Graduate Courses

Fundamentals of complex analysis, and further topics such asconformal mapping, hyperbolic geometry, canonical products, elliptic functions and modular forms.

Prerequisites::

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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Commutative Algebra lies at the foundations of Number Theory and Algebraic Geometry. It plays an important role in Algebraic Topology, Geometry and other fields. We will cover the main topics of Commutative Algebra and give a taste of its applications. Starting from generalities on rings, modules and ideals, localization and primary decomposition in Noetherian rings and modules, we then move to integral extensions, going-up and going-down, Noether normalization and Hilbert’s Nullstellensatz, dimension theory. The final part of the class will cover graded rings, Hilbert polynomials and homological methods (if time permits, including regular local rings).

Recommended Prep:

Mathematics 122, 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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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, 132, and 137.

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This is a continuation of the material covered in the first semester, namely a complex analytic introduction to algebraic geometry. Among other things, we will discuss: higher dimensional manifolds, Hodge structures, polarizations, complex tori and abelian varieties, deformations of complex structures.

Recommended Prep:

Knowledge of the material in Mathematics 232a.

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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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This will be a course on recent progress on constructing algebraic vector bundles on smooth affine varieties. I hope to get through a proof that on a general class (called “cellular”) of smooth affine varieties over C, every topological vector bundle has an algebraic structure. The proof exploits recent advances on the unstable motivic Steenrod and the unstable motivic Adams-Novikov spectral sequence.

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Hodge theory for smooth projective varieties over the complex numbers is a classical subject. It has been extended over the years in various ways to the setting of singular varieties. This course will (roughly) be devoted to one such extension, namely the theory of the Deligne-Du Bois complex, also known as the filtered De Rham complex. The emphasis will be on applications to birational geometry and to the theory of singularities. In particular it will discuss rational and Du Bois singularities, and recently introduced refinements of these notions.

Recommended Prep:

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This course will provide a friendly overview to some of the main ideas and constructions in motivic homotopy theory. We will cover some of the major theorems in the field (purity, localization, representability of algebraic K- theory, affine representability), and work towards the development of obstruction theory in the motivic setting.

Recommended Prep:

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Posets (partially ordered sets) are fundamental objects in algebraic combinatorics. On the one hand, this course will cover important partial orders on objects of interest in algebra, combinatorics, and geometry; such objects include permutations, integer partitions, set partitions, lattice paths, rooted plane trees, elements of Coxeter groups, faces of polytopes, and regions of hyperplane arrangements. On the other hand, we will discuss posets in their own right in topological, dynamical, and algebraic contexts. Particular topics we will discuss include order complexes, shellability, poset dynamics, combinatorial billiards, lattice theory, Garside theory, and polytopality.

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Recently, Fargues and Scholze attached a semi-simple L-parameter to any representation of a p-adic reductive group, showing that the local Langlands correspondence could be realized in terms of a geometric Langlands correspondence over the Fargues-Fontaine curve. This implies that the cohomology of certain p-adic shtukas is related to the L-parameter they construct. These p-adic shtukas can be related to global Shimura varieties via p- adic Hodge theory, and this builds a bridge between the classical and geometric Langlands correspondences. The goal of this course will be to explain some of the basic geometry and technical tools behind the Fargues- Scholze construction, as well as to illustrate the connection to the classical story of Shimura varieties in some particular examples.

Recommended Prep:

Some basic number theory (Dirichlet characters) and basic complex analysis (Cauchy residue theorem).

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The goal of the class is to offer an introduction into an emerging field of Integrable Probability that unites a variety of exactly solvable stochastic models under its umbrella. Such models usually have deep connections with representation theory and algebraic combinatorics on one side, and with quantum integrable systems on another one. Algebraic techniques end up being crucial for fine asymptotic analysis of these integrable probabilistic models, and its results often predict similar limiting behavior for much broader universality classes of stochastic systems.

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Symmetries are classically described by groups. However, it turns out that many objects of interests, in particular, topological field theories, possess symmetries that are “noninvertible”. To capture these generalized symmetries, we are naturally led to the notion of a fusion category. This course will focus on the algebraic theory of fusion categories and the associated higher categorical structures as motivated by the study of topological field theories. Time permitting, we may also discuss applications of these concepts to condensed matter and high energy Physics.

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This course will be an introduction to the combinatorics of symmetric functions at the graduate level. We will introduce the classical families of symmetric functions and some of their generalizations. We will discuss how these functions encode various combinatorial objects such as tableaux and combinatorial maps.

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This course will be an introduction to the relationship between function fields and number fields, through the lens of Tschirnausen bundles (for function fields) and lattices (for number fields). Tentative topics include: moduli spaces of covers of the projective line, Brill-Noether theory, number field counting, and class groups in number fields.

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This course will be an introduction to the symplectic duality in the context of 3D mirror symmetry. Symplectic duality is the observation that certain varieties (called symplectic singularities) come in pairs with matching properties. It turns out that symplectic duality is closely related to the 3D mirror symmetry, which predicts an equivalence between (A- and B-twists) of certain 3D TQFT’s. We will discuss the general theory and will see how it works in many examples (Higgs and Coulomb branches of 3D N=2 quiver gauge theories, Slodowy varieties, slices in affine Grassmannians, ADHM spaces). Tentative topics include: quantum cohomology of symplectic resolutions of singularities, quantizations of symplectic singularities, Koszul duality, and categories O. No physics background is required.

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This course is an invitation to resolution of singularities. I will introduce the classical methods and discuss the complications that arise in the classical algorithm. I will then discuss the recent work of Abramovich—Temkin— Włodarczyk and McQuillan. They demonstrated that complications in the classical algorithm can be better resolved by working more broadly with algebraic stacks instead of schemes. As a result, they were able to construct an iterative procedure to embedded resolution of singularities in characteristic zero, where at every step, one blows up the ​”most singular locus”, and immediately witnesses an improvement in singularities. This course will conclude with a discussion of Teissier’s proposal to resolution of singularities in arbitrary characteristic via torific embeddings.

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A discussion of topics in complexity theory using the tools of algebraic geometry and representation theory. Possible topics include rank and border rank of tensors and the complexity of matrix multiplication, circuit complexity of polynomials and the permanent vs determinant problem, and others depending on interest. A strong background in linear algebra is required. Some experience in algebraic geometry and/or representation theory would be helpful but is not required.

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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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