithmetic Ramsey theory is understanding which equations have the property that any finite colouring of the positive integers produces a solution where the variables all receive the same colour. We call such equations partition regular. Over the last century, ideas from combinatorics, analysis, dynamics, number theory and beyond have all been used to establish results on partition regularity. In this lecture, I will discuss the history of partition regularity and describe how different mathematical techniques may be used to study partition properties of integer equations.More information about our guest here
Nika Salia (King Fahd University)
Intersecting families of polynomials over finite fieldsAbstract
One of the most beautiful ideas in combinatorics is the Erdős–Ko–Rado theorem, which tells us how large a collection of sets can be if every pair of sets overlaps in 'many' elements. This idea has fascinated mathematicians for decades and has inspired many generalizations. In this talk, I’ll share how we recently discovered a version of this theorem for polynomials over finite fields—a surprising algebraic setting where similar questions can be asked. Instead of sets that intersect, we consider polynomials that share a common factor. We ask: how large can a family of such polynomials be if every pair (or even every triple!) shares a factor of some minimum degree? I’ll explain how ideas from algebra, combinatorics, and number theory come together to answer this question, affirming a recent conjecture. We'll also explore when these families must be “trivial”—that is, made in the most obvious way—and when more interesting structures can appear. No prior knowledge of finite fields or deep algebra is required.More information about our guest here
Maciej Dołęga (Polish Academy of Sciences)
Algebraic combinatorics of card shufflingAbstract
In this talk, we will explore the fascinating connections between algebraic structures and the mathematics of card shuffling. We will discuss how concepts from group theory, representation theory, and symmetric functions provide powerful tools for analyzing shuffling processes, such as the well-known riffle shuffle. Through concrete examples and combinatorial models, we will see how algebraic combinatorics helps answer questions about randomness, mixing times, and the surprising order hidden within seemingly chaotic shuffles.More information about our guest here
Štefko Miklavič (University of Primorska)
Algebraic combinatoricsAbstract
The main purpose of this lecture is to introduce the audience to Algebraic Combinatorics. We will start by stating the basic characteristics of this field, and continue with two well-known examples/problems, that beautifully illustrate what algebraic combinatorics (and its narrower subfield, algebraic graph theory) deals with.More information about our guest here