A complex algebraic variety can be naturally considered as a complex analytic space. The analytic category is often more flexible, but for this perspective to be useful in algebraic geometry it is necessary to know which analytic constructions produce algebraic objects. One powerful answer to this question is provided by Serre's celebrated GAGA theorem: it says that for a projective variety the algebraic and analytic categories of coherent sheaves are in fact naturally equivalent. This theorem easily fails for non-proper varieties.

In joint work with Y. Brunebarbe and J. Tsimerman, we show that a GAGA theorem holds in the non-proper case if one restricts to analytic structures that are "tame" in a sense made precise by the notion of o-minimality. We will also describe why this result has a number of important applications to Hodge theory.

D-modules allow us to study differential equations through the lens of algebraic geometry. They are widely studied and have been shown to be full of structure. In contrast, the case of difference equations is lacking some of the most basic constructions. We focus on the following question: D-modules have a clear notion of what it means to restrict to a (formal) neighborhood of a point, namely extension of scalars to a power series ring. However, what does it mean to restrict a difference equation to a neighborhood of a point? I will give an answer which encompasses the intuitive notions of a "zero" and a "pole" of a difference equation, but further it is consistent in two more ways. First of all, we can show that restricting a difference equation to a point and to its complement is enough to recover the difference equation. Secondly, there exists a local Mellin transform analogous to the local Fourier transform. The local Fourier transform describes singularities of a D-module on the affine line in terms of the singularities of its Fourier transform. Similarly, the Mellin transform is an equivalence between D-modules on the punctured affine line and difference modules on the line, and we can relate singularities on both sides via this local Mellin transform. I will also talk about how to apply the same ideas to other kinds of difference equations, such as elliptic equations, which generalize difference and differential equations at once.

The polynomial $x_1^5 + x_2^5 + . . . + x_5^5$ cuts out a smooth Calabi-Yau hypersurface in $\mathbb{P}^4$, called the quintic threefold. A standard object in Gromov-Witten theory, the moduli space of stable maps to the quintic is a proper Deligne-Mumford stack and carries virtual fundamental class. Recently Chang and Li embedded this moduli of stable maps in the zero section of the moduli space parameterizing maps to $\mathbb{P}^4$ with sections of $\mathcal{O}(5)$. With this view, they gave another construction of the fundamental class that, in many applications, is more useful than the traditional construction. After illustrating Chang and Li's construction, I will present a work joint with Qile Chen and Felix Janda in which we extend Chang-Li's construction to a complete intersection in a smooth projective variety. The argument uses a deformation to the normal cone to reduce the problem to a special case, and then localization to compute that case.

In this talk, we will discuss new results on projective normality associated to adjunction line bundles $K_X\otimes L^{\otimes n}$ where $L$ is an ample and globally generated line bundle for varieties with nef canonical bundle in dimension $n\geq 3$. As a corollary, we show some sharp bounds on very ampleness and projectve normality of pluricanonical line bundle in dimenson $3,4,5$ for $K_X$ ample. The case of hyperkähler $n$ folds is of particular interest to us for $4\leq n\leq 10$, here we will sketch the proof of new results on effective very ampleness and projectve normality for an ample and globally generated line bundle $L$.

A classical question asks whether the Abel-Jacobi map is universal among all regular homomorphisms. In this talk, we prove that we can construct a $4$-fold which gives the negative answer in codimension $3$ if the generalized Bloch conjecture holds for a $3$-fold constructed by Colliot-Thélène and Voisin in the context of the study of the defect of the integral Hodge conjecture in degree $4$.

Essential dimension of a geometric object is roughly the number of algebraically independent parameters needed to define the object. In this talk we give upper bounds for the essential dimension of parabolic bundles over a non-singular curve using Borne’s correspondence between parabolic bundles on a curve and vector bundles on a root stack. This is a generalization of the work of Biswas, Dhillon and Hoffmann on essential dimension of vector bundles by using their own techniques.

The Cayley-Salmon theorem states that every smooth cubic surface in $\mathbb{CP}^3$ has exactly 27 lines. Their proof is that marking a line on each cubic surface produces a 27-sheeted cover of the moduli space $M$ of smooth cubic surfaces. Similarly, marking a point produces a 'universal family' of cubic surfaces over $M$. One difficulty in understanding these spaces is that they are complements in affine space of incredibly singular hypersurfaces. In this talk I will explain how to compute the rational cohomology of these spaces. I'll then explain how these purely topological theorems have (via the machinery of the Weil Conjectures) purely arithmetic consequences: the typical smooth cubic surface over a finite field $\mathbb{F}_q$ contains 1 line and $q^2 + q + 1$ points.

The gonality of a curve C is the minimal degree of a dominant morphism C---> P^1. The (covering) gonality of a variety X is the minimal gonality of the normalization of an irreducible curve in X. In "Chow rings and gonality of general abelian varieties" Voisin studies the Chow group of zero-cycles of very general abelian varieties and deduces a lower bound on the gonality of a very general abelian variety of dimension g. This bound implies that the gonality of a very general abelian variety of dimension g goes to infinity with g, answering affirmatively a question of Bastianelli, De Poi, Ein, Lazarsfeld and Ullery. She also conjectures that a very general abelian variety of dimension greater or equal to 2g-1 has gonality at least g. We will explain a proof of Voisin's conjecture which is based on a generalization of her methods.

In the quest to understand curves, we often look at their divisors, that is, how many ways we can map them into complex projective space. In particular, we like to study the spaces $W^r_d(C)$ of such maps that have rank r and degree d. The scrollar invariants of a curve give us some notion of how the rank of each divisor changes as we repeatedly add it to itself, which in turn yields some insight into the behavior of $W^r_d(C)$. In this talk we'll introduce and motivate these ideas more carefully and give an overview of how we can use tropical tools to calculate scrollar invariants.