University of Primorska Faculty of Mathematics, Natural Sciences and Information Technologies
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Mathematical Research Seminar - Archive

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Datum in ura / Date and time: 25.3.24
(15:00-16:00)
Predavalnica / Location: FAMNIT-MP1
Predavatelj / Lecturer: Tea Štrekelj (University of Ljubljana)
Naslov / Title: Swap operators and the quantum max cut
Vsebina / Abstract:
 
Swap operators act on the space $(\mathbb{C}^d)^{\otimes{n}}$ of n qudits by exchanging tensor factors of $(\mathbb{C}^d)^{\otimes{n}}$. The algebra they generate (called the d-swap matrix algebra) is a subalgebra of $M_{d^n}(\mathbb{C}).$ Classically, in physics literature, the case d=2 of qubits has received the most attention. However, in this talk we discuss the properties of the d-swap matrix algebra for the case of a general d. This algebra is semisimple by Maschke's theorem and its block decomposition can be computed by the Schur-Weyl duality. We also give a precise presentation of the d-swap matrix algebra.
 
As an application, we introduce and discuss the Quantum Max d-Cut (d-QMC) problem. It is a generalization of the QMC (Quantum Max d-Cut with d=2) that has emerged as a test-problem for designing approximation algorithms in quantum physics. For fixed n and a graph G on n vertices, the objective function, the d-QMC Hamiltonian, is defined as a linear expression in the swap operators on $(\mathbb{C}^d)^{\otimes{n}}$. Using the block decomposition of the swap operators, we compute the maximum eigenvalue of the d-QMC Hamiltonian for a clique. Moreover, using a suitable clique decomposition we solve the d-QMC problem for a larger class of graphs, including star graphs
Datum in ura / Date and time: 18.3.24
(15:00 -- 16:00)
Predavalnica / Location: FAMNIT-MP1
Predavatelj / Lecturer: Matjaž Krnc (University of Primorska, Slovenia)
Naslov / Title: Toward characterizing locally common graphs
Vsebina / Abstract:

A graph H is common if the number of monochromatic copies of H in a 2-edge-coloring of the complete graph is asymptotically minimized by the random coloring. The classification of common graphs is one of the most intriguing problems in extremal graph theory. We study the notion of weakly locally common graphs considered by Csóka, Hubai, and Lovász [arXiv:1912.02926], where the graph is required to be the minimizer with respect to perturbations of the random 2-edge-coloring. We give a complete analysis of the 12 initial terms in the Taylor series
determining the number of monochromatic copies of H in such perturbations and classify graphs H based on this analysis into three categories:
   *   Graphs of Class I are weakly locally common.
   *   Graphs of Class II are not weakly locallycommon.
   *   Graphs of Class III cannot be determined to be weakly locally common or not based on the initial 12 terms.
As a corollary, we obtain new necessary conditions on a graph to be common and new sufficient conditions on a graph to be not common.
Joint work with Robert Hancock, Daniel Král’, and Jan Volec.

 Let's dive into the fascinating world of locally common graphs together!

Datum in ura / Date and time: 11.3.24
(15:00-16:00)
Predavalnica / Location: FAMNIT-VP1
Predavatelj / Lecturer: Ademir Hujdurović
Naslov / Title: Canonical double covers and their symmetries
Vsebina / Abstract:

Canonical double cover BX of a graph X is the direct product of X with K_2 (the complete graph on two vertices). Automorphisms of the base graph X naturally lift to automorphisms of BX. In addition, there is an obvious involutory automorphism of BX swapping the bipartition sets. Expected automorphisms of BX are those that can be obtained by combining the above two types, and generate a group isomorphic to Aut(X) × S_2. If BX has only the expected automorphisms, then X is called stable, and it is called unstable otherwise. Characterization of stable graphs is an open problem, even when restricted to special graph classes like circulant graphs. In this talk, I will present several constructions of unstable graphs and characterizations within certain graph families, with special emphasis on circulant graphs. I will show the connection of this problem with Schur rings.

 

Everyone is welcome and encouraged to attend!

Datum in ura / Date and time: 4.3.24
(15:00 -- 16:00)
Predavalnica / Location: FAMNIT-MP1
Predavatelj / Lecturer: Fabio Vlacci (University of Trieste)
Naslov / Title: On a matrix representation of (hyper)complex numbers and functions
Vsebina / Abstract:

We will present a recent approach on matrix representation of hypercomplex regular functions. This topic has been also considered independently by A. Altavilla and  C. de Fabritiis. In this talk, we will explore many applications and potential outcomes of these new representations.
We will start by recollecting ideas on several possible ways of representing complex and hypercomplex (namely quaternionic and octonionic) numbers and how these ideas can be - in some sense - transposed to (some classes of) hypercomplex regular functions.
The seminar is supposed to be self-contained with little knowledge of basic complex analysis.
This is a jont work with Jasna Prezelj.

Everyone is welcome and encouraged to attend!