Graduate Courses

Fundamentals of complex analysis, and further topics such as elliptic functions, canonical products, conformal mappings, the zeta function and prime number theorem, and Nevanlinna theory.

Prerequisites::

Basic complex analysis, topology of covering spaces, differential forms.

Recommended Prep:

Basic complex analysis, topology of covering spaces, differential forms.

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Fundamentals of algebraic curves as complex manifolds of dimension one. Topics may include branched coverings, sheaves and cohomology, potential theory, uniformization and moduli.

Recommended Prep:

Knowledge of the material in Mathematics 213a.

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This class is a graduate level course in Commutative Algebra, also aimed at undergraduates. Commutative Algebra lies at the foundation of many active research areas in mathematics like Algebraic Geometry, Number Theory and Algebraic Topology. It is also an important subject in itself. The different topics we will cover in this class will give a taste of these different directions. We will start by studying rings and ideals, associated prime ideals and primary decomposition in Noetherian rings. We will then study dimension theory and briefly touch upon Zariski topology. Then we will move to study integral extensions: going-up, going-down and Noether normalization theorems. The last part of the class will cover valuation rings, graded rings, Hilbert polynomials and homological methods in commutative algebra.

Recommended Prep:

Mathematics 123.

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Lie theory, including the classification of semi-simple Lie algebras and/or compact Lie groups and their representations.

Recommended Prep:

Knowledge of the material in Mathematics 114, 123 and 132.

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A graduate introduction to algebraic number theory. Topics: the structure of ideal class groups, groups of units, a study of zeta functions and L-functions, local fields, Galois cohomology, local class field theory, and local duality.

Recommended Prep:

Knowledge of the material in Mathematics 129.

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Continuation of Mathematics 223a. Topics: adeles, global class field theory, duality, cyclotomic fields. Other topics may include: Tate’s thesis or Euler systems.

Recommended Prep:

Knowledge of the material in Mathematics 223a.

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Complexes and homology. Homotopical category, its triangular structure. Projective and injective modules. Projective and injective resolutions. Derived functors. Functors Ext and Tor. Homological dimensions. Hilbert Syzygies Theorem. Local rings. Koszul complexes and their use. Homologies of groups. Tate cohomologies. Homologies of Lie algebras. Spectral sequences and their use. The spectral sequence of Lyndon/Hochschild-Serre. The spectral sequence of Künneth.Derived categories (if time permits)

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Fundamental methods, results, and problems of analytic number theory. Riemann zeta function and the Prime Number Theorem; Dirichlet’s theorem on primes in arithmetic progressions; lower bounds on discriminants from functional equations; sieve methods, analytic estimates on exponential sums, and their applications.

Recommended Prep:

Knowledge of the material in Mathematics 113 and 123.

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Basic properties and examples of smooth manifolds, Lie groups, and vector bundles; Riemannian geometry (metrics, geodesics, Levi-Civita connections, and Riemann curvature tensors); principal bundles and associated vector bundles with their connections and characteristic classes.

Recommended Prep:

Knowledge of the material in Mathematics 132 and 136.

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Covering spaces and fibrations. Simplicial and CW complexes, Homology and cohomology, universal coefficients and Künneth formulas. Hurewicz theorem. Manifolds and Poincaré duality.

Recommended Prep:

Knowledge of the material in Mathematics 131 and 132.

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Continuation of Mathematics 231a. Topics will be chosen from: Cohomology products, homotopy theory, bundles, obstruction theory, characteristic classes, spectral sequences, Postnikov towers, and topological applications.

Recommended Prep:

Knowledge of the material in Mathematics 231a

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Introduction to complex algebraic curves, surfaces, and varieties.

Recommended Prep:

Knowledge of the material in Mathematics 123 and 132 and 137.

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Introduction to the theory and language of schemes. Textbooks: Algebraic Geometry by Robin Hartshorne and Geometry of Schemes by David Eisenbud and Joe Harris. Weekly homework will constitute an important part of the course.

Recommended Prep:

Knowledge of the material in Mathematics 232a.

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This course introduces basic concepts of mathematical biology and evolutionary dynamics: reproduction, selection, mutation, genetic drift, quasi-species, finite and infinite population dynamics, game dynamics, evolution of cooperation, language, spatial models, evolutionary graph theory, infection dynamics, virus dynamics, somatic evolution of cancer.

Recommended Prep:

Mathematics 19a,b or 21a,b or 22a,b or 23a,b or 25a,b or 55a,b; or an equivalent background in mathematics.

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Research seminar on evolutionary dynamics, spanning mathematical and computational models of evolution in biological and social systems. Students attend a weekly lecture and conduct an original research project.

Recommended Prep:

Experience with mathematical biology at the level of Mathematics 153.

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The course will discuss representation theory of semisimple Lie algebras and (some) affine Kac-Moody algebras. Topics covered by this course include PBW theorem, Harish-Chandra isomorphism, category O, BGG resolution, Weyl character formula, Gelfand-Tsetlin theory, integrable representations of $\hat{sl_2}$ and other affine Kac-Moody algebras, as well as some applications to algebraic geometry and combinatorics.

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Kashiwara crystals are a combinatorial model of the tensor category of finite-dimensional representations of a semisimple complex Lie algebra $\mathfrak{g}$, where the weight basis vectors in the representation are represented by points (marked by the weights of the representation), and the action of the Chevalley generators by arrows (marked by simple roots). This can be also regarded as the $q\to 0$ limit of finite- dimensional $U_q(\mathfrak{g})$-modules. Crystals form a monoidal category in which the tensor product is not symmetric, but the tensor products of two crystals in different orders are still connected by some functorial isomorphism (similar to braiding but different). The role of the braid group here is played by the cactus group $J_n$ — the $S_n$-equivariant fundamental group of the Deligne-Mumford compactification $\overline{M_{0,n+1}}(\mathbb{R})$ moduli spaces of real stable rational curves with $n+1$ marked points. The famous combinatorial algorithms such as the Robinson-Schensted-Knuth correspondence and the Schützenberger involution become very natural in terms of this monoidal category. The main goal of this course is to explain how combinatorial structures like crystals arise from quantum integrable systems (more specifically, from the Gaudin magnet chain). This gives a new definition of the category of Kashiwara crystals for a given semisimple Lie algebra $\mathfrak{g}$in terms of $\mathfrak{g}^\vee$-opers (with respect to the Langlands dual of the Lie algebra $\mathfrak{g}^\vee$) on a rational curve with regular singularities at marked points. In particular, this explains the appearance of the Deligne-Mumford space in the theory of Kashiwara crystals.

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How much can one say about a symplectic manifold without having to know what a holomorphic disk is? For exact symplectic manifolds, the answer ought to be “everything.” This course will develop exact symplectic geometry from the perspective of microlocal sheaf theory, with a view toward applications in mirror symmetry and geometric representation theory. We will survey recent developments in the theory of sheaf quantization and “topological” approaches to the Fukaya category.

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The Kodaira dimension is a fundamental invariant in the classification of algebraic varieties. The course will focus on various current areas of study, like for instance understanding the Kodaira dimension of parameter spaces of varieties, or its behavior under morphisms. We will review the necessary tools from birational geometry, D-modules and Hodge theory.

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This course will survey the state-of-the-art on the theory of motives, motivic cohomology and algebraic K- theory for p-adic rings and schemes with p-adic coefficients. Topics include: the (logarithmic) de-Rham witt complexes (sheaves), the Geisser-Levine theorem and mod-p algebraic cycles, trace methods and topological cyclic homology, derived algebraic geometry methods, and prismatic methods. We will focus on examples and applications. Original results presented are all joint with Matthew Morrow. Prerequisites: first courses in algebraic geometry and algebraic topology, a curiosity with working at prime characteristics, and a voracious appetite for how abstract methods lead to concrete results.

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The focus of this course will be on derived (or “quantum”) invariants arising from symplectic geometry. We will particularly focus on connections between symplectic geometry and other fields (such as algebraic geometry, representation theory, homotopy theory, etc) which are invisible classically, but appear at the quantum level. Likely topics will include: symplectic geometry of Weinstein manifolds, Fukaya categories, microlocal sheaf theory, mirror symmetry, categorical entropy.

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The focus of the semester will be on complex-analytic techniques in dimensions bigger than one. Specifically, we will work through the basics of plurisubharmonic functions, currents on complex manifolds, and pluripotential theory. We will see how these tools are used in dynamical applications. If time permits, we will also study their p-adic (i.e., Berkovich-analytic) analogs. The topics covered in this course are motivated by connections between complex-algebraic dynamical systems and arithmetic geometry.

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The topic is the AD+ Duality Program and the Ultimate L Conjecture. The course will cover current developments after a survey of the basic notions, beginning with AD+ itself and continuing with large cardinals, the approximation and cover properties, and the HOD Dichotomy.

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Logarithmic geometry is a variant of algebraic geometry in which schemes are equipped with a sheaf of monoids. This so-called logarithmic structure can be thought of as a magic powder that reveals hidden smoothness in certain singular varieties. This class will serve as an introduction to logarithmic geometry with a focus on applications to compactifications and moduli spaces.

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This course will survey some examples and questions in low-dimensional topology, focusing on those questions where progress has been made using tools such as knot homology, Khovanov homology, and instanton Floer homology.

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Basics on vector bundles and Cohen-Macaulay modules. Problems of classification, the results of Grothendieck and Atiyah on the cases of projective line and elliptic curve. Vector bundles over chains of projective lines. Cohen-Macaulay modules over one-dimensional singularities (finite and tame cases). Two- dimensional singularities, quotient singularities, results of Auslander and Esnault. Kahn’s relation between Cohen-Macaulay modules and vector bundles. Minimal elliptic singularities. Gorenstein two-dimensional singularities and hypersurface singularities.

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This course will explore the interface between the geometry of numbers, sieve theory and analytic number theory with applications to arithmetic statistics.

Recommended Prep::

Knowledge of basic number theory and basic complex analysis.

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The non-abelian Chabauty method of Minhyong Kim is one of the premier methods for computing rational or integral points on curves in practice. This course will give a thorough description of the foundations of the method and the theoretical ingredients it combines, and will conclude with an overview of some computations of rational or integral points. Topics to be covered include: profinite and pro-unipotent etale fundamental groupoids; the Tannakian formalism; non-abelian cohomology and Selmer schemes; pro-unipotent de Rham and crystalline fundamental groupoids; Coleman functions and Coleman integrals; comparison isomorphisms for fundamental groupoids; applications of non-abelian Chabauty to the S-unit equation; quadratic Chabauty and the Mordell–Weil sieve; and recent applications to modular curves.

Recommended Prep:

First courses in algebraic geometry (including over non-algebraically closed fields), algebraic topology and category theory.

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This class will focus on understanding properties of spin glass models. We will review basic properties of the Sherrington-Kirkpatrick and related p-spin models. Topics to be discuss include the phase structures of these models and the Anderson-Thouless line. We will also cover basic stochastic analysis tools such as interpolation and concentration inequalities. If time permitted, we will discuss the gradient dynamics of these models.

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Become an effective instructor. This course focuses on observation, practice, feedback, and reflection providing insight into teaching and learning. Involves iterated videotaped micro-teaching sessions, accompanied by individual consultations. Required of all mathematics graduate students.

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