Graduate Courses

Fundamentals of complex analysis, and further topics such as conformal 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 223ar.

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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; asymptotics for arithmetic functions; sieve methods; analytic estimates of exponential sums and their applications.

Recommended Prep:

Knowledge of the material in Mathematics 113 and 123.

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The overarching theme is the Felix Klein dictum that places symmetry groups at the center of geometry. Topics include: affine geometry and symmetry types, smooth manifolds and Lie groups, Frobenius theorem; classical curves and surfaces; fiber bundles, principal bundles, connections, geodesics, curvature, torsion; Riemannian manifolds, Levi-Civita connection, Riemann curvature tensor; Gauss-Bonnet-Chern theorem; more Riemannian geometry.

Recommended Prep:

Knowledge of the material in Mathematics 132 and 136.

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A continuation of Mathematics 230a. Topics in complex differential geometry: Complex Manfifolds. Kahler metrics. Ricci curvature. Calabi Conjecture and its proof. Miyaoka-Yau Chern number inequalities and uniformization. Uniqueness of Kahler structure of projective spaces. Calabi-Yau manifolds and their moduli.

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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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This course is a general introduction to scheme theory and other foundational aspects of algebraic geometry. Occasionally this may be replaced by an introduction to the complex analytic side of algebraic geometry, via complex manifolds. See the course Canvas website for more about the semester’s course focus.

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, with a focus on coherent sheaves, cohomology, and their applications to the theory of curves and surfaces. Occasionally it may cover Hodge structures, Lefschetz theorems, or deformations. See the course Canvas website for more about the semester’s course focus.

Recommended Prep:

Knowledge of the material in Mathematics 232ar.

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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 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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This course will survey various aspects of Lagrangian Floer theory, starting with its introduction to address classical conjectures in symplectic geometry, and the rich algebraic structures captured by the Fukaya category. These will also be discussed through the lens of homological mirror symmetry. Further topics include family Floer cohomology and local-to-global principles for Fukaya categories.

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This course begins with a review of the fundamental properties of Brownian motion and martingales. The core curriculum covers Ito calculus and its connection to partial differential equations. Finally, we will apply these tools to derive the Wigner semicircle law and Dyson’s Brownian motion in random matrices.

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In this course we will introduce and develop the theory of dynamics and iteration in one complex variable. The Mandelbrot set, a fractal now famous both in mathematics and popular culture, is a central object of study in this field; we will focus on exploring this set and the modern techniques used to study it.

Prerequisites:

Basic complex analysis

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In number theory and algebraic geometry one often constructs an object of interest without giving a good way to compute it explicitly, even when the object is as down-to-earth as an integer or a polynomial. We develop and use some of the techniques, tools, tactics, and tricks that often let us exhibit and study such objects. Most of our motivating examples are low-dimensional moduli spaces of various kinds and the structures they parametrize.

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This class will be an introduction to topological and geometric combinatorics at the graduate level, covering several general areas:(1) Posets (2) Simplicial complexes (3) Matroids (4) Polytopes. One of the main themes of the class will be the question: “To what extent do combinatorial properties of an object determine its topology or geometry?” For example, to what extent does the face lattice of a simplicial or cell complex determine its homotopy type? To what extent does the graph of a polytope determine the polytope? And what kinds of combinatorial techniques can we use to then understand the topology of the object in question? Along the way we will discuss interesting examples coming from posets, polytopes, matroids, Coxeter groups, the Grassmannian and flag varieties, etc.

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Gromov-Witten invariants are numbers which can be interpreted as counts of holomorphic curves in symplectic manifolds.This course will be an introduction to Gromov-Witten theory for symplectic manifolds, with a view toward their connections to mirror symmetry and Fukaya categories. We will discuss the construction of genus zero Gromov-Witten invariants, i.e. counts of rational curves, as well as some of their classical applications to low-dimensional symplectic topology and Hamiltonian dynamics. In the second part of the class, we will explain the role of Gromov-Witten invariants in mirror symmetry, and survey some related recent developments. Potential topics include global Kuranishi charts, real and open Gromov-Witten invariants, and computations from Fukaya categories.

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This course will provide an introduction to the C*-algebraic formalism of quantum spin systems. Comfort with basic functional analysis of Banach and Hilbert spaces will be assumed. The theory of operator algebras will play a significant role and will be introduced as needed, but no prior knowledge of operator algebras is expected. We will cover Lieb-Robinson bounds and some of their many consequences for the ground states of such systems, such as exponential decay of correlations and quasi-adiabatic evolution. Following this, we will focus on entanglement properties and topological properties of ground states and their phases.

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This course aims to provide an introduction to the modern field of special holonomy geometry. After discussing what might now be regarded as the “classical” story of holonomy, culminating in Berger’s classification theorem, we will delve into a detailed study of the special and exceptional cases in the classification. The setting for the first third of the course will be Kähler and hyperKähler geometry, with an emphasis on the use of geometric gluing techniques to construct fundamental examples. In the second third of the course, we will discuss the exceptional G2 and Spin(7) geometries. Our focus will again be on non-trivial constructions (for which the aforementioned special geometries will often serve as building blocks) involving clever uses of symmetry and geometric analysis. Such constructions will often live near the boundary of certain degenerations and thus we will find that “spaces”, by which we mean more singular objects than smooth manifolds, play a central role in the theory. In the final third of the course, we will introduce Harvey and Lawson’s notion of a calibrated geometry and study the interplay between special holonomy and special minimal submanifolds known as calibrated submanifolds. The geometric analysis developed in the first two-thirds of the course will serve us in constructing non-trivial examples of compact calibrated submanifolds and if time permits at the end of the course we will discuss related open problems involving non-compactness questions and putative enumerative theories.

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This course aims to give a foundation in low-dimensional topology for graduate students and strong undergraduate students interested in the topic. We will focus on some pre-Floer theory developments in the field with an eye towards contemporary applications. An incomplete list of reference books includes Gompf and Stipsicz, Rolfsen, and Saveliev.

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A classical problem in homotopy theory is to understand the homotopy groups of spheres, $\pi_{n+k}(S^k)$. A classical theorem of Freudenthal says that these groups are independent of $k$ once $k>n+1$. The groups one obtains in this stable range are known as the stable homotopy groups of spheres, denoted $\π_n S$. By a theorem of Serre, these groups are finite abelian groups for $n>0$. Understanding the groups $\π_n S$ remains one of the central motivating problems in homotopy theory.The main approach to this problem is through the theory of spectra. Spectra are mathematical objects analogous to abelian groups in the homotopical world. Over the years, it has become clear that spectra are ubiquitous in mathematics: they play important roles not only in the study of stable homotopy groups of spheres, but also in the classification of manifolds, algebraic K-theory, p-adic geometry, condensed matter physics, algebraic geometry, representation theory, and many other areas.In this two-semester-long course, I will present an introduction to the theory of spectra, using the stable homotopy groups of spheres as a motivating problem. Particular emphasis will be placed on the methods of chromatic homotopy theory as an approach to the study of spectra.

Prerequisites:

A first-year graduate course in algebraic topology, such as Math 231a, or equivalent background plus basic category theory . Some familiarity with homological algebra would also be very helpful.

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A classical problem in homotopy theory is to understand the homotopy groups of spheres, $\pi_{n+k}(S^k)$. A classical theorem of Freudenthal says that these groups are independent of $k$ once $k>n+1$. The groups one obtains in this stable range are known as the stable homotopy groups of spheres, denoted $\π_n S$. By a theorem of Serre, these groups are finite abelian groups for $n>0$. Understanding the groups $\π_n S$ remains one of the central motivating problems in homotopy theory.The main approach to this problem is through the theory of spectra. Spectra are mathematical objects analogous to abelian groups in the homotopical world. Over the years, it has become clear that spectra are ubiquitous in mathematics: they play important roles not only in the study of stable homotopy groups of spheres, but also in the classification of manifolds, algebraic K-theory, p-adic geometry, condensed matter physics, algebraic geometry, representation theory, and many other areas.In this two-semester-long course, I will present an introduction to the theory of spectra, using the stable homotopy groups of spheres as a motivating problem. Particular emphasis will be placed on the methods of chromatic homotopy theory as an approach to the study of spectra.

Prerequisites::

A first-year graduate course in algebraic topology, such as Math 231a, or equivalent background plus basic category theory . Some familiarity with homological algebra would also be very helpful.

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Vertex algebras provide a rigorous mathematical framework for two-dimensional conformal field theory. This course offers a comprehensive introduction to the subject, bridging its algebraic and geometric aspects. The first half focuses on local constructions, covering the foundational axioms and key examples of vertex algebras (such as Kac-Moody and Virasoro algebras). The second half transitions to the global framework, introducing conformal blocks associated with a vertex algebra and vertex algebra bundles on a curve. We will then cover the fundamentals of chiral algebras following the work of Beilinson and Drinfeld. Time permitting, we will discuss advanced topics such as representation theory, chiral differential operators, AGT Correspondence etc.

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We will study recent advances in number theory. One of the aims of the class is to help participants improve their expository skills by giving talks on recent number theory papers, or on their own work in this subject.

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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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