# Raziskovalni matematični seminar - Arhiv

2019 | 2018 | 2017 | 2016 | 2015 | 2014 | 2013 | 2012 | 2011 | 2010 |

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**Datum in ura**/ Date and time: 26.4.19

**Predavalnica**/ Location: FAMNIT-MP6

**Predavatelj**/ Lecturer: Riste Škrekovski (UL FMF, UP FAMNIT, IMFM, FIŠ)

**Naslov**/ Title: Some results and problems on unique-maximum colorings of plane graphs

**Vsebina**/ Abstract:

A unique-maximum coloring of a plane graph G is a proper vertex coloring by natural numbers such that each face \alpha of G satisfies the property: the maximal color that appears on \alpha, appears precisely on one vertex of \alpha (or shortly, the maximal color on every face is unique on that face). Fabrici and G\"{o}ring proved that six colors are enough for any plane graph and conjectured that four colors suffice. Thus, this conjecture is a strengthening of the Four Color Theorem. Wendland later decreased the upper bound from six to five.

We first show that the conjecture holds for various subclasses of planar graphs but then we disprove it for planar graphs in general. Thus, the facial unique-maximum chromatic number of the sphere is not four but five. In the second part of the talk, we will consider various new directions and open problems.

(Joint work with Vesna Andova, Bernard Lidick\'y, Borut Lužar, and Kacy Messerschmidt)

**Datum in ura**/ Date and time: 15.4.19

**Predavalnica**/ Location: FAMNIT-MP1

**Predavatelj**/ Lecturer: Graziano Gentili (Universita degli studi Firenze, Italy)

**Naslov**/ Title: Regular functions of a quaternionic variable with applications

**Vsebina**/ Abstract:

The recent notion of quaternionic slice regularity has identified a new class of functions which can play the role of holomorphic (and meromorphic) functions in the quaternionic setting. The talk will present the main basic features of slice regular functions over the quaternions, as well as some of their most curious applications, including those concerning the study of orthogonal complex structures on subsets of the quaternionic space H.

**Datum in ura**/ Date and time: 8.4.19

**Predavalnica**/ Location: FAMNIT-MP1

**Predavatelj**/ Lecturer: Berenice Martinez-Barona (Universitat Politecnica de Catalunya and University of Primorska)

**Naslov**/ Title: Identifying codes in digraphs

**Vsebina**/ Abstract:

A (1,\le \ell)-identifying code in a digraph D is a dominating subset C of vertices of D such that all distinct subsets of vertices of cardinality at most \ell have distinct closed in-neighbourhoods within C.

In this talk, we give an upper bound on \ell and some sufficient conditions for a digraph of minimum in-degree \delta^-\ge 1 to admit a (1,\le \ell)-identifying code for \ell= \delta^-, \delta^-+1. We give also a new method to obtain an upper bound on \ell for digraphs and graphs using spectral graph theory. As particular cases, we give a characterization of the j-in-regular digraphs admitting a (1,\le \ell)-identifying code for j \in \{1,2\} and \ell\in\{1,2,3\}. We also prove that every k-iterated line digraph of minimum in-degree at least 2 admits a (1,\le \ell)-identifying code with \ell \le 2, and it does not admit a (1,\le \ell)-identifying code for \ell\ge 3. Moreover, we give some bounds of the identifying number of a line digraph, that is, the minimum size of a (1,\le 1)-identifying code.

**Datum in ura**/ Date and time: 1.4.19

**Predavalnica**/ Location: FAMNIT-MP1

**Predavatelj**/ Lecturer: Peter Muršič (University of Primorska)

**Naslov**/ Title: Hypergraph Nim

**Vsebina**/ Abstract:

Nim is an impartial game in which two players take turns choosing a pile and removing a positive tokens from it, with the player making the last move winning. Impartial games are solved by finding which positions are winning and losing, the winning move makes your winning position into a losing position for the opponent. A generalization of winning and losing positions is called the Sprague-Grundy function which partitions positions even further and is needed to know how to play the sum of games. Hypergraph Nim is a new generalization of the game Nim in which instead of a single pile we chose multiple, with our choice of piles denoted by a hypergraph. In this talk we will take a look at the Sprague-Grundy function of Hypergraph Nim.

**Datum in ura**/ Date and time: 25.3.19

**Predavalnica**/ Location: FAMNIT-Muzejski 1

**Predavatelj**/ Lecturer: Russ Woodroofe (University of Primorska)

**Naslov**/ Title: Binomial divisibility to 7 billion

**Vsebina**/ Abstract:

In earlier work with John Shareshian, we asked whether for every number n, there are primes p and r so that every nontrivial binomial coefficient n choose k is divisible by at least one of the two primes. The motivation for the question is from group theory: the question is equivalent to that of whether for every n there are primes p,r so that the alternating group A_n is generated by any Sylow subgroups at p and r. The answer is "yes" up to a set of asymptotic density zero (assuming the Riemann Hypothesis), and we verified a "yes" answer computationally out to 1 billion.

In more recent work, joined by Bob Guralnick, we have verified computationally that a stronger condition holds for all n out to 7 billion. This computation finishes in a few hours, which is a great improvement over the 2 weeks that our earlier computation out to 1 billion took.

In this talk, I'll overview the interplay between group theory, number theory, and computation allowing this computation.

**Datum in ura**/ Date and time: 21.3.19

**Predavalnica**/ Location: FAMNIT-MP4

**Predavatelj**/ Lecturer: Slobodan Filipovski (University of Primorska)

**Naslov**/ Title: A connection between a question of Bermond and Bollobas and Ramanujan graphs

**Vsebina**/ Abstract:

If we let n(k, d) denote the order of the largest undirected graphs of maximum degree k and diameter d, and let M(k,d) denote the corresponding Moore bound, then n(k,d)\leq M(k,d), for all k\geq 3, d\geq 2. Whilethe inequality has been proved strict for all but very few pairs k and d, the exact relation between the values n(k,d) and M(k,d) is unknown, and the uncertainty of the situation is captured by an open question of Bermond and Bollobas who asked whether it is true that for any positive integer c>0 there exist a pair k and d, such that n(k,d)\leq M(k,d)-c.

In this talk we present a connection of this question to the value 2\sqrt{k-1}, which is also essential in the definition of the Ramanujan graphs defined as k-regular graphs whose second largest eigenvalue (in modulus) does not exceed 2\sqrt{k-1}. We further reinforce this surprising connection by showing that if the answer to the question of Bermond and Bollobas were negative and there existed a c > 0 such that n(k,d) \geq M(k,d) - c, for all k\geq 3, d\geq 2, then, for any fixed k and all sufficiently large even d's, the largest undirected graphs of degree k and diameter d would have to be Ramanujan graphs. This would imply a positive answer to the open question whether infinitely many non-bipartite k-regular Ramanujan graphs exist for any degree k.

**Datum in ura**/ Date and time: 18.3.19

**Predavalnica**/ Location: FAMNIT-MP1

**Predavatelj**/ Lecturer: Samir Hodžić (University of Primorska)

**Naslov**/ Title: On secondary constructions of bent and plateaued functions

**Vsebina**/ Abstract:

In this talk, we consider secondary construction methods of bent and plateaued functions.

The first part of this talk focuses on plateaued functions, where we analyse the concept of its dual. In contrary to bent functions, for which the dual is well established, the dual of a plateaued functions never received enough attention. Using suitable ordering(s) in combination with signs of non-zero entries of the Walsh spectra of a plateaued function, we show that new characterization of plateaued functions can be derived. In addition, these results will lead to completely new approach (so-called Spectral method) for construction of cryptographically significant Boolean functions.

The second part of the talk focuses on secondary constructions of bent and plateaued functions. As we noticed that almost all secondary constructions deduced in the last four decades can be generalized using the composition of one Boolean (usually plateaued) and one vectorial function, we show how one can generalize almost all known secondary construction methods of bent and plateaued functions, by applying the concept of the dual developed for plateaued functions. The so-called “composite representation” of Boolean functions gives a better understanding of the existing secondary constructions and it also allows us to provide a general construction framework of these objects.

**Datum in ura**/ Date and time: 11.3.19

**Predavalnica**/ Location: FAMNIT-MP1

**Predavatelj**/ Lecturer: István Kovács (University of Primorska)

**Naslov**/ Title: Regular Cayley maps for dihedral groups

**Vsebina**/ Abstract:

A combinatorial map is a pair M=(X,r), where X is a finite simple connected graph and r is a permutation of its arcs (directed edges) such that the orbit of the arc (u,v) under r is the set of arcs emanating from u. Aut(M) consists of the automorphisms of X for which the induced permutations on the arcs commute with r. The order |Aut(M)| is less than equal to 2|E(G)|, and when equality holds M is called regular. M is a Cayley map for a finite group G if X=Cay(G,S), and r is defined by (h,g) -> (h,p(h^{-1}g)), where h^{-1} \in S and p is a fixed cyclic permutation of S. Regular Cayley maps for cyclic groups were classified by Conder and Tucker (Trans. Amer. Math. Soc., 2014). In this talk, I will show the classification of regular Cayley maps for dihedral groups.

**Datum in ura**/ Date and time: 4.3.19

**Predavalnica**/ Location: FAMNIT-MP1

**Predavatelj**/ Lecturer: Martin Strehler (Brandenburg University of Technology Cottbus–Senftenberg, Germany)

**Naslov**/ Title: Nash equilibria in network routing games

**Vsebina**/ Abstract:

The talk provides a brief introduction to routing games and related game theory concepts. We will discuss different variants and simple examples.

In the second part of the talk, we will look at a dynamic routing game with a special forwarding policy in which a player's priority depends on the previous edge of the chosen path. We present properties and efficiency of equilibria and we specify an algorithm for their calculation. Finally, we give some additional results and open problems.

**Datum in ura**/ Date and time: 25.2.19

**Predavalnica**/ Location: FAMNIT-MP1

**Predavatelj**/ Lecturer: Jasna Prezelj (University of Primorska and University of Ljubljana)

**Naslov**/ Title: On a class of automorphisms in \mathbb{H}^2 with volume

**Vsebina**/ Abstract:

We briefly present the theory of slice-regular functions of one quaternionic variable and compare this regularity properties with Fueter regular functions (the regularity used physics).

We give a possible extension for shears and overshears in the case of two non-commutative (quaternionic) variables in relation with the associated vector fields and flows. We present a possible definition of volume preserving automorphisms, even though there is no quaternionic volume form on \mathbb{H}^2.

Using this, we determine a class of quaternionic automorphisms for which the Andersen-Lempert theory applies. Finally, we exhibit an example of a quaternionic automorphism, which is not in the closure of the set of finite compositions of volume preserving quaternionic shears, though its restriction to complex subspace is in the closure of the set of finite compositions of volume preserving complex shears.

**Datum in ura**/ Date and time: 18.2.19

**Predavalnica**/ Location: FAMNIT-MP1

**Predavatelj**/ Lecturer: Medeleine Whybrow (Imperial College London & University of Primorska)

**Naslov**/ Title: Constructing Majorana representations

**Vsebina**/ Abstract:

Majorana theory was introduced by A. A. Ivanov as an axiomatic framework in which to study objects related to the Monster group and its 196884 dimensional representation, the Griess algebra. The objects at the centre of the theory are known as Majorana algebras and can be studied either in their own right or as Majorana representations of certain groups. I will give a brief introduction to the theory before presenting the methods and results of my recent work developing an algorithm to construct the Majorana representations of a given group.

This work is based on a paper by Á. Seress and is joint with M. Pfeiffer.

**Datum in ura**/ Date and time: 5.2.19

**Predavalnica**/ Location: FAMNIT-MP1

**Predavatelj**/ Lecturer: Tadeusz Januszkiewicz (Polish Academy of Sciences, Poland)

**Naslov**/ Title: On k-regular maps

**Vsebina**/ Abstract:

The concept of k-regular maps was introduced into topology by Karol Borsuk in 1950'. Long before it was investigeted in interpolation and approximation theory by (among others) Chebyshev and Kolmogorov, and had distinctly applied mathematics flavor.

A continuous map f: X\to V, from a topological space to a vector space is k-regular of for any k distinct points in X their images are linearly independent.

Example:

If f:X\to V is an embedding, then the map F: X\to R\oplus V, defned by F(x)=(1,f(x))

is 2-regular.

In analogy to theory of embeddings, one of central problems in the study of k-regular maps is (given k, X) to construct a k-regular map of X into space V of minimal possible dimension. Unlike for embeddings, this is interesting already for X=R^d I will discuss lower bounds (obstructions) to the existence to the existence of k-regular maps (this involves some algebraic topology of configuration spaces, following work of many people, most recent being Blagojevic, Cohen, Lueck and Ziegler) and upper bounds (constructions) (this involves some algebraic geometry of secant varieties and Hilbert schemes, following the work of Buczynski, Januszkiewicz, Jelisiejew and Michalek).

**Datum in ura**/ Date and time: 14.1.19

**Predavalnica**/ Location: FAMNIT-MP7 (formerly FAMNIT-POŠTA)

**Predavatelj**/ Lecturer: Nastja Cepak (UP IAM, UP FAMNIT)

**Naslov**/ Title: Blockchain

**Vsebina**/ Abstract:

In 2017 we have been hearing a lot about blockchain, in 2018 a bit less. But what is a blockchain, really? How is it connected to Bitcoin? Why is there the "crypto" in "cryptocurrencies"? And should we, as mathematicians, be even interested in it? We will be talking about these and similar questions during the seminar. We will briefly look at what kind of mathematics a usual blockchain is based on, how it started, and in what ways today's world is trying to use it.

**Datum in ura**/ Date and time: 7.1.19

**Predavalnica**/ Location: FAMNIT-MP7 (formerly FAMNIT-POŠTA)

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

**Naslov**/ Title: Computing Wiener index of a tree from its terminal matrix

**Vsebina**/ Abstract:

In this talk we present an algorithm for computing the Wiener index of a tree from the distance matrix of its leaves, known also as the terminal matrix of a tree. The algorithm performs better than the linear time algorithm for trees that have a large number of vertices of valence 2.