Mihail Hurmuzov

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Profile

Biography

I received my Master's (MMathPhil) from the University of Oxford, where I read Mathematics and Philosophy at St. Hilda's. I spent two years as a PhD student at UIC in Chicago before transferring to York in 2018. I completed a PhD in Mathematics in 2022 under the supervision of Brent Everitt. My thesis work was on a spectral sequence for sheaf cohomology. My first postdoc was as a Research Associate at York, working with Brent on applications of the thesis result and on understanding the grid homology of knots and links.

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Research

Overview

My interests are in algebra, topology, and combinatorics, in particular the combinatorial and homological aspects of knot invariants. Below are some past, current, and future projects that I am interested in.

Knots, graphs and the volume conjecture: while at UIC, I worked with Laura Schaposnik on some well-known connections between knot invariants and graph invariants. A connected planar graph can be turned into two different alternating links via the medial graph construction and François Jaeger's `double medial' construction. The Tutte polynomial of the starting graph then relates to the Kauffman bracket and the HOMFLY polynomial of the two links, respectively, giving a connection between the two invariants.

At the same time, I spent some time on the volume conjecture, under the supervision of Louis Kauffman. The conjecture claims a correspondence between a certain limit of the `coloured Jones polynomials' of a hyperbolic knot and the hyperbolic volume of its complement.

Fig. 1: The two-cells of a hyperbolic structure on the complement of the Borromean rings.

Khovanov homology – My PhD project was initially on converting the setup of Brent and Paul Turner's 2012 paper "Bundles of coloured posets and a Leray-Serre spectral sequence for Khovanov homology" into a cohomological setting. The theorem that came out of that work was, surprisingly, not applicable to Khovanov homology, but instead gave a general reduction property of sheaf cohomology on certain posets. In particular, it extends the usual cohomological reduction of posets with a unique maximum.

Fig. 2: The sheaf cohomology of any sheaf (E,F) on this poset is concentrated in E_≥x.

We are not aware of any attempts to calculate unknown values of Khovanov homology using the spectral sequence in the 2012 paper. One can conceivably use this machinery to produce at least bounds on which cells are non-zero in the Khovanov homology of certain families of links. Even more, the spectral sequence seems a good fit to explain the phenomenon where the Khovanov homologies of similar torus links coincide (up to a shift in q-degree) in the first several columns.

Fig. 3: Stability in the rational Khovanov homology of the torus knots T(8,5) and T(9,5).

Grid homology – I am currently considering, along with Brent, how a treatment similar to that of Khovanov homology in "Bundles of coloured posets" can be applied to grid homology, as presented in Ozsvath, Stipsicz, and Szabo's "Grid Homology for Knots and Links".

Teaching

Undergraduate

Since 2016 I have worked as a teaching assistant on a plethora of modules, including:

Real Analysis
Linear Algebra
Number Theory
Cryptography
Functions of a Complex Variable
Vector Calculus

I am particularly interested in innovative teaching methods, employing a variety of technological solutions (interactive online quizzes, real-time feedback) aimed at creating an engaging and inclusive learning environment for all students, regardless of their background or learning style. In 2020 I received an Associate Fellowship of HE with a teaching portfolio partly based on my teaching work during the pandemic.