Geometry of the South

Hülya Argüz, University of Georgia: The KSBA moduli space of log Calabi--Yau surfaces I

The KSBA moduli space, introduced by Kollár--Shepherd-Barron, and Alexeev, is a natural generalization of ``the moduli space of stable curves'' to higher dimensions. It parametrizes stable pairs (X,B), where X is a projective algebraic variety satisfying certain conditions and B is a divisor such that K_X+B is ample. This moduli space is described concretely only in a handful of situations: for instance, if X is a toric variety and B=D+\epsilonC, where D is the toric boundary divisor and C is an ample divisor, it is shown by Alexeev that this moduli space is the toric variety defined by the secondary fan. Generally, for a log Calabi--Yau variety (X,D) consisting of a projective variety X with B=D+\epsilonC, where D is an anticanonical divisor and C is an ample divisor, it has been an open question what this moduli space is, and it was conjectured by Hacking--Keel--Yu that the moduli space in this situation is still toric (up to passing to a finite cover). In joint work with Alexeev and Bousseau, we prove this conjecture for all log Calabi--Yau surfaces. This uses tools from the minimal model program, log smooth deformation theory and mirror symmetry.  Video

Andrei Okounkov, Columbia University: Enumerative Geometry and Special Functions I

Special functions are like threads which hold together many different pages of the history of mathematics. Functions that were important in mathematics and mathematical physics of the two previous centuries find new interpretations and important generalizations in the current research, including current research in enumerative geometry and number theory. In these two lectures, I will try to give an accessible introduction to some old and new ideas in the subject.  Video

Ludmil Katzarkov, University of Miami: Atoms, electrons and birational invariants

Recently a new A,B Hodge structures were introduced.   We will consider several non-rationality applications.  Video

Pierrick Bousseau, UGA: The KSBA moduli space of log Calabi--Yau surfaces  Video

Andrei Okounkov, Columbia University: Enumerative Geometry and Special Functions II

Special functions are like threads which hold together many different pages of the history of mathematics. Functions that were important in mathematics and mathematical physics of the two previous centuries find new interpretations and important generalizations in the current research, including current research in enumerative geometry and number theory. In these two lectures, I will try to give an accessible introduction to some old and new ideas in the subject.  Video

Jorge Lauret, Florida International University  Video

Leonardo Cavenaghi, University of Miami: Hodge structures 8-dimension Homotopy Hopf manifolds

In this talk, we report on ongoing work with Dr. Ludmil Katzarkov (University of Miami) and Dr. Lino Grama (Unicamp). We discuss the possible Hodge structure on manifolds of the form $X:=\Sigma^7\times\mathrm{S}^1$ where $\Sigma^7$ is one of the 28 possible homotopy spheres in dimension 7. Establishing a map from the (moduli space of) Hodge structures on $H:=\mathrm{S}^3\times\mathrm{S}^1$ (the standard 4-dimension Hopf manifold) to the moduli space of Hodge structures in $X:=\Sigma^7\times\mathrm{S}^1$, we show intriguing connections between $X$ and the K3-surface (obtained via $\mathrm{tmf}$), that may allow study Homological Mirror Symmetry on $X$ via $K3$.  Video

Jose Medel, Florida International University  Video

Rodolfo Aguilar, University of Miami: Bridging the gap between homology planes and Mazur manifolds.

A homology plane is an algebraic complex smooth surface with the same integral homology groups as the complex plane. A Mazur type manifold is a compact contractible smooth (real) 4-manifold built only with 0-,1- and 2- handles. We call a homology 3-sphere a Kirby-Ramanujam sphere if it bounds both a homology plane and a Mazur type manifold. In this talk, we present several infinite families of Kirby-Ramanujam spheres and some related topics if time permits. Such an interplay between complex surfaces and 4-manifolds was first observed by C. P. Ramanujam and R. Kirby. This is joint work with Oğuz Şavk.  Video