# Stony Brook MathematicsCapsule Talks

### Spring 2021

• Frederik Benirschke (Stony Brook University)
Boundary of linear subvarieties
May 5, 2021

Strata of differentials are moduli spaces of differential forms on Riemann surfaces with prescribed multiplicity of zeros and poles.Linear subvarieties are a special class of subvarieties of strata defined by linear equations among the periods of the 1-form. Examples of linear varieties arise from both algebraic geometry as well as Teichmüller dynamics. In this talk we study the behavior of linear subvarieties at infinity. Our main result is that the boundary of a linear subvariety in a suitable compactification is again given by linear equations among periods.

• Nathan Chen (Stony Brook University)
Abelian surfaces revisited
May 5, 2021

We will motivate and explain the irrationality problem for abelian varieties. Towards the end, we will explore the geometry of special families of algebraic curves on abelian surfaces.

• Aleksandar Milivojevic (Stony Brook University)
The rational homotopy types of closed almost complex manifolds
May 5, 2021

Surgery theory or constructive cobordisms from the 60's can be used to characterize the homotopy types of closed almost complex manifolds up to rational homological equivalence, along with their rational Chern classes. The talk will first explain the necessary conditions for a rational homotopy type to be that of a closed almost complex manifold. The Hirzebruch-Riemann-Roch theorem from algebraic geometry extends by Atiyah-Singer index theory to almost complex manifolds, giving necessary conditions that intertwine the rational Chern classes, the rational cohomology ring, and the fundamental homology class of any closed almost complex manifold. For simply connected homotopy types up to mappings inducing bijections on rational cohomology in complex dimensions three and greater, with constructive cobordisms we can conclude these conditions are also sufficient.

### Fall 2020

• Harold Blum (Stony Brook University)
Moduli spaces of Fano varieties and K-stability
August 25, 2020

Fano varieties are a class of positively curved algebraic varieties that form one of the three main building blocks of varieties in algebraic geometry. While moduli spaces parametrizing such objects are in general poorly behaved, it has been shown that there exist compact Hausdorff spaces parametrizing Fano varieties that are K-stable, which is an algebraic notion invented by differential geometers to characterize when a Fano variety admits a Kähler-Einstein metric. I will survey this topic and briefly discuss recent progress on constructing these moduli spaces algebraically.

• Matthew Romney (Stony Brook University)
Uniformization of metric surfaces
August 25, 2020

The classical uniformization theorem gives a conformal parametrization of any simply connected Riemannian 2-manifold by the disk, plane, or sphere. We discuss generalizations of this result to non-smooth metric spaces.

• Shengyuan Zhao (IMS/Stony Brook University)
Variation of Klenian Group
August 26, 2020

The Uniformization Theorem says that every hyperbolic Riemann surface can be realized as a quotient of the unit disk by a Fuchsian group. A Fuchsian group is a Kleinian group, i.e. a subgroup of the automorphism group of the complex projective line that acts properly discontinuously on a domain. I will discuss a generalization of this in complex dimension two where the complex projective line is replaced by an algebraic surface and the group is a group of birational transformations.

• Langte Ma (SCGP/Stony Brook University)
Exoticness and Seiberg-Witten Theory
August 26, 2020

The application of Yang-Mills theory to 4-dimensional topology was pioneered by Simon Donaldson in the 1980s. Since then numerous tools from gauge theory are developed to study classical topological problems. In this talk, I will discuss the tools used and problems I am interested in my research.

• Olivier Martin (Stony Brook University)
Abelian varieties and their subvarieties are far from rational
August 26, 2020

Algebraic geometry is the study of algebraic varieties: geometric objects cut out of $\mathbb{C}^n$ (or projective space) by polynomial equations. A key feature of the subject is that, unlike in the case of topological manifolds, open sets in algebraic varieties retain a lot of information about the variety. We say an n-dimensional variety is rational if it contains an open set which is isomorphic to an open set in $\mathbb{C}^n$. Determining if varieties are rational is a notoriously rich and subtle problem that has motivated the development of several tools in algebraic geometry. In recent years, renewed interest has been brought to the problem of measuring how far irrational varieties are from being rational. After presenting preliminary definitions and concepts, I will discuss three of my contributions to the subject. These results all point to the fact that general abelian varieties and their subvarieties are far from rational.

• Christina Karafyllia (IMS/Stony Brook University)
Conformal invariants and spaces of holomorphic functions
August 27, 2020

This talk is about some classical subjects in complex analysis and geometric function theory such as properties of conformal invariants and relations between them. In particular, we will talk about some recent results on the harmonic measure and its connection with the hyperbolic distance on simply connected domains. We also present geometric conditions for a conformal mapping of the unit disk to belong to some space of holomorphic functions such as the Hardy or Bergman space.

• Jacob Rooney (SCGP/Stony Brook University)
An overview of embedded contact homology
August 27, 2020

This talk will give an overview of embedded contact homology, a topological invariant of 3-manifolds developed by Michael Hutchings that is connected to the dynamics of certain vector fields. We will also discuss applications to the existence of periodic orbits of these vector fields and embeddings of symplectic 4-manifolds.

### Fall 2017

• Yu Li (Stony Brook University)
Hamilton's Ricci flow
August 29, 2017

Ricci flow has become an important tool to search classical metrics on manifolds since it first appeared in Hamilton's seminal 1982 paper. As an important evolutionary equation, it sets up a bridge between geometry and topology. In this talk, I will give a short introduction to some major achievements and recent progress of Ricci flow.

• Demetre Kazaras (Stony Brook University)
Minimal hypersurfaces with free boundary and psc-bordism
August 29, 2017

There is a well-known technique due to Schoen-Yau from the late 70s which uses (stable) minimal hypersurfaces to study the topological implications of a (closed) manifold's ability to support positive scalar curvature metrics. In this talk, I'll describe a version of this technique for manifolds with boundary and discuss how it can be used to study bordisms of positive scalar curvature metrics.

• Robert Silversmith (Stony Brook University)
Moduli spaces of curves on orbifolds
August 30, 2017

Enumerative problems involving curves on varieties have a long history in algebraic geometry — for example, one could ask how many rational plane cubic curves pass through 8 randomly chosen points. In thinking about such questions, one often considers a moduli space of curves (with some fixed discrete invariants) on the variety in question. I will discuss some examples of these spaces and their compactifications. I will also talk about how the story changes when one starts with an algebraic orbifold rather than a variety, and why one might care.

• Francois Greer (Stony Brook University)
Modular Forms in Enumerative Geometry
August 30, 2017

Enumerative geometry is a collection of techniques for counting algebro-geometric objects satisfying certain incidence conditions. First, we find a compact moduli space for the objects, and then we try to understand its cohomology ring. If the conditions are suitably transverse, then the enumerative answer will be given by the integral of a top class in the cohomology of the moduli space. Often we can assemble a family of such answers into a generating function, which satisfies differential equations or modular transformation laws. This knowledge allows us to extract infinitely many numbers from the few that are directly computable.

• Vardan Oganesyan (Stony Brook University)
Nonlinear partial differential equations, Schottky problem, Jacobian conjecture and commuting differential operators
August 31, 2017

We will talk about a very interesting and deep connection between all the tasks indicated in the title. I will try to briefly talk about the theory of commuting differential operators, to show you it's beauty and connections with other fields of math.

• Babak Modami (Stony Brook University)
Geometry and dynamics on moduli spaces
August 31, 2017

There are several naturally defined metrics on the moduli spaces of Riemann surfaces; each exhibiting its own geometric and dynamical features. In this talk after briefly introducing the Teichmüller and Weil-Petersson metrics, I explain how curve complexes can be used to develop a kind of symbolic coding for the geodesic flows of the two metrics.

• Jonguk Yang (Stony Brook University)
Applications of Renormalization to Rotational Complex Dynamics
August 31, 2017

The focus of my research is in dynamics of one and several complex variables. I am particularly interested in the applications of renormalization in this setting. Loosely speaking, the renormalization of a dynamical system is defined as a rescaled first return map on an appropriately chosen subset of the phase space. Iterating this procedure reveals the small-scale asymptotic behaviour of the dynamics, which is often universal and insensitive to the incidental details of the system.

Using the renormalization approach, I have studied rotational dynamics that occur in three different settings: non-polynomial quadratic rational maps, cubic polynomials, and dissipative Hénon maps. I will present a brief summary of my results, and discuss my current ongoing projects and research goals.