printer friendly pageMATH 227: Partial Differential Equations and Diffusion Processes
Parabolic and elliptic partial differential equations and their relation to diffusion processes. First order equations and optimal control. Emphasis is on applications to mathematical finance. Prerequisites:
MATH 131 and
MATH 136/
STATS 219, or equivalents.
Terms: Win
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Units: 3
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Grading: Letter or Credit/No Credit
Instructors:
Ryzhik, L. (PI)
MATH 228: Stochastic Methods in Engineering (CME 308)
Review of basic probability; Monte Carlo simulation; state space models and time series; parameter estimation, prediction, and filtering; Markov chains and processes; stochastic control; and stochastic differential equations. Examples from various engineering disciplines. Prerequisites: exposure to probability; background in real variables and analysis.
Terms: Spr
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Units: 3
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Grading: Letter or Credit/No Credit
Instructors:
Papanicolaou, G. (PI)
MATH 230A: Theory of Probability (STATS 310A)
Mathematical tools: sigma algebras, measure theory, connections between coin tossing and Lebesgue measure, basic convergence theorems. Probability: independence, Borel-Cantelli lemmas, almost sure and Lp convergence, weak and strong laws of large numbers. Large deviations. Weak convergence; central limit theorems; Poisson convergence; Stein's method. Prerequisites: 116,
MATH 171.
Terms: Aut
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Units: 2-4
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Grading: Letter or Credit/No Credit
Instructors:
Montanari, A. (PI)
MATH 230B: Theory of Probability (STATS 310B)
Conditional expectations, discrete time martingales, stopping times, uniform integrability, applications to 0-1 laws, Radon-Nikodym Theorem, ruin problems, etc. Other topics as time allows selected from (i) local limit theorems, (ii) renewal theory, (iii) discrete time Markov chains, (iv) random walk theory,nn(v) ergodic theory. Prerequisite: 310A or
MATH 230A.
Terms: Win
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Units: 2-3
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Grading: Letter or Credit/No Credit
MATH 230C: Theory of Probability (STATS 310C)
Continuous time stochastic processes: martingales, Brownian motion, stationary independent increments, Markov jump processes and Gaussian processes. Invariance principle, random walks, LIL and functional CLT. Markov and strong Markov property. Infinitely divisible laws. Some ergodic theory. Prerequisite: 310B or
MATH 230B.
Terms: Spr
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Units: 2-4
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Grading: Letter or Credit/No Credit
MATH 231A: An Introduction to Random Matrix Theory (STATS 351A)
Patterns in the eigenvalue distribution of typical large matrices, which also show up in physics (energy distribution in scattering experiments), combinatorics (length of longest increasing subsequence), first passage percolation and number theory (zeros of the zeta function). Classical compact ensembles (random orthogonal matrices). The tools of determinental point processes.
Terms: not given this year
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Units: 3
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Grading: Letter (ABCD/NP)
MATH 231C: Free Probability
Background from operator theory, addition and multiplication theorems for operators, spectral properties of infinite-dimensional operators, the free additive and multiplicative convolutions of probability measures and their classical counterparts, asymptotic freeness of large random matrices, and free entropy and free dimension. Prerequisite:
STATS 310B or equivalent.
Terms: not given this year
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Units: 3
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Grading: Letter or Credit/No Credit
MATH 232: Topics in Probability: Percolation Theory
An introduction to some of the most important theorems and open problems in percolation theory. Topics include some of the difficult early breakthroughs of Kesten, Menshikov, Aizenman and others, and recent fields-medal winning works of Schramm, Lawler, Werner and Smirnov. Prerequisites: graduate-level probability.
Terms: Win
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Units: 3
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Repeatable for credit
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Grading: Letter or Credit/No Credit
Instructors:
Chatterjee, S. (PI)
MATH 233: Probabilistic Methods in Analysis
Proofs and constructions in analysis obtained from basic results in Probability Theory and a 'probabilistic way of thinking.' Topics: Rademacher functions, Gaussian processes, entropy.
Terms: not given this year
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Units: 3
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Repeatable for credit
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Grading: Letter or Credit/No Credit
MATH 234: Large Deviations Theory (STATS 374)
Combinatorial estimates and the method of types. Large deviation probabilities for partial sums and for empirical distributions, Cramer's and Sanov's theorems and their Markov extensions. Applications in statistics, information theory, and statistical mechanics. Prerequisite:
MATH 230A or
STATS 310.
Terms: Win
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Units: 3
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Grading: Letter or Credit/No Credit
Instructors:
Dembo, A. (PI)
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