# Algebraic Geometry

Permanent faculty and their fields of interests

Valery AlexeevProfessor, 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.

Benjamin Bakker, 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.

William GrahamProfessor, 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.

Daniel KrashenProfessor, 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 spaces.

Dino LorenziniProfessor, 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.

Mitchell RothsteinProfessor, 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 SmithProfessor 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 VarleyProfessor 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

Asilata BapatLimited-Term Assistant Professor, Ph.D. University of Chicago, 2016. Representation theory and geometry, equivariant cohomology, perverse sheaves, Bernstein-Sato polynomials.

Anand DeopurkarLimited-Term Assistant Professor, Ph.D. Harvard University, 2012.  Geometry of moduli spaces and moduli stacks including Hilbert schemes, moduli of curves, Hurwitz spaces, moduli of vector bundles, and moduli of surfaces. Enumerative geometry and intersection theory. Classical and Geometric Invariant Theory. Connections with commutative algebra.

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.

Reza Seyyedali, Instructor, Ph.D. The Johns Hopkins University, 2009. Geometric analysis and complex geometry.

Alexander StathisRTG 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 A2.
Brandon Samples (William Graham), Components and Springer Fibers for the Exceptional Groups G2 and F4.
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.