# Algebraic Geometry

**Permanent faculty and their fields of interests**

Valery Alexeev, *Professor, Ph.D. Moscow University 1990.* Degenerations and compact moduli spaces of algebraic varieties, including curves, surfaces, abelian varieties, other varieties with group action. Birational geometry and Minimal Model Program. Singularities appearing in MMP. Toric and spherical varieites. Derived categories. Extremal metrics.

*Assistant Professor, Ph.D. Princeton University, 2010*. Geometric and arithmetic aspects of hyperbolicity in moduli spaces. Torsion structures on abelian varieties. Moduli of sheaves on curves and K3 surfaces. Birational geometry of hyperkahler varieties. Derived categories.

*Assistant Professor, Ph.D. Columbia 2015*. Complex surface singularities. Moduli of K3 surfaces and anticanonical pairs. Mirror symmetry, tropicalization, and integral-affine structures. Flat structures on Riemann surfaces. Enumeration of tilings and Hurwitz theory. Penrose tilings.

*Professor, Ph.D. Massachusetts Institute of Technology 1992.*Geometry related to algebraic groups: equivariant K-theory, cohomology, and Chow groups; flag varieties, Schubert calculus, and related combinatorics.

*Professor, Ph.D. University of Texas 2001*. Finite dimensional division algebras, quadratic forms, and their interplay with algebraic groups and homogeneous varieties. Algebraic cycles and motives. Moduli and configuration

*Assistant Professor, Ph.D. Stanford, 2015. (On leave at the IAS in 2018-2019.)*Interactions between algebraic and arithmetic geometry. Mixed Hodge structures and Galois actions on fundamental groups of algebraic varieties; iterated integrals and p-adic iterated integrals. Rational points on algebraic varieties. Positivity and vanishing theorems.

*Professor, Ph.D. U.C. Berkeley, 1988*. Rational points on algebraic varieties. Torsion points on abelian varieties. Néron models of abelian varieties. Modular curves and their jacobians. Models of curves and wild ramification. Wild quotient singularities of surfaces.

*Professor, Ph.D. UCLA, 1984.*Algebro-geometric methods in Mathematical Physics. Fourier-Mukai transforms, D-modules and integrable systems. Supervarieties.

**Adjunct and Emeritus Faculty**

Noah Giansiracusa, *Adjunct Assistant Professor, Ph.D. Brown University 2011*. Moduli space of marked rational curves and related objects/constructions; geometric invariant theory (GIT); Cox rings; algebro-geometric foundations of tropical geometry.

Elham Izadi, *Adjunct Professor, Ph.D. University of Utah, 1991*. Abelian varietes, curves and their moduli spaces, moduli of vector bundles on curves. Special constructions involving the cohomology of algebraic varieties, special cases of the Hodge conjecture involving abelian varieties.

Roy Smith, *Professor Emeritus, Ph.D. University of Utah, 1977.* Geometry of polarized abelian varieties and their moduli spaces, especially Jacobian and Prym varieties, Torelli problems, deformations of singularities.

Robert Varley, *Professor Emeritus, Ph.D. University of North Carolina, 1977*. Algebraic geometry, curves, abelian varieties, theta divisors, deformation theory, algebraic topology of varieties, mathematical aspects of quantum field theory.

**Post Doctoral Associates and their fields of interest**

Nikon Kurnosov, *Postdoctoral Research and Teaching Associate, Ph.D. Department of mathematics, Higher School of Economics, Moscow, 2016*. Geometry of hyperkähler manifolds. Cohomology of hyperkähler manifolds. Automorphisms of complex manifolds. Calabi-Yau manifolds and mirror symmetry.

Scott Mullane,* Limited-Term Assistant Professor, Ph.D. Boston College, 2016.* The strata of abelian differentials. Flat geometry. The birational geometry of the moduli space of curves.

Alexander Stathis, *RTG Postdoctoral Research and Teaching Associate, Ph.D. University of Illinois at Chicago, 2017.* Birational geometry and intersection theory of moduli spaces. Enumerative geometry.

**Recent graduates and their dissertations**

2017

**Brian Bonsignore** (Robert Varley), *Cohomological n-Equivalence in Differential Graded Algebras.***Natalie L. Hobson **(Angela Gibney), *Vector bundles of conformal blocks in types A and C from a combinatorial approach.***Luca Schaffler** (Valery Alexeev),* The KSBA compactification of a 4-dimensional family of polarized Enriques surfaces.*

2016

**Patrick K. McFaddin** (Daniel Krashen), *K-Cohomology of Generalized Severi-Brauer Varieties.***Matthew Zawodniak **(Robert Varley), *A Moduli Space for Rational Homotopy Types with the Same Homotopy Lie Algebra.*

2015

**Adrian Brunyate** (Valery Alexeev), *A Modular Compactification of the Space of Elliptic K3 Surfaces.*

2014

**Xiaoyan (Shannon) Hu** (Valery Alexeev), *The compactifications of moduli spaces of Burniat surfaces with $2/leqK^{2}\leq5$. ***Joseph Tenini** (Valery Alexeev), *Results on an Extended Torelli Map and Singularities of Degenerate Abelian Varieties. *

2013

**Jaeho Shin** (Valery Alexeev), *The reduction map for the moduli spaces of weighted hyperplane arrangements. ***David Krumm** (Dino Lorenzini), *Quadratic Points on Modular Curves. ***Maurice J. LeBlanc, III**, (Robert Varley), *Analyzing free quantum fields theories on the ax+b space-time and Wigner contractions to the Minkowski plane.*

2012

**Wenjing Li** (William Graham), *Spiral Schubert Varieties in type extended A _{2}. *

**Brandon Samples**(William Graham),

*Components and Springer Fibers for the Exceptional Groups G*

_{2}and F_{4}.**Ben Wyser**(William Graham),

*Symmetric Subgroup Orbit Closures on Flag Varieties: Their Equivariant Geometry, Combinatorics, and Connections With Degeneracy Loci.*UGA Presidential Fellowship 2006-11. NSF International Research Fellowship, Institut Fourier, Grenoble, France, 2013-2015.

**Jim Stankewicz**(Dino Lorenzini and Pete Clark),

*Twists of Shimura Curves.*

2011

**Maxim Arap** (Elham Izadi), *Tautological Rings of Prym Varieties. ***Justin Manning** (Robert Varley), *Axiomatic Quantum Fields on the de Sitter Surface with a Local Spectral Condition.*

2009

**Jeremiah Hower** (Dino Lorenzini), *On elliptic curves and arithmetical graphs.*

2007

**Michael Guy** (Valery Alexeev), *Moduli of Weighted Stable Maps and Their Gravitational Descendants. ***Peter Petrov** (Valery Alexeev), *Nash problem on spaces of arcs. ***Joe Rusinko** (Valery Alexeev), *Equivalence of Mirror Families Constructed from Toric Degenerations of Flag Varieties.*

2005

**Sungkon Chang** (Dino Lorenzini), *The arithmetic of twists of the jacobians of superelliptic curves.*

2004

**Tawanda Gwena** (Valery Alexeev), *Degenerations of Prym varieties and Cubic threefold.*

2003

**Vitaly Vologodsky** (Valery Alexeev), *The extended Jacobi and Prym maps. ***Daniele Arcara** (Elham Izadi) *Moduli Spaces of Vector Bundles on Curves.*

2002

**Dennis Wayne Tarrant** (Robert Varley) *Term Orders on the Polynomial Ring and the Grobner Fan of an Ideal. ***Janice Wethington** (Robert Varley) *On Computing The Thom-Boardman Symbols for Polynomial Multiplication Maps.*