Colloquia

The idea of this project comes from two related thoughts:

One of my favorite parts about MCSP were the colloquia. I liked them because I could get a bird's eye view of an area of mathematics rapidly, without getting bogged down into details and spending years staring at a mountain of prerequisites before seeing anything cool. It's been on my mind to recreate these colloquia on my own, and now it's finally happening.
Although I've spent the past couple of years deep diving into cryptography and representation theory, I have explored the equivalent of a speck of dust in all of the universe in both fields. I want to continue exploring more, and look at other areas of math too - but unfortunately time is limited and I won't have enough time to explore the entire universe. Fortunately, colloquia exist for experts to give bird's eye views of their area of expertise, which is perfect for someone like me trying to learn a tiny bit about everything.

The Plenary lectures from ICM 2018 and ICM 2022 are perfect for my goals, so that is probably where I will start.

To get the most out of each talk, I will be doing the Three Things Exercise and logging them on this page.

Knots and Quantum Theory (Edward Witten) - 3/15/24

Source. There is no paper.

I want to remember the following: Given a starting point, a quantum physicist has to "sum" all possible paths by which a particle might reach a destination. How to do this sum is what physicists learned in developing quantum theory and ultimately constructing what is now the Standard Model of particle physics.
I want to remember the connection between the Jones polynomial and quantum physics: if we regard a knot $K$ as the spacetime orbit of a charged particle then the Jones polynomial is the average value of the Wilson operator.
I want to remember that Khovanov homology is a refinement of the Jones polynomial in which the knot is a physical object rather than the path of a point particle. The main difference is that the theory is more abstract: while the Jones polynoial of a knot $K$ is a number $J_K$ , the Khovanov theory associates to a knot a "space of quantum states" $H_K$ .

Source and Slides. There is no base paper.

I want to remember this analogy: representations are like pieces of matter. Simple representations are elements, and semi-simple means the elements don't interact. In representation theory, we search for a classification (periodic table), character formulas (mass, number of neutrons, etc.)...
I want to remember the following conjecture:
(Kazhdan-Lusztig Conjecture, 1979)
$|\Delta_\lambda| = \sum_\mu P_{\lambda, \mu}(1) |L_\mu|$
where $\Delta_\lambda$ is the Verma module, $L_\lambda$ is the simple highest weight module, and $P_{\lambda, \mu} \in \mathbb{Z}[v]$ is a Kazhdan-Lusztig polynomial.
I also want to remember the analogue of Kazhdan-Lusztig for reductive algebraic groups in characteristic $p$ :
(Lusztig Conjecture, 1980)
$|\hat{\Delta}_A| = \sum_B q_{A,B}(1) |\hat{L}_B|.$

I want to remember what Macdonald-Cherednik theory is: Irreducible Lie group characters and, more generally, spherical functions, are eigenfunctions of invariant differential operators, that is, solutions to certain linear differential equations. In Macdonald-Cherednik theory, these are generalized to certain a difference equations associated to root systems and involving additional parameters. Solutions of these equations are remarkable multivariate generalizations of a hypergeometric functions, whose terminating cases are known as the Macdonald polynomials.
I want to remember what the quantum Knizhnik-Zamoldchikov equations are: $R$ -matrices with a spectral parameter define an action of an affine Weyl group of type A by q-difference operators. The lattice part of these q-difference operators are the quantum Knizhnik-Zamoldchikov equations and are among the most important linear equations in mathematical physics because solving theom generalizes the Bethe Ansatz problem.
I want to remember this connection between quivers and quantum groups: Maulik-Okounkov gave a geometric construction of the Yang-Baxter equation and relation equations using their theory of stable envelopes. This associates a new quantum loop group $U_\hbar \hat{g}$ -module. The corresponding Lie algebra $\mathfrak{g}$ is a generalization of the Kac-Moody Lie algebra constructed geometrically by Nakajima.

I want to remember the following idea: Cluster algebra can be interpreted as the ring of global functions on a non-compact Calabi-Yau variety obtained from a toric variety by a blow up construction.
I want to remember what mirror symmetry is: Mirror symmetry is a phenomenon arising in theoretical physics which predicts that Calabi-Yau varieties (together with a choice of Kähler form) come in mirror pairs $U$ and $V$ such that the symplectic geometry of $U$ is equivalent to the complex geometry of $V$ , and vice versa.
I want to remember the following conjecture:
Mirror symmetry defines an involution on the set of positive log Calabi-Yau varieties with maximal boundary. For a mirror pair $U$ and $V$ , there is a basis $\mathfrak{v}_q$ , $q\in U^{\text{trop}}(\mathbb{Z})$ of $H^0(V, \mathcal{O}_V)$ parametrized by the integral points of the tropical set of $U$ , which is canonically determined up to multiplication by scalars $\lambda_q \in \mathbb{C}^\times, q\in U^{\text{trop}}(\mathbb{Z})$ .

I want to remember the following theorem.
(Etingof, Nikshych, and Ostrik)
Up to equivalence, there is a finite number of fusion categories with a given Grothendieck ring.
I want to explore the proof of the following theorem.
(Calegari, Morrison, and Snyder)
Let $X$ be an object in a fusion category such that $\text{FPdim}X$ belongs to the interval $(2, 76/33]$ . Then $\text{FPdim}X$ is equal to one of the following:
$\dfrac{\sqrt{7}+\sqrt{3}}{2}, \sqrt{5}, 1+2\cos \left( \dfrac{2\pi}{7} \right), \dfrac{1+\sqrt{5}}{\sqrt{2}}, \dfrac{1+\sqrt{13}}{2}.$
Moreover, each of these numbers occurs as the Frobenius-Perron dimension of an object of a fusion category.
I want to remember the following definition
A braided fusion category $\Epsilon$ is called Tannakian if $\Epsilon \cong \text{Rep}G$ for some finite group $G$ as symmetric fusion categories.
because the current progress on the question: "Is every fusion category with integer Frobenius-Perron dimension weakly group-theoretical?" relies on the existence of Tannakian subcategories.