# Raziskovalni matematični seminar - Arhiv

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**Datum in ura**/ Date and time: 7.6.24

**Predavalnica**/ Location: FAMNIT-MP1

**Predavatelj**/ Lecturer: Besfort Shala (University of Bristol)

**Naslov**/ Title: A tour through multiplicative and probabilistic number theory

**Vsebina**/ Abstract:

I will give a general overview of recent developments in number theory, with many old and new results being proved through the lens of multiplicative functions. Many deep unsolved problems about primes such as the Riemann Hypothesis and the Twin Prime Conjecture are intimately connected with the Möbius function. Viewing the latter as simply one instance of a multiplicative function satisfying certain properties and developing a general theory of multiplicative functions has been very fruitful – most classical results in analytic number theory have been reproved in this framework, without appealing to the analytic continuation of the Riemann zeta function (or more general L-functions). In particular, one may adapt a probabilistic viewpoint and consider so called random multiplicative functions taking the values +1 and -1 on the primes randomly. This was initiated by Wintner, who proved the analogue of the Riemann Hypothesis in this simplified setting. Later this was further developed by Harper, who considered finer distributional questions. If time permits, I will present results on the Twin Prime analogue in this probabilistic setting, based on ongoing joint work with Jake Chinis.

**Datum in ura**/ Date and time: 3.6.24

**Predavalnica**/ Location: FAMNIT-MP1

**Predavatelj**/ Lecturer: Petr Golovach (University of Bergen)

**Naslov**/ Title: Detours in Directed Graphs

**Vsebina**/ Abstract:

**Datum in ura**/ Date and time: 27.5.24

**Predavalnica**/ Location: FAMNIT-MP1

**Predavatelj**/ Lecturer: Tara Abrishami (University of Hamburg)

**Naslov**/ Title: Induced matching treewidth and the maximum independent set problem

**Vsebina**/ Abstract:

Width parameters such as treewidth, the most prominent width parameter, are graph invariants that roughly measure how complicated a graph is. Several width parameters have algorithmic implications; for example, many NP-hard problems can be solved in polynomial time in graphs of bounded treewidth. On the other hand, there exist graph classes with unbounded treewidth but which do admit fast algorithms for hard problems, so in this sense treewidth does a poor job of capturing the solvability of hard problems. Several new width parameters have recently been introduced to better represent the solvability of the maximum independent set problem. In this talk, I will discuss results and problems related to one of these width parameters, called *induced matching treewidth*.

This talk is based on joint work with Marcin Briański, Jadwiga Czyżewska, Rose McCarty, Martin Milanič, Paweł Rzążewski, and Bartosz Walczak.

**Datum in ura**/ Date and time: 20.5.24

**Predavalnica**/ Location: FAMNIT-MP1

**Predavatelj**/ Lecturer: Dilawar Abbas Khan (University of Primorska)

**Naslov**/ Title: Optimal plateaued functions without linear structures

**Vsebina**/ Abstract:

In this talk, we address the algebraic method to design plateaued functions with desirable cryptographic properties (such as maximal algebraic degree and balancedness) by employing the generalized Maiorana-McFarland class (GMM) of Boolean functions. We consider functions in the GMM class of the form $f(x,y)=x \cdot \phi(y) \oplus h(y)$, where $x \in \F_2^{n/2+k}, y \in \F_2^{n/2 -k}$ and $\phi(y): \F_2^{n/2 -k} \rightarrow \F_2^{n/2 +k}$, and derive a set of sufficient conditions for designing optimal plateaued functions. We will show that under certain conditions designed optimal plateaued functions do not admit the linear structures. Furthermore, we will show that under specific conditions, the addition of an indicator ($1_{R}(x,y) = 1_{E_1}(x)1_{E_2}(y)$) to a function $g(x,y) = x \cdot \phi(y)$, we have that $f(x,y) = g(x,y) \oplus 1_{R}(x,y)$ is a plateaued function.

**Datum in ura**/ Date and time: 13.5.24

**Predavalnica**/ Location: FAMNIT-MP1

**Predavatelj**/ Lecturer: Draženka Višnjić (University of Primorska)

**Naslov**/ Title: Homomorphisms on the Coxeter-like graphs

**Vsebina**/ Abstract:

**Datum in ura**/ Date and time: 6.5.24

**Predavalnica**/ Location: FAMNIT-MP1

**Predavatelj**/ Lecturer: Endre Boros (Rutgers University, USA)

**Naslov**/ Title: Boole's problem and a zero-one lemma

**Vsebina**/ Abstract:

We introduce Boole's problem, and the reasonably large literature related to it. We then recall an old result of Renyi (1962) that we prefer to call "a zero-one lemma", and show that it can provide a simple, elementary (short, high school level) proof for most of the results in the extensive literature about this problem. We also derive a few new results with the help of this powerful lemma.

**Datum in ura**/ Date and time: 29.4.24

**Predavalnica**/ Location: FAMNIT-MP1

**Predavatelj**/ Lecturer: Aljaž Kosmač (University of Primorska)

**Naslov**/ Title: Isogeometric collocation for solving the biharmonic equation over planar multi-patch domains

**Vsebina**/ Abstract:

**Datum in ura**/ Date and time: 22.4.24

**Predavalnica**/ Location: FAMNIT-MP1

**Predavatelj**/ Lecturer: Tomaž Pisanski (University of Primorska, Slovenia)

**Naslov**/ Title: Geometric symmetry of graphs

**Vsebina**/ Abstract:

This work in progress explores graphs that can be drawn in the Euclidean plane exhibiting non-trivial geometric symmetry. We investigate the significance of semiregular and quasi-semiregular automorphisms for achieving such symmetric embeddings. We consider both cyclic and dihedral symmetries.

**Datum in ura**/ Date and time: 15.4.24

**Predavalnica**/ Location: FAMNIT-VP1

**Predavatelj**/ Lecturer: Marston Conder (University of Auckland, New Zeland)

**Naslov**/ Title: Magma Mini-Course

**Vsebina**/ Abstract:

Get ready for an exhilarating opportunity this April! **Professor Marston Conder** from New Zealand will grace FAMNIT with his presence to deliver an insightful mini-course on MAGMA, the powerful computational algebra system.**Course Outline:**

Overview of MAGMA and its applications, including graphs, digraphs, and Cayley graphs.

Handling permutation groups, matrix groups, and groups of small order.

Exploring finitely presented groups and their practical applications.

Plus, exciting demonstrations showcasing the practical usage of MAGMA will be included!

Mark your calendars! Here's the schedule:

**Monday, April 15th: FAMNIT-VP1**

- Opening: 9:50 - 10:00

- MAGMA 1: 10:00 - 11:00

- Coffee break: 11:00 - 11:30

- MAGMA 2: 11:30 - 12:30

- Lunch: 12:30 - 14:00

- MAGMA 3: 14:00 - 15:00

- Coffee break: 15:00 - 15:30

- MAGMA 4: 15:30 - 16:30**Tuesday, April 16th:FAMNIT-MP1**

- MAGMA 5: 10:00 - 11:00

- Coffee break: 11:00 - 11:30

- MAGMA 6: 11:30 - 12:30

- Closing remarks: 12:30 onwards

Who Should Attend?

These lectures are perfect for students specializing in Algebraic graph theory, young PhDs, postdocs, and anyone interested in cryptography!

**Datum in ura**/ Date and time: 8.4.24

**Predavalnica**/ Location: FAMNIT-MP1

**Predavatelj**/ Lecturer: Andrea Munaro (University of Parma)

**Naslov**/ Title: Polynomial-time approximation schemes for induced subgraph problems on fractionally tree-independence-number-fragile graphs

**Vsebina**/ Abstract:

**Datum in ura**/ Date and time: 25.3.24

**Predavalnica**/ Location: FAMNIT-MP1

**Predavatelj**/ Lecturer: Tea Štrekelj (University of Ljubljana)

**Naslov**/ Title: Swap operators and the quantum max cut

**Vsebina**/ Abstract:

**Datum in ura**/ Date and time: 18.3.24

**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

**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

**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

**Predavalnica**/ Location: FAMNIT-VP1

**Predavatelj**/ Lecturer: Joy Morris (University of Lethbridge, Canada)

**Naslov**/ Title: Detecting (Di)Graphical Regular Representations

**Vsebina**/ Abstract:

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

**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

**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

**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.