# Colloquia

Title: Challenges in High-dimension Genetic Data Analysis

Speaker: Dr. Shelley Bull, Lunenfeld-Tanenbaum Research Institute Sinai Health System

Date and time:
18 Mar 2020,
2:30pm -
3:30pm

Location: CLE A206

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DEPARTMENT LANSDOWNE LECTURE

Abstract

Progress in dissecting the genetic determinants of complex human traits requires cost-efficient genomic technologies, biological knowledge accessible in open data resources, and computationally feasible statistical methods that reliably infer association in the context of high-dimensional multiple testing. It has been estimated that there are 8 million common single nucleotide polymorphisms (SNPs) with variants occurring at a frequency of at least 1% in a population. Well-developed array technologies, widely used in genome-wide association studies (GWAS) of complex traits, can genotype individuals at 1 million SNP positions chosen to capture common variation. Emerging next generation sequencing (NGS) technologies generate genotypes at the nucleotide level, including rare variants, and enable imputation of 8-9 million high-quality unmeasured SNPs from 1 million genotyped SNPs. At the same time, investigators are focussing on use of population biobanks with many traits available on each individual. High-dimensional GWAS, NGS, and trait data, together with variant, gene, and biological pathway annotation resources are increasingly available for data mining and meta-analysis, but present major analytic challenges including joint analysis of multiple data types, use of biological knowledge to inform statistical modelling, and choice of study design. In this talk, I will highlight some of these challenges in two settings: joint modelling of multiple traits in a longitudinal study of unrelated individuals; and discovery of rare mutations in families ascertained on disease status.

Title: Statistical Approaches to Discovery of Genetic Association in Human Populations

Speaker: Dr. Shelley Bull, Lunenfeld-Tanenbaum Research Institute Sinai Health System

Date and time:
17 Mar 2020,
2:30pm -
3:30pm

Location: DTB A104

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PUBLIC LANSDOWNE LECTURE

Abstract

Human genomic DNA is composed of approximately 3 billion nucleotides in a linear string, organized into 23 chromosome pairs. Genome-wide association study (GWAS) analysis takes a comprehensive agnostic screening approach to discover genetic variants that are associated with complex traits such as physiological measures or disease status. Typically, the analysis proceeds by examining association of a trait with each genetic variant, one at a time. This involves making several million statistical tests across the genome. Motivated by genetic architectures in which analysis of sets of variants within a genomic region can uncover associations missed by single-variant analysis, in this talk I will describe a two-stage strategy for region-level genetic association testing. In the first stage, genomic regions are identified by computationally partitioning each chromosome into blocks according to local correlation structure known as linkage disequilibrium. Within each of the blocks, statistical association analysis then proceeds in stage two using multiple-variant methods. I will discuss some of the analytic issues we face in applying this strategy in association analysis of quantitative traits and disease status using dense GWAS genotype data.

Title: Combinatorics and representation theory of diagram algebras

Speaker: Dr. Zajj Daugherty, City College of New York

Date and time:
03 Feb 2020,
3:30pm -
5:00pm

Location: COR B112

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Abstract:
In his early work, Schur constructed a powerful link between the symmetric group and the general linear group via commuting actions on a common vector space. Much more recently, there has been a cascade of developments surrounding diagram algebras that share commuting actions with other Lie groups, Lie algebras, and deformations thereof, with consequences in many fields, including representation theory, combinatorics, knot theory, and statistical mechanics. We will take a brief tour of some important examples of diagram algebras, and discuss how seeing them from Schur's perspective can reveal a wealth of information.

Title: Determinants, Schubert varieties, and quiver representation varieties

Speaker: Dr. Jenna Rajchgot, University of Saskatchewan

Date and time:
31 Jan 2020,
3:30pm -
5:00pm

Location: COR B112

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Abstract:
Determinantal varieties are algebraic varieties defined by the simultaneous vanishing of minors of matrices. They are important in algebraic geometry because many naturally occurring algebraic varieties have determinantal structure.

After reviewing some of these ideas, I'll focus on two such families of algebraic varieties:
(i) Schubert varieties in flag varieties and multiple flag varieties; and
(ii) representation varieties of Dynkin quivers.
The study of each family is motivated by questions in algebraic geometry and representation theory and has led to beautiful combinatorics.
Furthermore, each family contains, as special cases, classes of determinantal varieties of independent interest.

I'll discuss an ongoing research program on unifying problems about the equivariant geometry and combinatorics of representation varieties of Dynkin quivers with the corresponding problems for Schubert varieties in multiple flag varieties (joint with R. Kinser), as well as some consequences on combinatorial formulas for degeneracy loci (joint with R.
Kinser and A. Knutson).

Title: Applications of Cayley Digraphs to Waring's Problem and Sum-Product Formulas

Speaker: Dr. Yesim Demiroglu Karabulut, Harvey Mudd College

Date and time:
30 Jan 2020,
3:30pm -
5:00pm

Location: CLE B007

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Abstract: In this talk, we first present some elementary new proofs (using Cayley digraphs and spectral graph theory) for Waring's problem over finite fields, and explain how in the process of proving these results, we obtain an original result that provides an analogue of S{\'a}rk{\"o}zy's theorem in the finite field setting (showing that any subset $E$ of a finite field $\Bbb F_q$ for which $|E| > \frac{qk}{\sqrt{q - 1}}$ must contain at least two distinct elements whose difference is a $k^{\text{\tiny th}}$ power). Once we have our results for finite fields, we can apply some classical mathematics to extend our Waring's problem results to the context of general (not necessarily commutative) finite rings. In the second half of our talk, we talk about some sum-product formulas related to matrix rings over finite fields, which can again be proven using Cayley digraphs and spectral graph theory in an efficient way.

Title: Bootstrap percolation and related graph processes

Speaker: Dr. Natasha Morrison, University of Cambridge, UK

Date and time:
27 Jan 2020,
3:30pm -
5:00pm

Location: COR B112

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Abstract:

Bootstrap percolation processes are a family of cellular automata that were originally introduced in 1979 by the physicists Chalupa, Leith and Reich as a model of ferromagnetism. Since then, it has been used throughout a variety of disciplines to model real world phenomena such as the spread of influence in a social network, information processing in neural networks or the spread of a computer virus.

The $r$-neighbour bootstrap percolation process on a graph $G$ starts with an initial set $A$ of \emph{infected} vertices and, at each step of the process, a \emph{healthy} vertex becomes infected if it has at least $r$ infected neighbours (once a vertex becomes infected, it remains infected forever). If every vertex of the graph eventually becomes infected, we say that $A$ \emph{percolates}.

In this talk I will discuss the solution to a conjecture of Balogh and Bollob\'{a}s that concerns the $r$-neighbour bootstrap process on the $d$-dimensional hypercube. I will also discuss results on related processes, including the weak saturation process introduced by Bollob\'{a}s in 1968, in both the extremal and probabilistic settings.

This talk contains work joint with Gal Kronenberg, Taisa Martins, Jonathan Noel and Alex Scott.