Univerza na Primorskem Fakulteta za matematiko, naravoslovje in informacijske tehnologije
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Raziskovalni matematični seminar - Arhiv

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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!

 


Datum in ura / Date and time: 27.2.24
(14:00 -- 15:00)
Predavalnica / Location: FAMNIT-VP1
Predavatelj / Lecturer: Joy Morris (University of Lethbridge, Canada)
Naslov / Title: Detecting (Di)Graphical Regular Representations
Vsebina / Abstract:
Graphical and Digraphical Regular Representations (GRRs and DRRs) are a concrete way to visualise the regular action of a group, using (di)graphs. More precisely, a GRR or DRR on the group $G$ is a (di)graph whose automorphism group is isomorphic to the regular action of $G$ on itself by right-multiplication. 
For a (di)graph to be a DRR or GRR on $G$, it must be a Cayley (di)graph on $G$. Whenever the group $G$ admits an automorphism that fixes the connection set of the Cayley (di)graph setwise, this induces a nontrivial graph automorphism that fixes the identity vertex, which means that the (di)graph is not a DRR or GRR. Checking whether or not there is any group automorphism that fixes a particular connection set can be done very quickly and easily compared with checking whether or not any nontrivial graph automorphism fixes some vertex, so it would be nice to know if there are circumstances under which the simpler test is enough to guarantee whether or not the Cayley graph is a GRR or DRR. I will present a number of results on this question.
 
Join Zoom Meeting

https://upr-si.zoom.us/j/85914318577

Meeting ID: 859 1431 8577

Don't miss out on this opportunity to delve into cutting-edge research and expand your mathematical horizons!

 

 


Datum in ura / Date and time: 19.2.24
(15:00 -- 16:00)
Predavalnica / Location: FAMNIT-VP1
Predavatelj / Lecturer: Marko Orel (University of Primorska)
Naslov / Title: Marko Orel
Vsebina / Abstract:

In this talk, a graph Γ = (V, E) is a finite simple graph, which means that V is a finite set and E is a family of its subsets that have two elements. Given two graphs Γ1 = (V1, E1) and Γ2 = (V2, E2), a map Φ : V1 → V2 is
a homomorphism if {Φ(u), Φ(v)} ∈ E2 whenever {u, v} ∈ E1. A bijective homomorphism is an isomorphism in the case {Φ(u), Φ(v)} ∈ E2 if and only if {u, v} ∈ E1. As usual, if Γ1 = Γ2, then a homomorphism/isomorphism is
an endomorphism/automorphism. A core of a graph Γ is any its subgraph Γ0 such that a) there exists some homomorphism from Γ to Γ0 and b) all endomorphisms of Γ0 are automorphisms.
In graph theory, Γ1 and Γ2 are usually treated as ‘equivalent’ if they are isomorphic, i.e. if there exists some isomorphism between them. Less frequent we meet the notion of homomorphically equivalent graphs Γ1, Γ2,
which means that there exists a graph homomorphism Φ : V1 → V2 and a graph homomorphism Ψ : V2 → V1. Here, the notion of a core appears very naturally because two graphs are homomorphically equivalent if and
only if they have isomorphic cores. Often, it is very difficult to decide if there exists a homomorphism between two graphs. In fact, this problem is related to many graph parameters that are hard to compute, such as the
chromatic/clique/independence number. As a result, the study of cores is challenging. In this talk, I will survey some properties of cores, with an emphasis on graphs that either admit a certain degree of ‘symmetry’ or have
‘nice’ combinatorial properties.

 

Everyone is welcome and encouraged to attend.

 


Datum in ura / Date and time: 15.1.24
(15:00 -- 16:00)
Predavalnica / Location: FAMNIT-MP1
Predavatelj / Lecturer: Hassan Cheraghpour (University of Primorska)
Naslov / Title: Vanishing Immanants
Vsebina / Abstract:

Let $ Q_{n}(\CC) $ be the space of all $ n\times n $ alternate matricesvover the complex field $ \CC $ and let $ d_{\chi}(A) $ denote the immanant of the matrix $ A \in Q_{n}(\CC) $ associated with the irreducible character $ \chi $ of the permutation group $ S_{n} $‎. ‎The main goal in this paper is to find all the irreducible characters
such that the induced immanant function $ d_{\chi} $ vanishes identically on $ Q_{n}(\CC) $‎.

This is a joint work with Bojan Kuzma.

 

Everyone is welcome and encouraged to attend.


Datum in ura / Date and time: 8.1.24
(15:00 -- 16:00)
Predavalnica / Location: FAMNIT-MP1
Predavatelj / Lecturer: Franc Forstnerič (University of Ljubljana, Slovenia)
Naslov / Title: Minimal surfaces with symmetries
Vsebina / Abstract:

A minimal surface in a Euclidean space $\mathbb R^n$ for $n\ge 3$ is an immersed surface which locally minimizes the area. Every oriented minimal surface is parameterized by a conformal harmonic immersion from an open Riemann surface, and vice versa. In this talk, I shall present a recent result on the existence of minimal surfaces of a given conformal type having a given finite group of symmetries induced by orthogonal transformations on $\mathbb R^n$.

 

Everyone is welcome and encouraged to attend.