Mathematical Research Seminar - Archive

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,…,F^d}. We describe the combinatorial structure of quotient-polynomial graphs with diameter 2 and 4 distinct eigenvalues.

This is joint work with Miquel À. Fiol.

This talk is based on one part of the preprint available at https://arxiv.org/abs/2009.05343

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.

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.

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/10.1101/2020.12.27.20232934

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.