Good moduli spaces, positivity, and rationality

Spring 2026, Boston College

When does an Artin stack admit a "reasonable" moduli space and when does this moduli space have nice properties (for example being (quasi-)projective, rational, etc)?

Geometric rep theory and universal centralizers

Jan 2025, Simons Center

Covers the geometry of complex semisimple Lie groups (Springer theory, wonderful/toroidal compactifications) before moving on to universal centralizers (and their geometry, mirror symmetry, etc).

Higher genus GW theory

Jan 2025, Simons Center

What does it take to prove enumerative mirror symmetry for the quintic 3-fold in higher genus? Find out more about recent breakthroughs here.

Blowup formula in GW theory

Spring 2024, Columbia

How do GW invariants change under birational transformations? Learn about Iritani's work on the subject here.

Integrability, enumerative geometry, quantization

Summer 2022, Simons Center

Covers enumerative geometry (GW/Hurwitz, GW/DT, quantum K-theory) and (quantum) integrable systems (KdV, W-constraints, etc).

Moduli spaces and hyperkähler manifolds

Fall 2021, Columbia

Covers hyperkähler manifolds, their geometry, some constructions, and moduli spaces of sheaves on K3 surfaces.

Minimal Model Program

Spring 2021, Zoom

Ever wondered how to classify algebraic varieties up to birational equivalence? The MMP is the answer! Covers the basics of the MMP and the MMP in dimension 3.

Class field theory

Spring 2021, Columbia

Covers local and global class field theory, including the proofs of the main theorems and some applications.

Algebraic topology II

Spring 2021, Columbia

Covers spectral sequences and applications to homotopy groups of spheres, characteristic classes, K-theory, and the Atiyah-Singer index theorem.

The Count of Instantons

2023-24, Columbia

On the mathematics (representation theory, geometry, probability theory) and physics (gauge theory) of instanton counting. Notes partially taken by Davis Lazowski.

Commutative algebra

Fall 2020, Columbia

Covers the first part of Matsumura's Commutative Algebra with a focus on dimension theory.

Simons Summer Math Workshop 2023

Summer 2023, Simons Center

Covers log GW theory in both the algebraic and symplectic settings, topological recursion and the spin GW/Hurwitz correspondence, and Givental-Teleman reconstruction of semisimple CohFTs.

Informal enumerative geometry seminar

Spring 2022, Columbia

Covers a variety of topics in enumerative geometry, including GW theory, DT theory, and quantum K-theory and its relationship to geometric representation theory.

Hyperbolicity

Spring 2022, Columbia

When is a variety hyperbolic? How many rational points does it have? How many rational curves lie on it? Find out here!

D-modules and localization

Spring 2022, Columbia

What is a D-module and how do they appear in representation theory? Learn about the Beilinson-Bernstein localization theorem and the Kazhdan-Lusztig conjectures.

Hodge theory

Fall 2021, Columbia

Covers the proof of the existence of mixed Hodge structures for smooth varieties and some applications.

Deformation theory

Fall 2021, Columbia

Covers how to deform schemes and sheaves, plus the moduli of stable curves and of coherent sheaves.

DAHA and knot homology

Fall 2021, Columbia

What on earth is a double affine Hecke algebra? How does it relate to more familiar mathematics? Find out here!

Category O

Fall 2021, Columbia

Covers the basic theory of category O for semisimple Lie algebras, including the Kazhdan-Lusztig conjectures.

Intersection theory

Spring 2021, Columbia

Ever wondered what a Chow ring is? Or how to compute intersection numbers on moduli spaces? Find out here!

Schemes

Spring 2021, Columbia

Covers schemes, sheaves, and cohomology, mostly following Hartshorne.

Lie groups and representations II

Spring 2021, Columbia

Covers invariant theory, Lie algebra cohomology, the cohomology of flag varieties, root systems, and Kac-Moody Lie algebras.

Geometric invariant theory

Fall 2020, Columbia

Covers the basics of reductive GIT and applications to moduli spaces of curves.

FGA explained

Fall 2020, Columbia

Ever wondered what Grothendieck meant in FGA? Find out here!

Lie groups and representations I

Fall 2020, Columbia

Covers the representation theory and classification of semisimple Lie groups.

Algebraic topology I

Fall 2020, Columbia

Covers homotopy groups, homology, cohomology, and Poincaré duality.

Singular spaces

Spring 2020, UMass

What is a singularity? How can we measure how bad it is and study its geometry? Find out here!

Symplectic topology

Spring 2020, UMass

Covers the basics of symplectic manifolds and some results in the topology of symplectic 4-manifolds.

Lie Algebras

Fall 2019, UMass

Covers the basic theory of Lie algebras and the classification and representation theory of semisimple Lie algebras.

Manifolds

Fall 2019, UMass

Covers the basic theory of smooth manifolds, differential forms, and de Rham cohomology.

Varieties

Spring 2019, UMass

Covers the basic theory of algebraic varieties, including dimension, tangent spaces, and divisors.

Algebra II

Spring 2019, UMass

Covers the basic theory of algebraic varieties, including dimension, tangent spaces, and divisors.

Analysis II

Spring 2019, UMass

Covers Hilbert and Banach spaces, convergence and compactness in infinite dimensions, and Fourier theory.

Representation theory

Fall 2018, UMass

Covers the representation theory of finite groups, Springer theory in type A, and the representation theory and classification of SLn.

Analysis I

Fall 2018, UMass

Covers measure theory, the Lebesgue integral, differentiation, and abstract measure theory.

Complex analysis

Spring 2018, UMass

Covers the basic theory of holomorphic functions, elliptic functions, and the Riemann mapping theorem.