Events

Aareyan Manzoor

Metrics on Polish groups

We investigate various compatible metrics that a Polish group admits. By Birkhoff-Kakutani, every Polish group admits a compatible left-invariant metric, and by being a Polish space, there is also a compatible metric which is complete. We discuss the relationship between these metrics, and introduce the class of CLI Polish groups, those Polish groups which admit a single compatible metric which is both left invariant and complete. We will mostly follow Section 2.2 of Gao's IDST book.

MC5403

Speaker: William Gollinger

"The Adams Spectral Sequence"

After having a week off, in this talk we will review concepts in homotopy theory. This will include: fibrations and cofibrations; CW complexes; Eilenberg-MacLane spaces; homotopy fibers and related constructions; the Postnikov tower of a space.

MC5417

William Gollinger

The Adams Spectral Sequence

After having a week off, in this talk we will review concepts in homotopy theory. This will include: fibrations and cofibrations; CW complexes; Eilenberg-MacLane spaces; homotopy fibers and related constructions; the Postnikov tower of a space.

MC5417

Speaker: Filip Milidrag, University of Waterloo

"The Classification of Irreducible Discrete Reflection Groups"

In this talk we will make a correspondence between irreducible discrete reflection groups and associated connected Coxeter diagrams. Then we will use this to classify all connected Coxeter diagrams and by extension every irreducible discrete reflection group.

MC 5501

Utkarsh Bajaj

Klein's icosahedral function

Can we define a rational function on the sphere? Sure we can. Can we define a rational function on the sphere so that it is invariant under the rotational symmetries under the icosahedron? Yes - by embedding the icosahedron in the Riemann sphere (and then doing some algebra). We then show how this beautiful function reveals connections between the symmetries of the icosahedron and the E8 lattice  - the lattice that gives the most efficient packing of spheres in 8 dimensions!

MC 5501

Keira Gunn (University of Calgary)

Some Problems on the Dynamics of Positive Characteristic Tori.

The real (or characteristic zero) torus is simply R/Z, or the "decimal part" of any real number with operations of addition and integer multiplication.  With the positive characteristic integers defined to be polynomials with coefficients from a finite field, and the positive characteristic reals their Laurent series counterparts, we can similarly construct the positive characteristic tori (each torus dependent on the choice of field).  At first glance there are many similarities to how operations work in both positive and zero characteristic, but these similarities break down quickly upon further inspection, particularly from a view of dynamics on the tori.

In this talk, we will discuss results on some orbital sets and dynamics formulae on the positive characteristic tori, including the Artin-Mazur zeta function and analogous hypothesis for Furstenberg's Orbital Theorem.

MC5403

Speaker: Utkarsh Bajaj

"Klein's icosahedral function"

Can we define a rational function on the sphere? Sure we can. Can we define a rational function on the sphere so that it is invariant under the rotational symmetries under the icosahedron? Yes - by embedding the icosahedron in the Riemann sphere (and then doing some algebra). We then show how this beautiful function reveals connections between the symmetries of the icosahedron and the E8 lattice  - the lattice that gives the most efficient packing of spheres in 8 dimensions!

MC 5501

Filip Milidrag

The Classification of Irreducible Discrete Reflection Groups

In this talk we will make a correspondence between irreducible discrete reflection groups and associated connected Coxeter diagrams. Then we will use this to classify all connected Coxeter diagrams and by extension every irreducible discrete reflection group.

MC 5501

Speaker: Keira Gunn, University of Calgary

"Some Problems on the Dynamics of Positive Characteristic Tori."

The real (or characteristic zero) torus is simply R/Z, or the "decimal part" of any real number with operations of addition and integer multiplication.  With the positive characteristic integers defined to be polynomials with coefficients from a finite field, and the positive characteristic reals their Laurent series counterparts, we can similarly construct the positive characteristic tori (each torus dependent on the choice of field).  At first glance there are many similarities to how operations work in both positive and zero characteristic, but these similarities break down quickly upon further inspection, particularly from a view of dynamics on the tori.

In this talk, we will discuss results on some orbital sets and dynamics formulae on the positive characteristic tori, including the Artin-Mazur zeta function and analogous hypothesis for Furstenberg's Orbital Theorem.

MC5403

Research Area: Algebraic Geometry/Number Theory

Speaker: Cynthia Dai

"Group Varieties"

Abstract: 

We will define group varieties and study their basic properties. 

You can access the handout at https://cynthiaddai.github.io/2024/05/17/HeightSeminar/

MC 5417