Department of Mathematics, Statistics, and Computer Science

851 S. Morgan St.

Chicago, IL 60607

I am an assistant professor of Mathematics at UIC.

UIC Algebraic Geometry Seminar

Hodge theory, period mappings, and local systems, a Zoom mini-workshop at CRM, December 3-7 2020.

**Definable structures on flat bundles**

with S. Mullane

preprint (pdf, arXiv)**Finiteness for self-dual classes in integral variations of Hodge structure**

with T. W. Grimm, C. Schnell, and J. Tsimerman

preprint (pdf, arXiv)**Quasiprojectivity of images of mixed period maps**

with Y. Brunebarbe and J. Tsimerman

submitted (pdf, arXiv)**The global moduli theory of symplectic varieties**

with C. Lehn

submitted (pdf, arXiv)**o-minimal GAGA and a conjecture of Griffiths**

with Y. Brunebarbe and J. Tsimerman

submitted (pdf, arXiv)**Algebraic approximation and the decomposition theorem for Kahler Calabi-Yau varieties**

with H. Guenancia and C. Lehn

*Invent. Math.*, to appear (pdf, arXiv)**Definability of mixed period maps**

with Y. Brunebarbe, B. Klingler, and J. Tsimerman

*J. Eur. Math. Soc.*, to appear (pdf, arXiv)**A global Torelli theorem for singular symplectic varieties**

with C. Lehn

*J. Eur. Math. Soc.*, Volume 23, Issue 3 (2021) (pdf, arXiv, journal)**Tame topology of arithmetic quotients and algebraicity of Hodge loci**

with B. Klingler and J. Tsimerman

*J. Amer. Math. Soc.*, Volume 33, No. 4 (2020) (pdf, arXiv, journal)**The Ax-Schanuel conjecture for variations of Hodge structures**

with J. Tsimerman

*Invent. Math.*, Volume 217, No. 1 (2019) (pdf, arXiv, journal)**The Mercat conjecture for stable rank 2 vector bundles on generic curves**

with G. Farkas

*Amer. J. Math.*, Volume 140, No. 5 (2018) (pdf, arXiv, journal)**The geometric torsion conjecture for abelian varieties with real multiplication**

with J. Tsimerman

*J. Differential Geom.*, Volume 109, No. 3 (2018) (pdf, arXiv, journal)**The Kodaira dimension of complex hyperbolic manifolds with cusps**

with J. Tsimerman

*Compos. Math.*, Volume 154, Issue 3 (2018) (pdf, arXiv, journal)**A classification of Lagrangian planes in holomorphic symplectic varieties**

*J. Inst. Math. Jussieu*, Volume 16, Issue 4 (2017) (pdf, arXiv, journal)**p-torsion monodromy representations of elliptic curves over geometric function fields**

with J. Tsimerman

*Ann. of Math.*184, No. 3 (2016) (pdf, arXiv, journal)**On the Frey-Mazur conjecture over low genus curves**

with J. Tsimerman

arXiv preprint (2013) (pdf, arXiv)**Lagrangian 4-planes in holomorphic symplectic varieties of K3^[4] type**

with A. Jorza

*Cent. Eur. J. Math.*, Volume 12, Issue 7 (2014) (pdf, arXiv, journal)

Computational appendix and Code**Higher rank stable pairs on K3 surfaces**

with A. Jorza

*Commun. Number Theory Phys.*, Volume 6, Number 4 (2012) (pdf, arXiv, journal)**Hodge polynomials of moduli spaces of stable pairs on K3 surfaces**

My thesis from Princeton University, under Rahul Pandharipande

June 2010 (pdf)

**Hodge theory and o-minimality**

Notes from the Felix Klein lecture series in Bonn, May 2019 (pdf)**Lectures on the Ax-Schanuel Conjecture**

with J. Tsimerman

*Arithmetic Geometry of Logarithmic Pairs and Hyperbolicity of Moduli Spaces*, CRM Short Courses, Springer (2020) (pdf, book)

**Period integrals of algebraic varieties**

University of Chicago, November 2021.

Abstract.A period integral of a complex algebraic variety is the integral of an algebraic differential form along a topological cycle. While such integrals are transcendental in nature, they have had pervasive importance in algebraic and arithmetic geometry going back to the foundational work of Abel and Jacobi. The functions one obtains from taking period integrals in families of algebraic varieties moreover recur in many areas of mathematics. In this talk I will first survey how algebraic information can be extracted from period integrals, as well as their connection to various conjectural frameworks. I will then discuss recent progress on understanding the algebraic properties of period integrals in families and recent applications to Hodge theory, algebraic geometry, arithmetic geometry, and logic.**Algebraic approximation of Calabi-Yau varieties and the decomposition theorem**

Geometry and TACoS, November 2020.

Abstract.Calabi-Yau manifolds are built out of simple pieces by the Beauville–Bogomolov decomposition theorem: any Calabi–Yau Kahler manifold up to an etale cover is a product of complex tori, irreducible holomorphic symplectic manifolds, and strict Calabi-Yau manifolds (which have no holomorphic forms except a holomorphic volume form). Work of Druel-Guenancia-Greb-Horing-Kebekus-Peternell over the last decade has culminated in a generalization of this result to projective Calabi-Yau varieties with the kinds of singularities that arise in the MMP, and the proofs critically use algebraic methods. In this talk I will describe joint work with H. Guenancia and C. Lehn extending the decomposition theorem to nonprojective varieties via deformation theory. A crucial step in the proof is the resolution of the K-trivial case of a conjecture of Peternell asserting that any minimal Kahler variety can be approximated by algebraic varieties.**Quasiprojectivity of images of mixed period maps**

Princeton, October 2020.

Abstract.Families of smooth proper algebraic varieties give rise to variations of pure Hodge structures; general algebraic families yield variations of mixed Hodge structures. It was conjectured by Griffiths and proven in joint work with Y. Brunebarbe and J. Tsimerman that the closure of the image of the classifying map to the moduli space of Hodge structures is a quasiprojective algebraic variety in the pure case. In this talk I will explain how to extend this result to the mixed setting. As in the pure case, the proof heavily uses techniques from o-minimal geometry, and we will also discuss some related applications.**Recent developments in the moduli of singular symplectic varieties**

Categories, Cycles and Cohomology of Hyperkahler Varieties, September 2020.

Abstract.A recurring feature of hyperkahler manifolds is that their geometry is closely tied to the Hodge structure on their weight 2 cohomology. Verbitsky's global Torelli theorem, for instance, essentially shows that for any hyperkahler manifold, all deformations of the Hodge structure are realized geometrically exactly once, at least up to bimeromorphism. In this talk I will survey joint work with C. Lehn extending these moduli-theoretic results to the case of singular symplectic varieties, including a generalization of the global Torelli theorem. We will also discuss some more recent work with C. Lehn and H. Guenancia extending the decomposition of projective symplectic varieties by holonomy type due to Druel--Greb--Guenancia--Horing--Kebekus--Peternell to the nonprojective case.**Hodge theory and o-minimality**

Brown, February 2020.

Abstract.The cohomology groups of complex algebraic varieties come equipped with a powerful but intrinsically analytic invariant called a Hodge structure. Hodge structures of certain very special algebraic varieties are nonetheless parametrized by algebraic varieties, and while this leads to many important applications in algebraic and arithmetic geometry it fails badly in general. Recent joint work with Y. Brunebarbe, B. Klingler, and J. Tsimerman remedies this failure by showing that parameter spaces of Hodge structures always admit "tame" analytic structures in a sense made precise using ideas from model theory. A salient feature of the resulting tame analytic geometry is that it allows for the local flexibility of the full analytic category while preserving the global behavior of the algebraic category.

In this talk I will explain this perspective as well as some important applications, including an easy proof of a celebrated theorem of Cattani--Deligne--Kaplan on the algebraicity of Hodge loci and the resolution of a longstanding conjecture of Griffiths on the quasiprojectivity of the images of period maps.

My course websites are maintained here.

- UIC Spring 2022: Hyperkahler manifolds (Math 571).

- UIC Fall 2021: Second Course in Abstract Algebra I (Math 516).
- UIC Fall 2021: Definable Complex Analytic Geometry (Math 571).
- UIC Spring 2021: Linear Algebra (Math 320) -- (10am, 1pm).
- UGA Spring 2020: Moduli spaces (Math 8330).
- UGA Fall 2019: Calculus II for Science and Engineering (Math 2260)
- UGA Fall 2018: Differential equations (Math 2700).
- UGA Fall 2018: (Counter)examples in char. p geometry (Math 8330).
- UGA Spring 2018: Commutative algebra (Math 8020).
- UGA 2017-2018: Topics in Hodge theory (VRG) (Math 8850).
- UGA Fall 2017: Differential equations (Math 2700).
- UGA Spring 2017: Modern algebra and geometry I (Math 4000/6000).
- UGA Spring 2017: Abelian varieties (Math 8330).
- HU Summer 2016: Berkovich spaces.
- HU Winter 2015: Literature Seminar.
- HU Summer 2015: Faltings theorem.
- HU Winter 2014: Abelian varieties and Fourier-Mukai transforms.
- NYU Spring 2013: Number Theory (MATH-GA 2210.001).
- NYU Fall 2012: Theory of Numbers (MATH-UA 248.001).
- NYU Fall 2011: Algebra I (MATH-GA 2130.001).
- NYU Spring 2011: Topology II (MATH-GA 2320.001).
- NYU Fall 2010: Calculus I (V63.0121.026.FA10).

**Ben Tighe****Zhehao Li**