Transformation Groups Seminar

Speaker: Tao Gong (Western)

"Affine toric varieties and cones"

Time: 10:30

Room: MC 204

In this lecture, I will give an introduction to toric varieties and cones, and propositions and relations about them.

Algebraic Geometry

Speaker: Kumar Shukla (Western)

"WAG: Degree and dimension"

Time: 15:30

Room: MC 107

Geometry and Topology

Speaker: Sterling Ebel (Western)

"Synthetic approach to the Quillen model structure on spaces"

Time: 15:30

Room: MC 107

Quillen's construction of a model structure on the category of topological spaces is a fundamental result in homotopy theory. This construction has since been applied to several related categories, such as k-spaces, and the importance of many model categories is justified by their equivalence with Quillen's structure on spaces.

In this talk, we will present an axiomatic approach to constructing Quillen's model structure on spaces to apply it to a wider range of settings. As special cases we recover several existing model structures, such as on the categories of sober spaces and of pseudotopological spaces. We also use this approach to construct a novel model structure on the category of locales, making the coreflection to sober spaces a Quillen adjunction.

This is joint work with Chris Kapulkin (arXiv:2310.14235).

Transformation Groups Seminar

Speaker: Tao Gong (Western)

"Affine toric varieties and cones (Part 2)"

Time: 10:30

Room: MC 204

In this lecture, I will give an introduction to toric varieties and cones, and propositions and relations about them.

Algebraic Geometry

Speaker: Nickolas Rieke (Western)

"WAG: Affine Varieties"

Time: 15:30

Room: MC 107

Pizza Seminar

Speaker: Chris Kapulkin (Western)

"An invitation to constructive mathematics"

Time: 17:00

Room: MC 107

The classical understanding of logic and, by extension, of mathematics is based upon the notion of truth. In that view, the role of a mathematical proof is to establish the truth of some statement, at which point the proof itself can be discarded. For instance, when computing a derivative, we apply the chain rule, not its proof.

Constructive mathematics reverses this approach, focusing primarily on the notion of a proof itself. It asks that all proofs be effective, thus rejecting classical principles such as the law of excluded middle. Initially seen as a largely philosophical position, constructive mathematics has come back to the forefront, since it is precisely the kind of mathematics that computers can understand.

In this talk, we will explore the main principles of constructive mathematics as developed by early constructivists: Brouwer and Heyting, and see how it is used these days by mathematicians around the world who wish to verify the correctness of their proofs using computers.

Geometry and Topology

Speaker: Ajneet Dhillon (Western)

"Grothendieck-Lefschetz theorems in algebraic geometry"

Time: 15:30

Room: MC 107

This talk is an overview of a genre of theorems called "Grothendieck-Lefschetz theorems". I will start by introducing Lefschetz's original theorem. Our next, stop is Grothendieck's theorem for divisors and line bundles which can be though of as a codimension one Lefschetz theorem in the algebraic category.

From this point there are two natural paths to generalisation. The first is to higher codimension subvarieties. In this direction there are some infinitesimal results due to Girivaru-Patel.

The second path considers higher rank bundles. The first results in this direction are due to Hartshorne. In joint work with R. Girivaru, Hartshorne's work has been generalised to principal bundles. The talk will end with some applications of our result to splitting criteria.

Graduate Seminar

Speaker: Michelle Hatzel (Western)

"Continuation Methods for Numerical Problem Solving"

Time: 16:30

Room: MC 107

Informally, if two functions can be “continuously deformed” from one to the other, this is called a homotopy. Homotopy emerged from theory more than a century ago and was introduced as a numerical method for solving non-linear problems in the 1960s. The basic components of these early continuation algorithms built on earlier path-tracking methods, which exist in today’s “black box” solvers. We will look at the building blocks of continuation algorithms, how they work (or don’t), and how key insights from the 1970s and 1980s contributed to some powerful polynomial-solving software packages.

Transformation Groups Seminar

Speaker: Larry So (Western)

"Toric varieties of fans"

Time: 10:30

Room: MC 204

I will define toric varieties associated to fans and give a few examples.

Algebraic Geometry

Speaker: Brett Nasserden (Western)

"WAG: Projective Varities"

Time: 15:30

Room: MC 107

Graduate Seminar

Speaker: Tao Gong (Western)

"What is the space of conjugacy classes?"

Time: 16:30

Room: MC 107

When G is a topological group, G acts on itself by conjugation. This leads to the natural question: what is the quotient space of conjugacy classes? In this talk, we will work out the answer to this question in the special case when G is a connected compact simple Lie group – the space is contractible!

Algebraic Geometry

Speaker: Ajneet Dhillon (Western)

"WAG: Low dimensional varieties"

Time: 15:30

Room: MC 107

Transformation Groups Seminar

Speaker: Larry So (Western)

"The quotient construction and homogeneous coordinates of toric varieties"

Time: 10:30

Room: MC 108

Algebraic Geometry

Speaker: Kate Kim (Western)

"WAG: 0-dimensional ideals"

Time: 15:30

Room: MC 108

Graduate Seminar

Speaker: Marwa Tuffaha (Western)

"Mutator dynamics cannot be explained by mutation rates alone"

Time: 16:30

Room: MC 107

Mutators, cells with elevated mutation rates, are common in both natural microbial populations and in human cancers. Recent experiments have shown that mutators can invade a population, but the invasion dynamics and probability couldn’t be explained by mutation rates alone. Here we show, analytically and in simulation, that mutation bias (which types of mutations are likely to occur) can play an important role in the emergence of mutators. A mutator that reduces or reverses the historically prevailing mutational bias is shown to have an increased chance of invasion, while chances are reduced when the bias is reinforced. These findings are important when trying to understand natural populations or competition experiments with mutators.