Let T be an n-dimensional torus acting on a ‘nice’ 2n-manifold M effectively, with nonempty set of fixed points and suppose that all the isotropy groups are connected. If the action satisfies another hypothesis (equivariant formality) then the quotient space M/T has the structure of a homology cell complex and is in fact a homology disk. In this talk we will give first discuss some general facts about orbits of torus actions on manifolds and about locally standard actions. Then using the Atiyah-Bredon-Franz-Puppe sequence we give a detailed proof of the fact that under the assumptions stated above, M/T is a homology disk.

A compact $K\subset\mathbb{C}^n$ is called rationally convex if for every point $p\not\in K$ there is a polynomial $P$ with $P(z)=0$ but $P^{-1}(0)\cap K=\varnothing$. Rational convexity is important in view of the Oka--Weil theorem, which states that a holomorphic function defined in a neighbourhood of a rationally convex compact $K$ is the uniform limit on $K$ of a sequence of rational functions. $$ $$ It is not obvious how to generalize rational convexity to the setting of a Stein manifold $X$. For instance, what should play the role of the polynomial? A first guess would be to replace the polynomial with a holomorphic function on $X$, but a second guess would be to replace the zero set of the polynomial with a general analytic hypersurface in $X$. In Euclidean space, it is well known that every analytic hypersurface has a global representation as the zero set of an entire function, so these notions coincide. $$ $$ In this talk, we will compare and contrast these different notions of rational convexity and flesh out their relevant properties. In particular, each generalization has its own version of an Oka--Weil theorem. We will also explore connections to weak and strong meromorphic functions. This is joint work with Rasul Shafikov.

Discrete homotopy theory is a homotopy theory designed for studying simple graphs, detecting combinatorial, rather than topological, "holes." Central to this theory are the discrete homotopy groups which, just like their continuous counterparts, are easy to define but generally hard to compute. A discrete analogue of the Seifert-van Kampen theorem is thus a crucial tool to have in our computational toolbox. However, the version found in literature turns out to be too restrictive and is not applicable to several examples of interest. In this talk, we will state and sketch a proof of a new version that applies to a wider range of examples, and along the way, introduce some techniques with broader applicability. This talk is based on joint work with C. Kapulkin (arxiv:2303.06029).

Associated to a family of curves $C\to S$ are $\ell$-adic monodromy representations, which generalize Galois representations. I will discuss part my ongoing thesis work demonstrating a big monodromy result for the moduli space of superelliptic curves. This result uses an arithmeticity result of reduced Burau representations of Venkataramana and clutching methods of Achter and Pries. Time permitting, I will also describe applications of this big monodromy result in other parts of my thesis --- it can be used to prove a Cohen-Lenstra result for function fields and to prove a result on the vanishing of zeta functions for Kummer curves over the projective line over finite fields.