### Courses Fall 2016

MATH MA | Introduction to Functions and Calculus I | Kelly | Sectioned | The study of functions and their rates of change. Fundamental ideas of calculus are introduced early and used to provide a framework for the study of mathematical modeling involving algebraic, exponential, and logarithmic functions. Thorough understanding of differential calculus promoted by year long reinforcement. Applications to biology and economics emphasized according to the interests of our students. |

MATH 1A | Introduction to Calculus | Cain | Sectioned | The development of calculus by Newton and Leibniz ranks among the greatest achievements of the past millennium. This course will help you see why by introducing: how differential calculus treats rates of change; how integral calculus treats accumulation; and how the fundamental theorem of calculus links the two. These ideas will be applied to problems from many other disciplines. |

MATH 1B | Calculus, Series, and Differential Equations | Penner | Sectioned | Speaking the language of modern mathematics requires fluency with the topics of this course: infinite series, integration, and differential equations. Model practical situations using integrals and differential equations. Learn how to represent interesting functions using series and find qualitative, numerical, and analytic ways of studying differential equations. Develop both conceptual understanding and the ability to apply it. |

MATH 18 | Multivariable Calculus for Social Sciences | Hsu | MWF 11, SC 216 | Focus on concepts and techniques of multivariable calculus most useful to those studying the social sciences, particularly economics: functions of several variables; partial derivatives; directional derivatives and the gradient; constrained and unconstrained optimization, including the method of Lagrange multipliers. Covers linear and polynomial approximation and integrals for single variable and multivariable functions; modeling with derivatives. Covers topics from Math 21a most useful to social sciences. |

MATH 19A | Modeling and Differential Equations for the Life Sciences | Cain | MWF 1, SC Hall E | Considers the construction and analysis of mathematical models that arise in the life sciences, ecology and environmental life science. Introduces mathematics that include multivariable calculus, differential equations in one or more variables, vectors, matrices, and linear and non-linear dynamical systems. Taught via examples from current literature (both good and bad). |

MATH 21A | Multivariable Calculus | Knill | Sectioned | To see how calculus applies in practical situations described by more than one variable, we study: Vectors, lines, planes, parameterization of curves and surfaces, partial derivatives, directional derivatives and the gradient, optimization and critical point analysis, including constrained optimization and the Method of Lagrange Multipliers, integration over curves, surfaces and solid regions using Cartesian, polar, cylindrical, and spherical coordinates, divergence and curl of vector fields. |

MATH 21B | Linear Algebra and Differential Equations | Chen | Sectioned | Matrices provide the algebraic structure for solving myriad problems across the sciences. We study matrices and related topics such as linear transformations and linear spaces, determinants, eigenvalues, and eigenvectors. Applications include dynamical systems, ordinary and partial differential equations, and an introduction to Fourier series. |

MATH 23A | Linear Algebra and Real Analysis I | Patel | TTh 1:30-3, SC Hall A | Linear algebra: vectors, linear transformations and matrices, scalar and vector products, basis and dimension, eigenvectors and eigenvalues, including an introduction to the R scripting language. Single-variable real analysis: sequences and series, limits and continuity, derivatives, inverse functions, power series and Taylor series. Multivariable real analysis and calculus: topology of Euclidean space, limits, continuity, and differentiation in n dimensions, inverse and implicit functions |

MATH 25A | Honors Linear Algebra and Real Analysis I | Peterson | MWF 10, SC Hall E | A rigorous treatment of linear algebra. Topics include: Construction of number systems; fields, vector spaces and linear transformations; eigenvalues and eigenvectors, determinants and inner products. Metric spaces, compactness and connectedness. |

MATH 55A | Honors Abstract Algebra | Elkies | MWF 11, SC 507 | A rigorous treatment of abstract algebra including linear algebra and group theory. |

MATH 60R | Reading Course for Senior Honors Candidates | Lurie | TBA | Advanced reading in topics not covered in courses. |

MATH 99R | Tutorial | Lurie | TBA | Supervised small group tutorial. In the fall semester, the title is "Arithmetic of elliptic curves" taught by Zijian Yao More information. |

MATH 101 | Sets, Groups and Knots | McMullen | TuTh 10-11:30, SC 216 | An introduction to rigorous mathematics, axioms, and proofs, via topics such as set theory, symmetry groups, and low-dimensional topology. |

MATH 114 | Analysis II: Measure, Integration and Banach Spaces | Haiden | TTh 1-2:30, SC 310 | Lebesgue measure and integration; general topology; introduction to Lp spaces, Banach and Hilbert spaces, and duality. |

MATH 116 | Real Analysis, Convexity, and Optimization | Bamberg | MW 12:30-2, SC 222 | Develops the theory of convex sets, normed infinite-dimensional vector spaces, and convex functionals and applies it as a unifying principle to a variety of optimization problems such as resource allocation, production planning, and optimal control. Topics include Hilbert space, dual spaces, the Hahn-Banach theorem, the Riesz representation theorem, calculus of variations, and Fenchel duality. Students will be expected to understand and invent proofs of theorems in real and functional analysis. |

MATH 117 | Probability and Random Processes with Economic Applications | Bamberg | TuTh 2:30-4, SC 104 | A self-contained treatment of the theory of probability and random processes with specific application to the theory of option pricing. Topics: axioms for probability, calculation of expectation by means of Lebesgue integration, conditional probability and conditional expectation, martingales, random walks and Wiener processes, and the Black-Scholes formula for option pricing. Students will work in small groups to investigate applications of the theory and to prove key results. |

MATH 121 | Linear Algebra and Applications | Ullery | MWF 11, SC 411 | Real and complex vector spaces, linear transformations, determinants, inner products, dual spaces, and eigenvalue problems. Applications to geometry, systems of linear differential equations, optimization, and Markov processes. |

MATH 122 | Algebra I: Theory of Groups and Vector Spaces | Hopkins | MWF 2, SC Hall E | Groups and group actions, vector spaces and their linear transformations, bilinear forms and linear representations of finite groups. |

MATH 124 | Number Theory | Tripathy | TTh 10-11:30, SC 411 | Factorization and the primes; congruences; quadratic residues and reciprocity; continued fractions and approximations; Pell's equation; selected Diophantine equations; theory of integral quadratic forms. |

MATH 131 | Topology I: Topological Spaces and the Fundamental Group | Harris | MWF 10, SC 507 | First, an introduction to abstract topological spaces and their properties; and then, an introduction to algebraic topology and in particular homotopy theory, fundamental groups and covering spaces. |

MATH 136 | Differential Geometry | Taubes | MWF 12, SC 507 | The course is an introduction to Riemannian geometry with the focus (for the most part) being the Riemannian geometry of curves and surfaces in space where the fundamental notions can be visualized. |

MATH 145A | Set Theory I | Boney | TTh 1-2:30, SC 411 | An introduction to set theory covering the fundamentals of ZFC (cardinal arithmetic, combinatorics, descriptive set theory) and the independence techniques (the constructible universe, forcing, the Solovay model). We will demonstrate the independence of CH (the Continuum Hypothesis), SH (Suslin's Hypothesis), and some of the central statements of classical descriptive set theory. |

MATH 152 | Discrete Mathematics | Bamberg | TTh 11:30-1, NW Build B-104 | An introduction to sets, logic, finite groups, finite fields, finite geometry, finite topology, combinatorics and graph theory. A recurring theme of the course is the symmetry group of the regular icosahedron. Elementary category theory will be introduced as a unifying principle. Taught in a seminar format: students will gain experience in presenting proofs at the blackboard. |

MATH 153 | Mathematical Biology-Evolutionary Dynamics | Nowak | TTh 2:30-4, 1 Brattle Sq 616 | Introduces basic concepts of mathematical biology and evolutionary dynamics: evolution of genomes, quasi-species, finite and infinite population dynamics, chaos, game dynamics, evolution of cooperation and language, spatial models, evolutionary graph theory, infection dynamics, somatic evolution of cancer. |

MATH 212A | Real Analysis | HT Yau | MWF 10, SC 310 | Measure theory, functional analysis, Sobolev spaces and introduction to harmonic analysis. |

MATH 213A | Complex Analysis | Schmid | TuTh 2:30-4, SC 216 | A second course in complex analysis: series, product and partial fraction expansions of holomorphic functions; Hadamard's theorem; conformal mapping and the Riemann mapping theorem; elliptic functions; Picard's theorem and Nevanlinna Theory. |

MATH 221 | Algebra | Pasten Vasquez | MWF 11, SC 310 | A first course in Algebra: Noetherian rings and modules, Hilbert basis theorem, Cayley-Hamilton theorem, integral dependence, Galois theory, Noether normalization, the Nullstellensatz, localization, primary decomposition. Representation theory of finite groups. Introduction to Lie groups and Lie algebras: definitions, the exponential maps, semi-simple Lie algebras, examples. |

MATH 230A | Differential Geometry | Collins | MWF 1, SC 507 | Smooth manifolds (vector fields, differential forms, and their algebraic structures; Frobenius theorem), Riemannian geometry (metrics, connections, curvatures, geodesics, flatness, and manifolds of constant curvature), symplectic geometry, Lie groups, principal bundles. |

MATH 231A | Algebraic Topology | Kronheimer | MWF 2, SC 507 | Covering spaces and fibrations. Simplicial and CW complexes, Homology and cohomology, universal coefficients and Künneth formulas. Hurewicz theorem. Manifolds and Poincare duality. |

MATH 232A | Introduction to Algebraic Geometry I | Engel | MWF 12, SC 310 | Introduction to complex algebraic curves, surfaces, and varieties. |

MATH 256X | Heisenberg Calculus in Quantum Topology | McLellan | MWF 12, SC 411 | The main goal of this course is to provide a hands-on introduction to the Heisenberg calculus of Beals-Greiner and Taylor. This pseudodifferential calculus provides a natural method for studying non-elliptic problems where the classical abelian symbol calculus is not applicable. This theory is of indenpendent interest and also makes connections with diversive topics such as harmonic analysis, noncommutative geometry, number theory, partial differential equations, representation theory. |

MATH 258Y | Degenerations in Algebraic Geometry | Patel | MWF 2, SC ??? | Many results in algebraic geometry can be proved by degenerating the varieties under consideration. The construction of useful degenerations is more an art rather than a science, highly dependent on context, but it is helpful to have an understanding of some "standard" examples. This course is intended to study examples which serve to illustrate common technical issues that arise under degeneration. We will attempt to cover instances of degeneration in three areas. |

MATH 263 | Analytic Techniques in Algebraic Geometry | Collins | TTh 10-11:30, SC 310 | This course will discuss applications of analysis to algebraic geometry. Topics: the d-bar equations, plurpotential theory and techniques of multiplier ideal sheaves with applications to Kodaira embedding, the Fujita conjecture, invariance of plurigenera (after Siu) and asymptotic invariants of line bundles. Possible further topics: applications of harmonic maps to the Frankel conjecture (after Siu-Yau), and applications to Yang-Mills theory to classification of class VII |

MATH 274 | 3d-3d Duality | Tripathy | SC 310 | An introduction to three-dimensional mirror symmetry, known as symplectic duality in the mathematical literature. Possible topics: the foundational papers of Braden-Licata-Proudfoot-Webster, the proposed construction of the Coulomb branch in the moduli space of vacua by Braverman-Finkelbert-Nakajima, and relations to geometric Langlands and two-dimensional mirror symmetry, especially quantum cohomology of symplectic resolutions and mirror symmetry for hypertoric varieties. |

MATH 283 | Geometric Langlands Correspondence in Characteristic p | Gaitsgory | TTh 11:30-1, SC 310 | The genuine geometric Langlands correspondence is a difficult open problem. But it has a variant that is much more easily accessible (but still very interesting): when we consider D-modules in characteristic p. We will cover recent developments in this field, surveying the works of Braverman, Bezrukanikov, Chen, Travkin, and Zhu. |

MATH 286 | Nonlinear Analysis in Geometry | ST. Yau | TTh 11:30-1, SC 507 | We shall cover nonlinear equations appearing in complex geometry and their applications to algebraic geometry and physics. |

MATH 300 | Teaching Undergraduate Mathematics | Al-Aidroos | Tue 1-2:30, SC 507 | 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. |

MATH 303 | Topics in Diophantine Problems | Pasten Vasquez | TBA | NA |

MATH 304 | Topics in Algebraic Topology | Hopkins | TBA | NA |

MATH 308 | Topics in Number Theory and Modular Forms | Gross | TBA | NA |

MATH 314 | Topics in Differential Geometry and Mathematical Physics | Sternberg | TBA | NA |

MATH 316 | Topics in Algebraic Geometry | TBA | TBA | NA |

MATH 318 | Topics in Number Theory | Mazur | TBA | NA |

MATH 321 | Topics in Mathematical Physics | Jaffe | TBA | NA |

MATH 327 | Topics in Several Complex Variables | Siu | TBA | NA |

MATH 333 | Topics in Complex Analysis, Dynamics and Geometry | McMullen | TBA | NA |

MATH 335 | Topics in Differential Geometry and Analysis | Taubes | TBA | NA |

MATH 343 | Topics in Complex Geometry | Collins | TBA | NA |

MATH 345 | Topics in Geometry and Topology | Kronheimer | TBA | NA |

MATH 346Y | Topics in Analysis: Quantum Dynamics | Yau | TBA | NA |

MATH 348 | Topics in Representation Theory | Haiden | TBA | NA |

MATH 352 | Topics in Algebraic Number Theory | Kisin | TBA | NA |

MATH 356 | Topics in Harmonic Analysis | Schmid | TBA | NA |

MATH 357 | Topics in Model Theory | Boney | TBA | NA |

MATH 361 | Topics in Differential Geometry and Analysis | Canzani | TBA | NA |

MATH 362 | Topics in Number Theory | Miller | TBA | NA |

MATH 364 | Topics in Algebraic Geometry | Ullery | TBA | NA |

MATH 365 | Topics in Differential Geometry | S.T. Yau | TBA | NA |

MATH 368 | Topics in Algebraic Topology | Peterson | TBA | NA |

MATH 373 | Topics in Algebraic Topology | Lurie | TBA | NA |

MATH 381 | Introduction to Geometric Representation Theory | Gaitsgory | TBA | NA |

MATH 382 | Topics in Algebraic Geometry | Harris | TBA | NA |

MATH 385 | Topics in Set Theory | Woodin | TBA | NA |

MATH 387 | Topics in Mathematical Physics: Bridgeland Stability Conditions | Tanaka | TBA | NA |

MATH 388 | Topics in Mathematics and Biology | Nowak | TBA | NA |

MATH 389 | Topics in Number Theory | Elkies | TBA | NA |

### Courses Spring 2017

MATH MB | Introduction to Functions and Calculus II | Kelly | Sectioned | Continued investigation of functions and differential calculus through modeling; an introduction to integration with applications; an introduction to differential equations. Solid preparation for Mathematics 1b. |

MATH 1A | Introduction to Calculus | Gupta | Sectioned | The development of calculus by Newton and Leibniz ranks among the greatest achievements of the past millennium. This course will help you see why by introducing: how differential calculus treats rates of change; how integral calculus treats accumulation; and how the fundamental theorem of calculus links the two. These ideas will be applied to problems from many other disciplines. |

MATH 1B | Calculus, Series, and Differential Equations | Gottlieb | Sectioned | Speaking the language of modern mathematics requires fluency with the topics of this course: infinite series, integration, and differential equations. Model practical situations using integrals and differential equations. Learn how to represent interesting functions using series and find qualitative, numerical, and analytic ways of studying differential equations. Develop both conceptual understanding and the ability to apply it. |

MATH 19B | Linear Algebra, Probability, and Statistics for the Life Sciences | Belanger-Rioux | TBA | Probability, statistics and linear algebra with applications to life sciences, chemistry, and environmental life sciences. Linear algebra includes matrices, eigenvalues, eigenvectors, determinants, and applications to probability, statistics, dynamical systems. Basic probability and statistics are introduced, as are standard models, techniques, and their uses including the central limit theorem, Markov chains, curve fitting, regression, and pattern analysis. |

MATH 21A | Multivariable Calculus | Chen | Sectioned | To see how calculus applies in practical situations described by more than one variable, we study: Vectors, lines, planes, parameterization of curves and surfaces, partial derivatives, directional derivatives and the gradient, optimization and critical point analysis, including constrained optimization and the Method of Lagrange Multipliers, integration over curves, surfaces and solid regions using Cartesian, polar, cylindrical, and spherical coordinates, divergence and curl of vector fields. |

MATH 21B | Linear Algebra and Differential Equations | Knill | Sectioned | Matrices provide the algebraic structure for solving myriad problems across the sciences. We study matrices and related topics such as linear transformations and linear spaces, determinants, eigenvalues, and eigenvectors. Applications include dynamical systems, ordinary and partial differential equations, and an introduction to Fourier series. |

MATH 23B | Linear Algebra and Real Analysis II | Patel | TBA | A rigorous, integrated treatment of linear algebra and multivariable calculus. Topics: Riemann and Lebesgue integration, determinants, change of variables, volume of manifolds, differential forms, and exterior derivative. Stokes's theorem is presented both in the language of vector analysis (div, grad, and curl) and in the language of differential forms. |

MATH 25B | Honors Linear Algebra and Real Analysis II | Peterson | TBA | A rigorous treatment of basic analysis. Topics include: convergence, continuity, differentiation, the Riemann integral, uniform convergence, the Stone-Weierstrass theorem, Fourier series, differentiation in several variables. Additional topics, including the classical results of vector calculus in two and three dimensions, as time allows. |

MATH 55B | Honors Real and Complex Analysis | Elkies | TBA | A rigorous treatment of real and complex analysis. |

MATH 60R | Reading Course for Senior Honors Candidates | Lurie | TBA | Advanced reading in topics not covered in courses. |

MATH 91R | Supervised Reading and Research | Lurie | TBA | Programs of directed study supervised by a person approved by the Department. |

MATH 99R | Tutorial | Lurie | TBA | Supervised small group tutorial. Topics to be arranged. |

MATH 101 | Sets, Groups and Topology | Grundmeier | TBA | An introduction to rigorous mathematics, axioms, and proofs, via topics including set theory, symmetry groups, and low-dimensional topology. |

MATH 110 | Vector Space Methods for Differential Equations | McLellan | TBA | Develops the theory of inner product spaces, both finite-dimensional and infinite-dimensional, and applies it to a variety of ordinary and partial differential equations. Topics: existence and uniqueness theorems, Sturm-Liouville systems, orthogonal polynomials, Fourier series, Fourier and Laplace transforms, eigenvalue problems, and solutions of Laplace's equation and the wave equation in the various coordinate systems. |

MATH 112 | Introductory Real Analysis | Haiden | TBA | An introduction to mathematical analysis and the theory behind calculus. An emphasis on learning to understand and construct proofs. Covers limits and continuity in metric spaces, uniform convergence and spaces of functions, the Riemann integral. |

MATH 113 | Analysis I: Complex Function Theory | Siu | TBA | Analytic functions of one complex variable: power series expansions, contour integrals, Cauchy's theorem, Laurent series and the residue theorem. Some applications to real analysis, including the evaluation of indefinite integrals. An introduction to some special functions. |

MATH 115 | Methods of Analysis | TBA | TBA | Complex functions; Fourier analysis; Hilbert spaces and operators; Laplace's equations; Bessel and Legendre functions; symmetries; Sturm-Liouville theory. |

MATH 118R | Dynamical Systems | Cain | TBA | Introduction to dynamical systems theory with a view toward applications. Topics include existence and uniqueness theorems for flows, qualitative study of equilibria and attractors, iterated maps, and bifurcation theory. |

MATH 123 | Algebra II: Theory of Rings and Fields | Mazur | TBA | Rings and modules. Polynomial rings. Field extensions and the basic theorems of Galois theory. Structure theorems for modules. |

MATH 129 | Number Fields | Kisin | TBA | Algebraic number theory: number fields, unique factorization of ideals, finiteness of class group, structure of unit group, Frobenius elements, local fields, ramification, weak approximation, adeles, and ideles. |

MATH 130 | Classical Geometry | Hopkins | TBA | Presents several classical geometries, these being the affine, projective, Euclidean, spherical and hyperbolic geometries. They are viewed from many different perspectives, some historical and some very topical. Emphasis on reading and writing proofs. |

MATH 132 | Topology II: Smooth Manifolds | Melvin | TBA | Differential manifolds, smooth maps and transversality. Winding numbers, vector fields, index and degree. Differential forms, Stokes' theorem, introduction to cohomology. |

MATH 137 | Algebraic Geometry | Kronheimer | TBA | Affine and projective spaces, plane curves, Bezout's theorem, singularities and genus of a plane curve, Riemann-Roch theorem. |

MATH 154 | Probability Theory | Taubes | TBA | An introduction to probability theory. Discrete and continuous random variables; distribution and density functions for one and two random variables; conditional probability. Generating functions, weak and strong laws of large numbers, and the central limit theorem. Geometrical probability, random walks, and Markov processes. |

MATH 155R | Combinatorics | Harris | TBA | An introduction to counting techniques and other methods in finite mathematics. Possible topics include: the inclusion-exclusion principle and Mobius inversion, graph theory, generating functions, Ramsey's theorem and its variants, probabilistic methods. |

MATH 157 | Mathematics in the World | Harris | TBA | An interactive introduction to problem solving with an emphasis on subjects with comprehensive applications. Each class will be focused around a group of questions with a common topic: logic, information, number theory, probability, and algorithms. |

MATH 212BR | Advanced Real Analysis | TBA | TBA | Functional analysis related to quantum mechanics. Topics include (but not limited to) The Stone-von Neumann theorem, Gruenwald-van Hove theorem, Ruelle's theorem on the continuous spectrum and scattering states, Agmon's theorem on the exponential decay of bound states, scattering theory. |

MATH 222 | Lie Groups and Lie Algebras | Haiden | TBA | Lie theory, including the classification of semi-simple Lie algebras and/or compact Lie groups and their representations. |

MATH 224 | Representations of Reductive Lie Groups | Gaitsgory | TBA | Structure theory of reductive Lie groups, unitary representations, Harish Chandra modules, characters, the discrete series, Plancherel theorem. |

MATH 229X | Introduction to Analytic Number Theory | Pasten Vasquez | TBA | 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. |

MATH 230BR | Advanced Differential Geometry | ST Yau | TBA | A continuation of Mathematics 230a. Topics in differential geometry: Analysis on manifolds. Laplacians. Hodge theory. Spin structures. Clifford algebras. Dirac operators. Index theorems. Applications. |

MATH 231BR | Advanced Algebraic Topology | Peterson | TBA | Continuation of Mathematics 231a. Vector bundles and characteristic classes. Bott periodicity. K-theory, cobordism and stable cohomotopy as examples of cohomology theories. |

MATH 232BR | Algebraic Geometry II | TBA | TBA | The course will cover the classification of complex algebraic surfaces. |

MATH 233A | Theory of Schemes I | TBA | TBA | An 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. |

MATH 243 | Evolutionary Dynamics | Nowak | TBA | Advanced topics of evolutionary dynamics. Seminars and research projects. |

MATH 252 | Linear Series and Positivity of Vector Bundles | Ullery | TBA | In this course, we will cover a variety of topics related to positivity in algebraic geometry. We will start with the basics of divisors, line bundles, vector bundles, and linear series and their cohomological and numerical properties. Other topics may include but are not limitied to: syzygies and Castelnuovo-Mumford regularity, vanishing theorems, asymptotic linear series, degeneracy loci, Fujita's conjecture and Reider's Theorem, and stability of vector bundles |

MATH 260 | Low Dimensional Topology: Mapping Class Groups | Tshishiku | TBA | An introduction to mapping class groups and their cohomology. Possible topics: surface bundles, monodromy Earle-Eells theorem; Miller-Morita-Mumford classes, nontriviality, Atiyah-Kodaira constructions; curve complexes and homological stability; flat surface bundles and Nielsen realization problems; Morita's nonllifting theorem, Bott vanishing, Thurston stability; sections of surface bundles and Milnor-Wood inequalities. |

MATH 268Y | Diophantine Approximation | Pasten Vasquez | TBA | Roth's theorem. The subspace theorem. Arithmetic discriminants. The abc conjecture: partial results and applications. |

MATH 275X | Topics in Geometry and Dynamics | McMullen | TBA | A survey of fundamental results and current research. Topics may include: Riemann surfaces, hyperbolic 3-manifolds, moduli spaces, complex dynamics and rigidity. |

MATH 284X | Canonical Bases in Representation Theory | Melvin | TBA | An introduction to (dual) canonical bases in representation theory and some of their applications/appearance in geometry. Topics: Kashiwara crystals and their realisations, total positivity, geometric crystals of Berenstein-Kazhdan, toric degenerations of Schubert varieties, mirror constructions. |

MATH 288 | Probability Theory and Stochastic Process | H.T. Yau | TBA | We will cover the construction of Brownian motions and develop the Ito calculus. We will review discrete martingale and stopping time. |

MATH 303 | Topics in Diophantine Problems | Pasten Vasquez | TBA | NA |

MATH 304 | Topics in Algebraic Topology | Hopkins | TBA | NA |

MATH 308 | Topics in Number Theory and Modular Forms | Gross | TBA | NA |

MATH 314 | Topics in Differential Geometry and Mathematical Physics | Sternberg | TBA | NA |

MATH 316 | Topics in Algebraic Geometry | TBA | TBA | NA |

MATH 318 | Topics in Number Theory | Mazur | TBA | NA |

MATH 321 | Topics in Mathematical Physics | Jaffe | TBA | NA |

MATH 327 | Topics in Several Complex Variables | Siu | TBA | NA |

MATH 333 | Topics in Complex Analysis, Dynamics and Geometry | McMullen | TBA | NA |

MATH 335 | Topics in Differential Geometry and Analysis | Taubes | TBA | NA |

MATH 343 | Topics in Complex Geometry | Collins | TBA | NA |

MATH 345 | Topics in Geometry and Topology | Kronheimer | TBA | NA |

MATH 346Y | Topics in Analysis: Quantum Dynamics | Yau | TBA | NA |

MATH 348 | Topics in Representation Theory | Haiden | TBA | NA |

MATH 352 | Topics in Algebraic Number Theory | Kisin | TBA | NA |

MATH 356 | Topics in Harmonic Analysis | Schmid | TBA | NA |

MATH 357 | Topics in Model Theory | Boney | TBA | NA |

MATH 361 | Topics in Differential Geometry and Analysis | Canzani | TBA | NA |

MATH 362 | Topics in Number Theory | Miller | TBA | NA |

MATH 364 | Topics in Algebraic Geometry | Ullery | TBA | NA |

MATH 365 | Topics in Differential Geometry | S.T. Yau | TBA | NA |

MATH 368 | Topics in Algebraic Topology | Peterson | TBA | NA |

MATH 373 | Topics in Algebraic Topology | Lurie | TBA | NA |

MATH 381 | Introduction to Geometric Representation Theory | Gaitsgory | TBA | NA |

MATH 382 | Topics in Algebraic Geometry | Harris | TBA | NA |

MATH 385 | Topics in Set Theory | Woodin | TBA | NA |

MATH 387 | Topics in Mathematical Physics: Bridgeland Stability Conditions | Tanaka | TBA | NA |

MATH 388 | Topics in Mathematics and Biology | Nowak | TBA | NA |

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