Mathematical Institute University of Oxford

TBA

I will describe a method that deforms analytic cycles to algebraic cycles. This will be done using global derived analytic geometry and analytic (Efimov) K-theory. I will conjecture how this is related to the Tate and Hodge conjectures. If time permits, I will speculate about a global arithmetic site. This is joint work in progress with Federico Bambozzi, Jack Kelly, and Devarshi Mukherjee.

The Kervaire conjecture was formulated around 1963 after a conversation between Kervaire and Baumslag. It states that adding a generator and then a relator to a non-trivial group always yields a non-trivial group. To this day, the conjecture remains unproven in its most general form; however, it has been shown under certain additional hypotheses, either on the new relator or on the original group. For instance, the result holds for locally indicable groups and for locally residually finite groups. In this talk, I will explain Klyachko’s proof of the conjecture for torsion-free groups, which uses a funny property of the sphere known as the Car Crash Theorem, and van Kampen pictures. I will also discuss how these techniques were generalised by Fenn and Rourke to study equations over torsion-free groups defined by a large class of words (amenable words).

In 1982, Conway introduced the angel-devil game, which is played on an infinite chess board.  For fixed k, the angel moves at most distance k from its current position on its turn.  The devil then blocks a square permanently.  The devil wins if the angel eventually has no legal moves left.  Berlekamp showed the devil wins against the 1-angel.  Conway asked whether there exists k such that the k-angel has a winning strategy against the devil.  This was resolved independently by Kloster, Máthé, and Bowditch in 2006.  Bowditch proposed playing the game on Cayley graphs of finitely generated groups.  A group for which the devil beats the k-angel for every k is called diabolical.  We will explore the ends of these diabolical groups.

Gromov-hyperbolic groups are classically defined geometrically, by the negative curvature of their Cayley graphs. Interestingly, an algorithmic characterization of hyperbolicity is possible in terms of properties of the formal languages of quasigeodesics (geodesics up to bounded error) in their Cayley graphs. Holt and Rees proved, roughly speaking, that these formal languages are regular in the case of hyperbolic groups. More recently Hughes, Nairne, and Spriano established the converse. In this talk, I will discuss progress towards a conjectured strengthening of the result, where we consider context-free quasigeodesic languages. This is based on my summer project, supervised by Joseph MacManus and Davide Sprianoc

In this talk, I will introduce the concept of Archimedean closedness - a concept from real non-commutative algebraic geometry which determines when "positivity" of an element (captured through *-representations) in a *-algebra can be completely certified algebraically. On the other hand, property FD is a representation theoretic property of groups depicting when any representation of a group can be approximated by finite representations in the unitary dual. I will try to connect these two seemingly very different concepts through some examples and speculations. This is a work in progress.

Skein lasanga modules are a smooth 4-manifold invariant that was introduced by Morrison, Walker and Wedrich using Khovanov homology. This invariant was recently used by Ren and Willis to give the first analysis free proof of the existence of exotic 4-manifolds. However, even for simple handlebodies it remains difficult to compute. A generalisation was introduced by Chen using Knot Floer homology, which in principle should be easier to compute due to cabling formulas for knot Floer homology. I will give a general introduction to lasagna modules assuming no knowledge of Khovanov or knot Floer homology, and then explain some methods, from upcoming work, for computing Floer Lasagna modules.

We give an introduction to a rigorous renormalization group analysis of the sine-Gordon model with a focus on deriving the lowest-order beta function.