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 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; 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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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 232ar.

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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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This course will survey recent developments in arithmetic statistics over function fields. Specifically, we will focus on three conjectures. The first is the Cohen-Lenstra heuristics about class groups of quadratic extensions. The second is Malle’s conjecture pertaining to counting Galois extensions. The third is the Poonen-Rains heuristics about Selmer groups and ranks of elliptic curves. The general plan for the course will be to explain how these conjectures are related to various forms of homological stability for Hurwitz spaces and explain what is known about the homology of these Hurwitz spaces.

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The course will be an introduction to some of the tools in low-dimensional topology, and some examples of how they are used.

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This course will describe categorifications of the Reshetikhin-Turaev invariant of links and their topological applications, both as originally developed by Khovanov and Rozansky via matrix factorizations and via the Roberts-Wagner foam evaluation. Particular goals will be to study the general theory of matrix factorizations in some detail and understand the connections between these two approaches. Once this background is established, we will discuss functoriality under cobordisms, and the many resulting obstructions to concordance.

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This course aims to provide an introduction to Khovanov homology with a focus on a non-exhaustive survey of its applications in the world of topology. A topologist can hope to leave this course with a better idea of when a tool like Khovanov homology could be helpful/how they might use it. On the other side of the spectrum, a categorification expert can hope to leave this course with a better idea of what types of questions topologists might hope to use Khovanov-like invariants for as well as the current gaps between what a topologist might want and what the theories are capable of doing.

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A Markov chain is a random process in which states change in a “memoryless” manner that only depends on the current state. Markov chains are ubiquitous objects in probability theory that permeate through both pure and applied mathematics. When one runs an irreducible Markov chain for a long time, the distribution of the states will converge to a stationary distribution. This course will develop tools for computing this stationary distribution and estimating how quickly the Markov chain converges to it.

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This class will focus on the theory of parabolic PDEs, namely well-posedness, regularity, and long-time behavior. Tentative topics include linear PDEs with variable coefficients, some theory for nonlinear PDEs, and Li-Yau Harnack inequalities. Important applications to other areas, such as stochastic processes and geometry, will be discussed throughout the semester according to students’ interests.

Pre-requisites:

Math 114 or Math 212, i.e. familiarity with Lebesgue integration, Banach spaces, Hilbert spaces, and Fourier analysis.

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We will review f-vector theory, from classical results to recent developments, focusing on face numbers of polytopes, simplicial spheres and other manifolds. Tools used combine Combinatorics (e.g.  graph theory and shelling), Algebra (e.g. framework rigidity, Stanley-Reisner rings and anisotropy), Algebraic Geometry (e.g. toric varieties and hard Lefschetz), and PL-topology.

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This course will be an introduction to mapping class groups with an eye towards 4-manifold topology. The mapping class group of a manifold is the group of isotopy classes of its diffeomorphisms or homeomorphisms. We will first discuss the case of surfaces, for which the mapping class group is often studied via its action on related spaces (e.g. curve complex, Teichmüller space) and has applications to 4-manifold topology (e.g. surface bundles, Lefschetz fibrations). We will also discuss some of what is known about the (smooth, topological) mapping class groups of 4-manifolds. Some tentative topics include the Nielsen—Thurston classification theorem, surface bundles and Lefschetz fibrations, basic 4-manifold topology, finite group actions on 4-manifolds.

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A discussion of topics in higher dimensional algebraic geometry over a field of characteristic p>0, with an emphasis on pathological examples. Possible topics: F-split singularities and their relation to singularities of the Minimal Model Program, Witt liftability, the theorem of Deligne and Illusie, vanishing and extension theorems, subadditivity of Kodaira dimension, stable families.

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This course will survey one of the most exciting recent developments in algebraic combinatorics, namely, Fomin and Zelevinsky’s theory of cluster algebras. Cluster algebras are a class of combinatorially defined commutative rings that provide a unifying structure for phenomena in a variety of algebraic and geometric contexts. Introduced in 2001, cluster algebras have already been shown to be related to a host of other fields of math, such as quiver representations, Teichmuller theory, Poisson geometry, and total positivity. Cluster structures in Grassmannians have in particular been linked to integrable systems and physics. In the first part of the course I will cover the basics of cluster algebras and total positivity. In the second part of the class I will discuss recent developments and applications of the theory (topics could include the positive Grassmannian, the amplituhedron, KP solitons, etc). I will assume that people have some familiarity with combinatorics. Familiarity with root systems would also be helpful. I will not assume prior knowledge of total positivity or cluster algebras.

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Will present important techniques in several complex variables developed over the last few decades with motivations from, and applications to, related fields such as partial differential equations, differential geometry, algebraic geometry and arithmetic geometry. Some examples are: (i) Solvability and regularity problems of partial differential equations. (ii) Global non-deformability and strong/super rigidity problems. (iii) Effective theorems such as the Fujita conjecture, especially its very am- pleness part. (iv) Skoda’s division and finite generation of canonical ring for any general compact complex manifold. (v) Hyperbolicity problem and Nevanlinna theory for generic complex hyper- surface. (vi) Gelfond-Schneider’s technique for Hilbert’s 7th problem, Lang-Bombieri’s theory of algebraic values of meromorphic maps, Bombieri-Pila’s counting of integral points on arcs, Pila-Wilkie’s o-minimal geometry, together with fur- ther later developments by many others, to treat the Andr ́e-Oort conjecture

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The concept of “BPS state” in quantum field theory is connected to a vast array of rich mathematics ranging from PDEs to category theory and homological algebra. In this course we will uncover this gradually and systematically by working through examples connected to field theories in two, three and four dimensions. Along the way we will aim to connect this notion to the mathematics of Fukaya-Seidel categories, Fueter 2-categories and cohomological Hall algebras.

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We will survey foundational methods in arithmetic geometry and Diophantine equations. Topics may include the arithmetic of elliptic curves, abelian varieties, heights, Diophantine approximation, Siegel’s Theorem on integral points, Vojta’s proof of Faltings’ Theorem, and arithmetic dynamics.

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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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