**11 October**- SPEAKER: Thorsten Hertl
- TITLE:
*Meromorphic Differentials.*- ABSTRACT: In this talk, we will introduce meromorphic differentials on Riemann surfaces. After providing local modules for them, we compare the moduli space of these differentials (with prescribed poles and zeros) to its holomorphic counterpart. We introduce a topological invariant that allows us to find infinitely many components in the genus 1 case.

**4 October**- SPEAKER: Quan Nguyen
- TITLE:
*Affine invariant manifolds.*- ABSTRACT: In this talk, we will define affine invariant manifolds, which are interesting linear subspaces in the stratum of differentials. We give examples and discuss some of their properties, including a theorem by Eskin-Mirzakhani-Mohammadi which allows us to redefine affine invariant manifolds as $\text{GL}^+(2,\mathbb{R})$-invariant subspaces. Affine invariant manifolds also give rise to Veech surfaces, which have some interesting dynamical properties. Some other elementary constructions of affine invariant manifolds come from real multiplication in genus 2, which relate to special properties of the Jacobian of a Riemann surface.
**20 September**- SPEAKER: Marcel Dang
- TITLE: Compactifications of strata of differentials.
- ABSTRACT: Whenever one encounters noncompact spaces in nature, one can ask the question of how to compactify them. But why do we care? One can think of it as a generalization of the concept of fini teness. These spaces tend to be more well behaved than their noncompact counterparts, and they allow new techniques and constructions. Putting it into the words of Angelo Vistoli: "Working with noncompact spaces is like trying to keep change with holes in your pockets." We have seen that a typical stratum of differentials is not compact. As there is no canonical construction to compactify a space, we will present multiple compactifications: Deligne-Mumford, "You get what you see", incidence variety and themultiscale compactification.
**13 September**- SPEAKER: Paul Norbury
- TITLE:
*Square-tiled surfaces.*- ABSTRACT: I will discuss square-tiled surfaces Veech groups commensurable to SL(2, Z) and conditions on a flat surface to be a square-tiled surface. Enumeration of square-tiled surfaces produces generating functions with modular properties and leads to calculations of Masur-Veech volumes.
**6 September, 2024**- SPEAKER: Scott Mullane
- TITLE:
*Saddle connections and Veech groups for flat surfaces.*- ABSTRACT: After discussing three examples of different sources of non-compactness in the strata of differentials, we'll state Masur's compactness criterion, the action of GL
^{+}(2,R) on the set of saddle connections of a flat surface, and define and discuss properties of the Veech group, the stabiliser of a surface under the GL^{+}(2,R) action.**30 August, 2024**- SPEAKER: Marcel Dang
- TITLE:
*The GL*^{+}(2,R)-action on the strata of differentials.- ABSTRACT: Originally motivated by dynamics, GL
^{+}(2,R) acts on the strata of differentials naturally by acting on the polygons in any polygon presentation of a flat surface in the plane (the first definition). After defining this action and showing it is well-defined, we'll discuss how square-tiled surfaces have closed orbits in the strata of differentials, as well as how some interesting 1-parameter subgroups of GL^{+}(2,R) act on the strata.**23 August, 2024**- SPEAKER: Thorsten Hertl
- TITLE:
*Connected components of the strata of differentials II.*- ABSTRACT: Building on the last lecture, I will introduce spin structures from different perspectives. I will define and give an example of computing the Arf invariant of a flat surface. Then I will define the spin structure from an algebraic perspective, given by the parity of the number of global sections of the associated theta divisor. Finally I will state Kontsevich and Zorich's theorem.
**16 August, 2024**- SPEAKER: Scott Mullane
- TITLE:
*Connected components of the strata of differentials I.*- ABSTRACT: A seminal theorem of Kontsevich and Zorich classifies the number of connected components of each stratum of differentials showing that each stratum of differentials has up to three connected components due to spin and hyperelliptic components. In this talk, we will introduce the concepts of hyperellipticity and spin structures and how they are used to classify the connected components in the example of the stratum H(4).
**9 August, 2024**- SPEAKER: Paul Norbury
- TITLE:
*Moduli spaces of flat structures, period coordinates and volumes.*- ABSTRACT: In previous lectures we have described how a collection of n vectors in R^2, or equivalently n points in C, describe 2n-sided polygons in R^2. This gives a rather elementary description of the period coordinates of a flat structure, defined as the periods of a holomorphic differential on a Riemann surface. These equivalent viewpoints lead to natural volume measures on the moduli spaces of flat structures. I will describe this measure, and also give calculations of (finite) volumes of the moduli spaces of unit area flat structures.
**2 August, 2024**- SPEAKER: Scott Mullane
- TITLE:
*Flat Surfaces*- ABSTRACT: This talk will give three definitions of a flat surface and a proof of equivalence of these definitions. Flat surfaces produce concrete constructions of Riemann surfaces equipped with a holomorphic differentials. The concrete constructions include: square tiled surfaces, translation coverings, and slit torus and slit cylinder constructions.
**26 July, 2024**- SPEAKER: Paul Norbury
- TITLE:
*Abelian differentials and flat structures.*- ABSTRACT: In this talk I will give an introduction to the topic of abelian differentials and flat structures. To describe billiards bouncing off an edge of a rectangular table, one can instead continue along a straight line path past the edge to return from the opposite edge (as in some old video games). This describes a path on a flat torus. Equivalently, the surface is equipped with a conformal structure = well-defined angles, plus further structure, which is neatly captured by the notion of a Riemann surface equipped with an abelian differential. This viewpoint allows for more general shaped billiard tables, and gives a concrete approach to these algebraic and geometric objects.