Mathematical Research Seminar - Archive

Let \F_q be a finite field with q elements, where q is odd, and let A=A^{\top}\in GL_n(\F_q) be an invertible symmetric matrix of size n with coefficients in \F_q. The talk will be about the maps \Phi :\F_q^n\to \F_q^n that satisfy the implication
(x-y)^{\top}A(x-y)=0,  x\neq y
\Longrightarrow
\big(\Phi(x)-\Phi(y)\big)^{\top}A\big(\Phi(x)-\Phi(y)\big)=0, \Phi(x)\neq \Phi(y)
for all column vectors x,y\in \F_q^n. The classification problem of such maps is partially related to some physics. The results and the techniques applied in the proofs are related to finite geometry and graph theory. Primarily, this is a typical `preserver problem' studied in matrix theory.

The electrical activity generated by nerve cells of the brain can be measured through electrodes attached to various locations on the scalp.  This technique is known as electroencephalography (EEG) and is helpful not only in clinical work, but also in studying  various cognitive processes. In the talk we present how graph theory can be used to analyse EEG data. The efficiency of the graph-theoretical approach is illustrated in the case of a mild cognitive impairment.

Properties of fullerenes are critically dependent on the distribution of their 12 pentagonal faces. It is well known that there are infinitely many IPR-fullerenes. IPR-fullerenes can be described as fullerenes in which each connected cluster of pentagons has size 1.

We studied the combinations of cluster sizes that can occur in fullerenes (and whether the clusters can be at an arbitrarily large distance from each other). For each possible partition of the number 12, we are able to decide whether the partition describes the sizes of pentagon clusters in a possible fullerene.

This is joint work with Gunnar Brinkmann, Patrick W. Fowler, Tomaž Pisanski and Nico Van Cleemput.

We shall recall of basic principles of the non-commutative harmonic analysis, discuss the case of classical groups, and how problems of non-commutative harmonic analysis in this case are linked with some important problems of modern theory of automorphic forms.

In our talk, a highly symmetric graph will mean a vertex-transitive graph whose automorphism group is much larger than the order of the graph. We will introduce two numerical parameters - the Cayley and quasi-Cayley deficiency - that will allow us to divide the class of highly symmetric graphs into disjoint subfamilies. We will discuss several known families of highly symmetric graphs, determine their Cayley and quasi-Cayley deficiencies, and try to ask the question which graphs are extremal with regard to these parameters.

Kahler manifolds appear naturally in both Algebraic and Differential Geometry: projective complex varieties from Algebraic Geometry, and smooth manifolds with riemannian metrics with holonomy U(n). Understanding when a manifold may admit a Kahler structure and how far it is from that is a key problem in geometry. Classically, tools from global analysis (harmonic forms and Hodge theory) provide striking topological properties of Kahler manifolds, linking topology and analysis. We shall review this circle of ideas, and recent results on the construction of manifolds admitting complex structures (and even both complex and symplectic structures simultaenously) but not admitting Kahler structures.

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