Title: Compactified Universal Jacobians over \( \overline{\mathcal{M}}_{g,n} \) via GIT.

Abstract: Associated to any smooth projective curve \( C \) is its degree \( d \) Jacobian variety, parametrising isomorphism classes of degree \( d \) line bundles on \( C \). Letting the curve vary as well, one is led to the universal Jacobian stack. This stack admits several compactifications over the stack of marked stable curves \( \overline{\mathcal{M}}_{g,n} \), depending on the choice of a stability condition. In this talk I will introduce these compactified universal Jacobians, and explain how their moduli spaces can be constructed using Geometric Invariant Theory (GIT). This talk is based on arXiv:2210.11457.

Title: Quiver Representations and Moduli of Sheaves.

Abstract: We give an introduction to moduli problems in algebraic geometry and focus on the cases of quiver representations and coherent sheaves on a projective scheme. We outline King's construction of moduli spaces of quiver representations and the main ideas of the functorial approach to construction of moduli of sheaves by embedding its moduli functor into the functor of moduli of quiver representations. Lastly, we will see how Greb, Ross and Toma extended these ideas and performed variation of GIT for moduli spaces of sheaves.

Title: Derived blow-ups using Rees algebras and virtual Cartier divisors.

Abstract: The blow-up \( B \) of a scheme \( X \) in a closed subscheme \( Z \) enjoys the universal property that for any scheme \( X' \) over \( X \) such that the pullback of \( Z \) to \( X' \) is an effective Cartier divisor, there is a unique morphism of \( X' \) into \( B \) over \( X \). It is well-known that the blow-up commutes along flat base change. In this talk, I will discuss a derived enhancement \( B' \) of \( B \), namely the derived blow-up, which enjoys a universal property against all schemes over \( X \), satisfies arbitrary (derived) base-change, and contains \( B \) as a closed subscheme. To this end, we will need some elements from derived algebraic geometry, which I will review along the way. This will allow us to construct the derived blow-up as the projective spectrum of the derived Rees algebra, and state its functor of points in terms of virtual Cartier divisors, using Weil restrictions. This is based on ongoing joint work with Adeel Khan and David Rydh.

Title: Gauge theories in 4, 8 and 5 dimensions.

Abstract: In the 1980s, gauge theory was used to provide new invariants (up to diffeomorphism) of orientable four dimensional manifolds, by counting solutions of certain equations up to to a choice of gauge. More recently, similar techniques have been used to study manifolds of different dimensions, most notably on \( \mathrm{Spin}(7) \) and \( G_2 \) manifolds. Using dimensional reduction, one can find candidates for gauge theoretic equations on manifolds of lower dimension. The talk will give an overview of gauge theory in the 4 and 8 dimensional cases, and how gauge theory on \( \mathrm{Spin}(7) \) manifolds could be used to develop a gauge theory on 5 dimensional manifolds.

Title: A logarithmic version of projective space.

Abstract: Fix \( X \) a toric surface and \( \beta \) the homology class of a curve on \( X \). The moduli space of curves of class \( \beta \) in \( X \) is a projective space. In this talk, we explore the logarithmic version of this statement. Logarithmic geometry is a way of imposing boundary conditions in algebraic geometry. Our moduli space illustrates the link between logarithmic geometry and the geometry of cone complexes - so-called tropical geometry. Along the way, we will give one answer to the question, "what is a tropical curve and why should one care?".

Title: Torsion points on varieties and the Pila-Zannier method.

Abstract: In 2008 Pila and Zannier used a Theorem coming from Logic, proven by Pila and Wilkie, to give a new proof of the Manin-Mumford Conjecture, creating a new, powerful way to prove theorems in Diophantine Geometry. The Pila-Wilkie Theorem gives an upper bound on the number of rational points on analytic varieties which are not algebraic; this bound usually contradicts a Galois-theoretic bound obtained by arithmetic considerations. We show how this technique can be applied to the following problem of Lang: given an irreducible polynomial \( f(x,y) \) in \( \mathbb{C}[x,y] \), if for infinitely many pairs of roots of unity \( (a,b) \) we have \( f(a,b) = 0 \), then \( f(x,y) \) is either of the form \( x^m y^n - c \) or \( x^m - cy^n \) for \( c \) a root of unity.

Title: Ricci curvature lower bounds for metric measure spaces.

Abstract: In the '80s, Gromov proved that sequences of Riemannian manifold with a lower bound on the Ricci curvature and an upper bound on the dimension are precompact in the measured Gromov--Hausdorff topology (mGH for short). Since then, much attention has been given to the limits of such sequences, called Ricci limit spaces. A way to study these limits is to introduce a synthetic definition of Ricci curvature lower bounds and dimension upper bounds. A synthetic definition should not rely on an underlying smooth structure and should be stable when passing to the limit in the mGH topology. In this talk, I will briefly introduce CD spaces, which are a generalization of Ricci limit spaces.