# Mathematical Research Seminar - Archive

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**Datum in ura**/ Date and time: 1.3.21

**Predavalnica**/ Location: Zoom

**Predavatelj**/ Lecturer: Safet Penjić (UP FAMNIT and UP IAM, Slovenia)

**Naslov**/ Title: On a certain families of a symmetric association schemes with d+1 classes

**Vsebina**/ Abstract:

Let Γ denote an undirected, connected, regular graph with vertex set X, adjacency matrix A, and d+1 distinct eigenvalues. Let {\cal A}={\cal A}(Γ) denote the subalgebra of Mat_X(C) generated by A. We refer to {\cal A} as the {\em adjacency algebra} of Γ. In this talk, we investigate the algebraic and combinatorial structure of Γ for which the adjacency algebra {\cal A} is closed under Hadamard multiplication. In particular, under this simple assumption, we show the following: (i) {\cal A} has a standard basis {I,F_1,…,F_d}; (ii) for every vertex there exists identical distance-faithful intersection diagram of Γ with d+1 cells; (iii) the graph Γ is quotient-polynomial; and (iv) if we pick F∈{I,F_1,…,F_d} then F has d+1 distinct eigenvalues if and only if span{I,F_1,…,F_d}=span{I,F,…,

This is joint work with Miquel À. Fiol.

This talk is based on one part of the preprint available at https://arxiv.org/abs/2009.

We are looking forward to meeting at the video-conference. Join Zoom Meeting HERE!

Everyone is welcome and encouraged to attend.

For more info visit our YouTube Channel.

**Datum in ura**/ Date and time: 22.2.21

**Predavalnica**/ Location: Zoom (See link below)

**Predavatelj**/ Lecturer: Luke Morgan (UP FAMNIT)

**Naslov**/ Title: Shuffling cards with groups

**Vsebina**/ Abstract:

There are two standard ways to shuffle a deck of cards, the in and out shuffles. For the in shuffle, divide the deck into two piles, hold one pile in each hand and then perfectly interlace the piles, with the top card from the left hand pile being on top of the resulting stack of cards. For the out shuffle, the top card from the right hand pile ends up on top of the resulting stack.

Standard card tricks are based on knowing what permutations of the deck of cards may be achieved just by performing the in and out shuffles. Mathematicians answer this question by solving the problem of what permutation group is generated by these two shuffles. Diaconis, Graham and Kantor were the first to solve this problem in full generality - for decks of size 2*n. The answer is usually “as big as possible”, but with some rather beautiful and surprising exceptions. In this talk, I’ll explain how the number of permutations is limited, and give some hints about how to obtain different permutations of the deck. I’ll also present a more general question about a “many handed dealer” who shuffles k*n cards divided into k piles.

We are looking forward to meeting at the video-conference. Join Zoom Meeting HERE!

Everyone is welcome and encouraged to attend.

For more info visit our YouTube Channel.

**Datum in ura**/ Date and time: 15.2.21

**Predavalnica**/ Location: ZOOM (See link below)

**Predavatelj**/ Lecturer: Francesca Scarabel (York University, Canada)

**Naslov**/ Title: A renewal equation model for disease transmission dynamics with contact tracing

**Vsebina**/ Abstract:

I will present a deterministic model for disease transmission dynamics including diagnosis of symptomatic individuals and contact tracing. The model is structured by time since infection and formulated in terms of the individual disease rates and the parameters characterising diagnosis and contact tracing processes. By incorporating a mechanistic formulation of the processes at the individual level, we obtain an integral equation (delayed in calendar time and advanced in the time since infection) for the probability that an infected individual is detected and isolated at any point in time. This is then coupled with a renewal equation for the total incidence to form a closed system describing the transmission dynamics involving contact tracing. After presenting the derivation of the model, I will conclude with some applications of public health relevance, especially in the context of the ongoing COVID-19 pandemic.

Joint work with Lorenzo Pellis (University of Manchester), Nicholas H Ogden (PHAC, Public Health Agency of Canada) and Jianhong Wu (York University).

Reference: Scarabel F, Pellis L, Ogden NH, Wu J. A renewal equation model to assess roles and limitations of contact tracing for disease outbreak control, available on medRxiv: https://doi.org/10.1101/2020.1

We are looking forward to meeting at the video-conference. Join Zoom Meeting HERE!

Everyone is welcome and encouraged to attend.

For more info visit our YouTube Channel.

**Datum in ura**/ Date and time: 11.1.21

**Predavalnica**/ Location: ZOOM (See link below)

**Predavatelj**/ Lecturer: Petr Golovach (Bergen University, Norway)

**Naslov**/ Title: Paths and Cycles Above Combinatorial Bounds

**Vsebina**/ Abstract:

*An undirected graph G d-degenerate if every subgraph of G has a vertex of degree at most d, and the degeneracy is the minimum d such that G is d-degenerate. By the classical theorem of Erdös and Gallai from 1959, every graph of degeneracy d>1 contains a cycle of length at least d+1. The proof of Erdös and Gallai is constructive and can be turned into a polynomial time algorithm constructing a cycle of length at least d+1. But can we decide in polynomial time whether a graph contains a cycle of length at least d+2? An easy reduction from Hamiltonian Cycle provides a negative answer to this question: Deciding whether a graph has a cycle of length at least d+2 is NP-complete. Surprisingly, the complexity of the problem changes drastically when the input graph is 2-connected. In this case we prove that deciding whether G contains a cycle of length at least d+k can be done in time 2^{O(k)}|V(G)|^{O(1)}. Further, we briefly consider similar problem arising from the classical result of Dirac from 1952 that every 2-connected n-vertex graph has a cycle with at least min{2\delta(G),n} vertices. *

*The talk is based on the joint work with Fedor V. Fomin, Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, Meirav Zehavi, Danil Sagunov, and Kirill Simonov.*

We are looking forward to meeting at the video-conference. Join Zoom Meeting HERE!

Everyone is welcome and encouraged to attend.

For more info visit our YouTube Channel.

**Datum in ura**/ Date and time: 4.1.21

**Predavalnica**/ Location: ZOOM (See link below)

**Predavatelj**/ Lecturer: Ted Dobson (UP FAMNIT, Slovenia)

**Naslov**/ Title: Recognizing vertex-transitive digraphs which are wreath products, double coset digraphs, and generalized wreath products

**Vsebina**/ Abstract:

It is known that a Cayley digraph $\Cay(A,S)$ of an abelian group $A$ is isomorphic to a nontrivial wreath product if and only if there is a proper nontrivial subgroup $B\le A$ such that $S\setminus B$ is a union of cosets of $B$ in $A$. We generalize this result to Cayley digraphs $\Cay(G,S)$ to nonabelian groups $G$ by showing that such a digraph is isomorphic to a nontrivial wreath product if and only if there is a proper nontrivial subgroup $H\le G$ such that $S\setminus H$ is a union of double cosets of $H$ in $G$. This result is proven in the more general situation of a double coset digraph (also known as a Sabidussi coset digraph.) We then give applications of this result which include showing the problem of determining automorphism groups of vertex-transitive digraphs is equivalent to the problem of determining automorphism groups of Cayley digraphs, and extending the definition of generalized wreath product digraphs to double coset digraphs of all groups $G$.

Join Zoom Meeting HERE!

Everyone is welcome and encouraged to attend.

For more info visit our YouTube Channel.