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.

This report is based on the paper by Prof. Hristo Sendov and a joint work in preparation. The aim is to discuss the results presented in the paper in the light of the notions of polar convexity and convince the reader of the usefulness of this approach in studying polynomials. We start by recalling some relevant concepts and basic Definitions from polar convexity. We will then set up the stage for the main theorem with some supporting results and then finally prove it. Then we go on to discuss my own work in extending the notion of polar convexity to any finite dimension Euclidean space. The problem holds interesting consequences regarding non-linear optimization and somehow polar convex sets turn up very naturally when considering zeroes or critical points of polynomials.

There are two generalizations of the notion of rational convexity on $\mathbb{C}^n$ to a general Stein manifold $X$. For a given compact $K\subset X$, the two associated hulls are $$ h(K)=\left\{z\in X\,:\,f^{-1}(0)\cap K\neq\varnothing\text{ for every }f\in\mathcal{O}(X)\text{ satisfying }f(z)=0\right\} $$ and $$ H(K)=\left\{z\in X\,:\,\text{every $\mathbb{C}$-hypersurface in $X$ passing through $z$ intersects $K$}\right\}. $$ In the last talk, it was shown these hulls are equal to the hulls with respect to meromorphic and "strong" meromorphic functions on $X$, respectively. In this talk we continue this comparison by showing a generalization of Runge's theorem and a theorem of Duval-Sibony for strong meromorphically convex compacts.

A metric space X is said to be injective if any collection of pairwise intersecting balls admits a total intersection. Equivalently, injective metric spaces are exactly the injective objects in the category of metric spaces with respect to 1-Lipshitz map. In the talk I will:

1) Define injective spaces and investigate some of their properties, 2) Discuss what can be learnt about a group G that admits a ``nice" action on an injective metric space, and 3) Give a criterion for building actions on injective metric spaces with some applications.

The talk is based on two joint works, one is with Sisto and the other is with Petyt and Spriano.

In this talk, I will tell the story of 'computational enumerative geometry', the area of research which uses computers to study enumerative problems. An enumerative problem asks how many geometric figures have a prescribed position to a given set of fixed, generic, geometric figures. For example: "how many conics are tangent to five conics in the plane". The answer to this question is famously 3264, but actually computing 3264 conics is another story. This task, and several others, is a job for the emerging field of computational enumerative geometry!

We investigate the question: for which functions $f(x_1,...,x_n),~g(x_1,...,x_n)$ the asymptotic expansion of the integral $\int g(x_1,...,x_n) e^{\frac{f(x_1,...,x_n)+x_1y_1+...+x_ny_n}{\hbar}}dx_1...dx_n$ consists only of the first term. We reveal a hidden projective invariance of the problem which establishes its relation with geometry of projective hypersurfaces of the form $\{(1:x_1:...:x_n:f)\}$. We also construct various examples, in particular we prove that Kummer surface in $\P^3$ gives a solution to our problem. This is a joint work with Maxim Kontsevich.

In this talk I will introduce the notation of a matrix ensemble with a focus on the Gaussian Unitary Ensemble (GUE) as an example. I will introduce its basic properties in connection with map enumeration. In particular, I will outline a proof of the Genus Expansion Formula for moments of the GUE. Time permitting we will discuss the famous Harer-Zagier formula.