Events

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

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

Speaker: Benoit Charbonneau

"Maple for differential geometry"

While we are certainly competent to do with pen and paper the myriad of computations required by our research, refereeing and our supervision work, I find that using tools can improve speed and accuracy and reduce frustration. I will share some principles and illustrate using Maple, including packages useful for differential geometry: difforms, DifferentialGeometry, and Clifford. Code displayed for this presentation can be found at https://git.uwaterloo.ca/bcharbon/maple-demos

MC 5417

Speaker: Cynthia Dai

"Elliptic curves"

We will study elliptic curves, derive Weistrass short form, and study the law of addition.

You can access the handout at Height Study Seminar | Doing Meow-thematics (cynthiaddai.github.io)

MC 5417

Jashan Bal

Non-Archimedean Polish Groups

We consider the class of Polish non-Archimedean groups, those groups admitting a base at the identity of clopen subgroups. We give a complete characterization of these groups as those groups isomorphic to automorphism groups of countable, first-order structures. Time permitting, we will also discuss van Dantzig's theorem. References include Section 2.4 of Gao's IDST along with Chapter 1 of Becker and Kechris's DST of PGA.

MC 5403

William Gollinger

The Adams Spectral Sequence: Construction of the Adams Spectral Sequence

Now that we are equipped with the context of the stable homotopy category we can perform our construction. We will introduce resolutions of spectra and show how a resolution produces a spectral sequence. Identifying the terms of the resulting spectral sequence is unapproachable without additional assumptions, and we define an Adams Resolution to satisfy some homotopically exhaustive conditions. Using these conditions we can identify our E_2 page in terms of the Ext functor, and the E_\infty page in terms of the (p-completed) homotopy groups.

MC 5417

Faisal Romshoo

The Ebin Slice Theorem

The Ebin Slice Theorem shows the existence of a "slice" for the action of the group of diffeomorphisms $\textrm{Diff}(M)$ on the space of Riemannian metrics $\mathcal{R}(M)$ for a closed smooth manifold $M$. We will see a proof of the existence of a slice in the finite-dimensional case and if time permits, we will go through the generalization of the proof to the infinite-dimensional setting.

MC 5417

Scott Wilson & Joana Cirici

Higher-homotopical BV-structures on the differential forms of symplectic and complex manifolds.

In 1985 Koszul showed that the differential forms of a symplectic manifold have an additional second order operator; part of what is now called a differential BV-algebra. Subsequent work by Getzler, Barannikov-Kontsevich, and Manin describe this structure as a (genus zero) cohomological field theory on the de Rham cohomology, i.e. an action of the compactified moduli space of (genus zero) Riemann surfaces with marked points. Such structures, also known as (formal) Frobenius manifolds, or hypercommutative algebras, have numerous connections with the A-model and mirror symmetry.

In this talk I'll explain a natural generalization of this to (almost) symplectic and complex manifolds using a higher-homotopical notion of BV-algebras. This relies on generalizations of the Kahler identities to these cases. I'll explain the setup, establish the existence of the higher-homotopy BV-structure, and give some explicit examples of almost symplectic and complex manifolds where these higher operations on cohomology are non-zero. Some examples suggest a relationship with ABC-Massey products, defined for complex manifolds.

MC 5417

Peter Crooks, Utah State University

Lie-theoretic constructions in the Moore-Tachikawa category

I will briefly review the Moore-Tachikawa conjecture, as well as the representation theory underlying its formulation. This will lead to an outline of recent, affirmative evidence for the conjecture. I will also detail a systematic association of topological quantum field theories to Lie-theoretic data. Distinguished roles will be played by the partial Grothendieck-Springer resolutions and their Poisson-geometric relatives. This represents joint work with Maxence Mayrand. 

MC5417