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Math 72: Topics in Geometry
Title: An Invitation to Differential Geometry
- Description: Differential geometry is a vast subject that employs techniques from advanced calculus and linear algebra in the study of geometry. After laying some foundational work, we'll explore two subfields of differential geometry: Riemannian geometry and symplectic geometry. These subfields are widely studied for their intrinsic mathematical beauty and their numerous connections with physics. Our exposition will sit somewhere between a traditional undergraduate course on curves and surfaces and an introductory graduate-level course in geometry that assumes familiarity with differential topology. The price for eschewing differential topology is that we'll need to defer a discussion of the intriguing relationships between topology and geometry. Our reward for climbing further up the mountain than a traditional curves and surfaces course will be a more modern/sophisticated understanding of geometric tools which will position interested students to take up more advanced study and research in geometry. Topics will include some of the following:
- (1) A Little Linear algebra: inner product spaces and linear symplectic forms;
- (2) Revisiting Calculus: Differentiation & Linear approximation; geometric and analytic definitions of the tangent space; implicit function theorem; vector fields & integral curves;
- (3) Differential forms and tensors;
- (4) Riemannian Geometry: Riemannian metrics; connections, parallelism and geodesics; curvature; isometries;
- (5) Symplectic Geometry: Hamiltonian Mechanics; symplectic forms; symplectomorphism; symplectic and Hamiltonian vector fields;
Audience: This course might be of interest to students interested in mathematics and/or (theoretical) physics.
- (1) Math 24
- (2) Math 35/63 or 31/71
- Or, permission of the instructor (Note: the ORC lists Math 71 as the prerequisite.)
Math 106: Topics in Applied Mathematics
Title: Stochastic Processes with Applications
Instructor: Feng Fu
Description: Stochastic models are central to the study of many problems in physics, engineering, finance, evolutionary biology, and medicine. This course introduces concepts and techniques in probability theory and key methods for stochastic processes, along with their applications to the natural sciences.
Textbook: The Elements of Stochastic Processes, Norman T. J. Bailey, John Wiley & Sons, Inc. (1963)
- Week 1: Basic concepts of probability & generating function Day 1: Introduction & examples Day 2: Definitions & elementary results Day 3: Generating functions
- Week 2: Random walks Day 4: Gambler’s ruin Day 5: Extensions & recurrence Day 6: Random walks on graphs
- Week 3: Markov chains Day 7: Transition matrices, classification of states of a Markov Day 8: Recurrent Markov chains & limit theorems Day 9: Martingales
- Week 4: Branching processes Day 10: Discrete branching processes Day 11: Generating function approach & extinction probabilitiDay 12: Multi-type branching processes
- Week 5: Markov processes in continuous time Day 13: The Poisson process Day 14: Random-variable technique Day 15: General theory
- Week 6: Birth and death processes I Day 16: Homogeneous birth and death processes Day 17: The effect of immigration Day 18: General multiplicative processes
- Week 7: Birth and death processes II Day 19: The Pólya process Day 20: Non-homogeneous birth-and-birth processes Day 21: General stochastic population growth models
- Week 8: Diffusion processes Day 22: Diffusion limit of a random walk Day 23: Diffusion limit of a discrete branching process Day 24: Applications to population growth
- Week 9: Non-Markov processes Day 25: Renewal theory and related concepts Day 26: Renewal equations and generalizations Day 27: Applications of renewal processes
Starting in the fall of 2011, certain curricular changes were implemented which affected the content and focus of Math 73, 74, and 81. Prior to those changes, these courses were primarily advanced undergraduate courses which first-year graduate students took as necessary for review, or to fill in gaps in their own backgrounds. After these changes there were both content changes and a change in the intended audience, becoming graduate courses numbered 103, 114, 111 which were cross-listed as 73, 74, 81 and which advanced undergraduates were allowed to take.
Math 73/103: Measure Theory and Complex AnalysisThis course is an introduction to graduate level analysis. Divided roughly in half, the first part of the course covers abstract measure theory. The second half of the course covers complex analysis.
Further details in the ORC listings.
Math 74/114: Algebraic TopologyThis course provides a foundation in algebraic topology, including both homotopy theory and homology theory. Topics may include: the fundamental group, covering spaces, calculation of the fundamental group, singular homology theory, Eilenberg-Steenrod axioms, Mayer-Vietoris sequence, computations, applications to fixed points and vector fields.
Further details in the ORC listings.
Math 81/111: Abstract AlgebraThis course provides a foundation in core areas in the theory of rings and fields. Specifically, it provides an introduction to commutative ring theory with a particular emphasis on polynomial rings and their applications to unique factorization and to finite and algebraic extensions of fields. The study of fields continues with an introduction to Galois Theory, including the fundamental theorem of Galois Theory and numerous applications.
Further details in the ORC listings.
Math 116: Applied Mathematics
"Great papers in numerical analysis and computational algorithms"
How are we able to compute the massive things that we can? How are we able to find eigenvalues of large matrices, solve PDEs with error 0.0000000001, optimize a function of thousands of variables, or solve linear systems with a million unknowns? Although faster hardware is part of the story, much of the credit goes to the invention (discovery?) of vastly more efficent numerical algorithms, say O(N log N) instead of O(N^2).
This spring I will run Math 116 as a research-paper-based "journal club" where students will present some of the great numerical analysis and computational algorithms (either classic or recent). For the first 2 weeks, I will lecture on key background concepts in numerical computation. In weeks 3-8 each student will prepare two lectures (ie one week) on their chosen paper. We will also code up some algorithms, to see how, why, and when they work. Your workload will mainly be reading for, then preparing, your two lectures; the rest of the time you will assess presentation skills (something we will all learn), be an active audience member, and possibly "LaTeX scribe". You will leave with a broad appreciation of numerical topics and research areas.
- You can bring papers of your own choosing/interest, assuming they involve continuous (as opposed to discrete, about which I know little) computational problems, and I can understand them!
- The course should be accessible to advanced undergraduates and graduate students in math, CS, as well as engineering and the sciences (assuming they have a mathematical bent). Come and talk if unsure.
- Prerequisites: some programming, real/complex analysis, linear algebra and diff eq, some graduate analysis will help ... and a positive energy to learn, and craft clear explanations.
- This course is largely orthogonal to the Math 116/126 I have taught before, so please retake!
Math 56: Computational and Experimental Mathematics
Time: 13S, 14S: (to be arranged)
Computations have always played a key role in mathematical progress both pure and applied, generating conjectures (e.g. distibution of primes), and nu- merically solving models of the real world (e.g. climate change). An exponential growth in computing power has made this game-changing. This course surveys computational methods, algorithms, and software environments that are an essential part of the modern mathematician’s toolkit. Possible topics include: the fast Fourier transform, visualization, computer-assisted proofs, numerical integration, high-precision computing, computational combinatorics and number theory.
Prerequisites: Math 22/24 (linear algebra), Computer Science 1 or permission of the instructor.
Math 96: Mathematical Finance II
Time: Tu/Th 2 (2A)
Description: In the study of ordinary differential equations the parameters and coefficients of the equations are assumed to be deterministic. In contrast, a stochastic differential equation is a differential equation in which one or more of its terms is governed by a stochastic (or random) process. As one might imagine, stochastic differential equations (SDEs) arise naturally in the study of finance, engineering, economics, physics and mathematics. This term Math 96 will serve as an introduction to the theory and applications of SDEs with an eye towards finance. Topics may include some of the following:
- Probability spaces, Stochastic processes & Brownian Motion
- Ito Integrals, the Ito formula and the Martingale Representation Theorem
- Existence & Uniqueness of Solutions
- Basic Properties of Diffusions
- Feynman-Kac Theorem & the Girsanov Theorem
- Optimal Stopping Time
- Applications to financial derivatives on equities and fixed-income securities
- Applications to Boundary-Value Problems & Control Theory
- 1. Math 86 or Permission of the instructor
- 2. Some knowledge of analysis at the level of Math 63/35 or 103 will be useful