Things I Learned From CUMC 2022

“Why did the two algebraic geometers get arrested at the airport? Because they were talking about blowing up six points on the plane.”

Anonymous

I spent my last week in Quebec attending the Canadian Undergraduate Mathematics Conference 2022. It was really an intense week of learning. I learned a lot and met many smart and interesting people. So I wanted to write about the things I learned and the people I met during the conference. There were so many excellent talks but unfortunately I could only attend some of them. In particular, there are some very good ones in geometry, topology, representation theory, algebra, logic theory, etc.

1. When Polynomials don’t Commute: An Introduction to Ore Extensions by Nick Priebe. Non-commutative algebra is a central topic in modern algebra research, which concerns objects such as non-commutative rings. Suppose \(R\) is a ring, \(\sigma:R\rightarrow R\) a homomorphism, \(\delta:R\rightarrow R\) a \(\sigma\)-derivation i.e. a homomorphism s.t. \(\delta(rs)=\sigma(r)\delta(s)+\delta(r)s\), then an Ore extension or a skew polynomial ring is the noncommutative ring \(R[x;\sigma,\delta]\) where the multiplication satisfies \(xr=\sigma(r)x+\delta(r)\). Nick provided some examples of noncommutative rings that are useful in physics. He also introduced some basic results and open problems about their representations. I am actually quite curious about the relationship between noncommutative algebra and noncommutative geometry.
2. A Brief Introduction to Homotopy Type Theory by Jacob Ender. Homotopy type theory is a new foundation of mathematics which is developed from intuitionist type theory. In HoTT, the fundamental objects are types, and we can build dependent types with \(\sum_{(x:A)}B(x)\) (analogue of \(\exists x\in A,B(x)\)) and \(\prod_{(x:A)}B(x)\) (analogue of \(\forall x\in A,B(x)\)). To avoid Russell’s paradox, we can build a tower of universes. Jacob also introduced Voevodsky’s Univalence axiom \((A= B)\simeq (A\simeq B)\) so isomorphic objects are equal. HoTT is very analogous to homotopy theory in the sense of its \(\infty\)-groupoid structure. Currently there is a lot of research in their higher-categorical models and their application in computer proof assistants.
3. Waring Squaring by Maya Gusak. A classical result in number theory is that each natural number is the sum of at most four squares. Waring’s problem asks for which \(k\in\mathbb N\) exists \(s\in\mathbb Z^+\) s.t. each \(n\in\mathbb N\) can be written as a sum \(n=n_1^k+n_2^k+\cdots+n_s^k\), where \(n_1,\dots,n_s\in\mathbb N\). To tackle this problem, mathematicians used so called circle method. Additionally, this method can be used to prove Vinogradov’s theorem (every odd natural number \(n>5\) is the sum of three primes), but fails to solve Goldbach conjecture.
4. ADE Classification for Quivers of Finite Representation Type by Xinrui You. A quiver is a multidigraph (i.e. a directed graph with multiple edges and loops allowed) which simulates underlying structure of a category. A representation of a quiver sends each vertex to a vector space or a module and each edge to a morphism. These representations decompose just as representations of finite groups. One fundamental result in the classification of quivers is Gabriel’s theorem. Gabriel’s theorem says that a quiver of finite representation type is a union of Dynkin graphs of type A, D, or E.
5. The Nash Embedding Theorem for Tori in \(3\)-space by Diba Heydary. A fundamental result in Riemannian geometry is that every Riemannian manifold can be isometrically (preserving the length of every path) embedded into a Euclidean space. Diba talked about a specific example of an embedding of a torus in the Euclidean \(3\)-space. A torus has a polygonal representation which can be used to create this embedding. Whitney embedding theorem is a similar result for differentiable manifolds.
6. Dissection of Polyhedra and the Dehn invariant by James Bona-Landry. The Wallace–Bolyai–Gerwien theorem says that that two polygons can be dissected and transformed to each other iff they have the same area. The natural generalization of this question into \(3\) dimensions does not hold. This is due to the Dehn invariant \(\sum_i\ell_i\otimes \theta_i\in \mathbb R\otimes (\mathbb R/2\pi \mathbb Z)\). We can show that there are polyhedra that share same volume but has different Dehn invariant, which means they can not transform to each other via dissection. In fact the Dehn-Sydler theorem says that they can transform to each other via dissection iff they have the same volume and same Dehn invariant. The generalization for higher dimensions is still open.
7. An Introduction to Linear Logic by Amélie Comtois. Linear logic is different from classical logic in that if \(A\rightarrow B\) is used then \(A\) is no longer available anymore. She also talked about sequent calculus, which is a formal calculus of logic. Linear logic uses drastically different logical symbols than classical logic as well. Linear logic is an extension of classical logic, and classical logic can be realized in linear logic. Linear logic is thought of as the language of quantum theory. Moreover, I discussed with Amélie about category theory, and I learned from her about multicategories and higher categories.
8. Exploring Quivers, Representations, and Varieties via Multisegments by Iretomiwa Ajala. This is another talk about quivers, and much more technical than that of Xinrui. Iretomiwa introduced multisegments, and discussed how they related to quivers of type A. He also mentions parabolic induction, which is a method of obtaining a representation of a reductive group from the representations of its parabolic subgroups.
9. An Introduction to Coxeter Groups and the Properties of their Weak Order by Kimia Shaban. A very interesting talk on the combinatorial side of Coxeter groups. Coxeter groups are a type of groups generated by reflections. She discussed their classification, and the Sperner property of the weak order on Coxeter groups. In the end, she also talked about a conjecture she and her supervisor wish to disprove about the Sperner property of the weak order.
10. Univariate Tropical Polynomials by Charlotte Lavoie-Bel. Tropical polynomical replace addition with the minimum function and multiplication by the usual addition, which makes it a semifield. Tropical polynomials can be graphed and they will have a cell-looking structure. They can be thought of as the limit of logarithm under base \(0\). They also have a different notion of roots than regular polynomials. We can consider zero locus of tropical polynomials i.e. tropical varieties. The tropical version of some classical algebraic geometry theorems such as Riemann—Roch theorem, can be proven by combinatorial arguments. I like her quote that “tropical geometry is the combinatorial shadow of classical algebraic geometry”.

There are many other excellent talks that I unfortunately did not attend, and I only wrote about some of the talks I went to. I also gave my own talk on Schubert calculus and cohomology of Grassmannians as the last student speaker of the conference, which you can view in a previous post.