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Blogs about: 254a Ergodic Theory

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254A, Lecture 17: A Ratner-type theorem for SL_2(R) orbits

In this final lecture, we establish a Ratner-type theorem for actions of the special linear group on homogeneous spaces. More precisely, we show: Theorem 1. Let G be a Lie group, let be a discrete sub… more →

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254A, Lecture 17: A Ratner-type theorem for SL_2(R) orbits — 2 comments

Terence Tao wrote 1 year ago: In this final lecture, we establish a Ratner-type theorem for actions of the special linear group on … more →

Tags: math.DS, math.RT, Ratner's theorem, unipotence, Mautner phenomenon, Lie Groups, Lie Algebras

254A, Lecture 16: A Ratner-type theorem for nilmanifolds — 3 comments

Terence Tao wrote 1 year ago: The last two lectures of this course will be on Ratner’s theorems on equidistribution of orbit … more →

Tags: math.DS, math.GR, Structure, Ratner's theorem, nilmanifolds, ergodicity, Dichotomy, nilpotent groups

254A, Lecture 15: The Furstenberg-Zimmer structure theorem and the Furstenberg recurrence theorem — 6 comments

Terence Tao wrote 1 year ago: In this lecture – the final one on general measure-preserving dynamics – we put together … more →

Tags: math.DS, Structure, randomness, Zorn's lemma, Furstenberg recurrence theorem

254A, Lecture 14: Weakly mixing extensions — 9 comments

Terence Tao wrote 1 year ago: Having studied compact extensions in the previous lecture, we now consider the opposite type of exte … more →

Tags: math.DS, math.OA, randomness, almost periodicity, weak mixing, Hilbert-Schmidt operators

254A, Lecture 13: Compact extensions — 7 comments

Terence Tao wrote 1 year ago: In Lecture 11, we studied compact measure-preserving systems – those systems in which every fu … more →

Tags: math.DS, math.OA, Structure, almost periodicity, van der Waerden's theorem, multiple recurrence, compact extensions, Hilbert modules

254A, Lecture 12: Weakly mixing systems — 14 comments

Terence Tao wrote 1 year ago: In the previous lecture, we studied the recurrence properties of compact systems, which are systems … more →

Tags: math.DS, math.FA, randomness, Roth's theorem, compactness, weak mixing, strong mixing, van der Corput, cauchy-schwarz

254A, Lecture 11: Compact systems — 13 comments

Terence Tao wrote 1 year ago: The primary objective of this lecture and the next few will be to give a proof of the Furstenberg re … more →

Tags: math.CA, math.DS, almost periodicity, multiple recurrence, compact systems, Kronecker systems

254A, Lecture 10: The Furstenberg correspondence principle — 10 comments

Terence Tao wrote 1 year ago: In this lecture, we describe the simple but fundamental Furstenberg correspondence principle which c … more →

Tags: math.CO, math.DS, correspondence principle, Szemeredi's theorem, multiple recurrence, density Hales-Jewett, polynomial recurrence, averaging argument

254A, Lecture 9: Ergodicity — 29 comments

Terence Tao wrote 1 year ago: We continue our study of basic ergodic theorems, establishing the maximal and pointwise ergodic theo … more →

Tags: math.DS, ergodic theorem, ergodic decomposition, ergodicity, probability kernel, disintegration, maximal inequality

254A, Lecture 8: The mean ergodic theorem — 29 comments

Terence Tao wrote 1 year ago: We now begin our study of measure-preserving systems , i.e. a probability space together with a prob … more →

Tags: math.DS, mean ergodic theorem, von Neumann ergodic theorem, spectral theorem, dual functions, conditional expectation, Hilbert spaces

254A, Lecture 7: Structural theory of topological dynamical systems — 10 comments

Terence Tao wrote 1 year ago: In our final lecture on topological dynamics, we discuss a remarkable theorem of Furstenberg that cl … more →

Tags: math.DS, Structure, distal systems, weak mixing, Furstenberg structure theorem

254A, Lecture 6: Isometric systems and isometric extensions — 35 comments

Terence Tao wrote 1 year ago: In this lecture, we move away from recurrence, and instead focus on the structure of topological dyn … more →

Tags: math.DS, math.MG, math.GR, math.AT, Structure, Ratner's theorem, polynomial sequences, Isometric extensions, Kronecker factor

254A, Lecture 5: Other topological recurrence results — 39 comments

Terence Tao wrote 1 year ago: In this lecture, we use topological dynamics methods to prove some other Ramsey-type theorems, and m … more →

Tags: math.CO, math.DS, hypergraphs, Ramsey theory, Hales-Jewett theorem, polynomial van der Waerden theorem, Hindman's theorem, Ramsey's theorem, idempotents

254A, Lecture 4: Multiple recurrence — 23 comments

Terence Tao wrote 1 year ago: In the previous lecture, we established single recurrence properties for both open sets and for sequ … more →

Tags: math.CO, math.DS, van der Waerden's theorem, multiple recurrence

254A, Lecture 3: Minimal dynamical systems, recurrence, and the Stone-Čech compactification — 45 comments

Terence Tao wrote 1 year ago: We now begin the study of recurrence in topological dynamical systems – how often a non-empty … more →

Tags: math.DS, math.LO, math.GN, lamplighter group, Recurrence, almost periodicity, syndetic sets, Stone-Cech compactification, ultrafilter

254A, Lecture 2: Three categories of dynamical systems — 27 comments

Terence Tao wrote 1 year ago: Before we begin or study of dynamical systems, topological dynamical systems, and measure-preserving … more →

Tags: math.CT, math.DS, topological dynamics, minimal system, Zorn's lemma, Morse sequence

254A, Lecture 1: Overview — 23 comments

Terence Tao wrote 1 year ago: In this lecture, I define the basic notion of a dynamical system (as well as the more structured not … more →

Tags: math.DS, ergodic theory, Ratner's theorem, course overview, recurrence theorem

254A: Topics in Ergodic Theory — 4 comments

Terence Tao wrote 2 years ago: Next quarter, starting on Wednesday January 9, I will be teaching a graduate course entitled “ … more →

Tags: math.DS, ergodic theory


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