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Foto: Matthias Friel

Gaussian processes - Einzelansicht

Veranstaltungsart Vorlesung/Übung Veranstaltungsnummer 18206
SWS 6 Semester SoSe 2020
Einrichtung Institut für Mathematik   Sprache englisch
Belegungsfrist 20.04.2020 - 10.05.2020

Belegung über PULS
Gruppe 1:
     jetzt belegen / abmelden
    Tag Zeit Rhythmus Dauer Raum Lehrperson fällt aus am Max. Teilnehmer/-innen
Einzeltermine anzeigen
Vorlesung Mo 14:15 bis 15:45 wöchentlich 13.04.2020 bis 20.07.2020  2.10.0.26 Jun. Prof. Dr. Lie  
Einzeltermine anzeigen
Vorlesung Mo 16:15 bis 17:45 wöchentlich 13.04.2020 bis 20.07.2020  2.09.0.14 Jun. Prof. Dr. Lie  
Einzeltermine anzeigen
Übung Do 16:15 bis 17:45 wöchentlich 16.04.2020 bis 23.07.2020  2.09.0.13 Jun. Prof. Dr. Lie  
Kurzkommentar

This is an advanced course for students with very strong mathematical ability and very strong interest in theoretical mathematics, especially analysis and probability theory.

The lectures will be given in English. Participants can write up their homework solutions in English or German.

Kommentar

This is an advanced course that will survey some parts of the theory of Gaussian processes. Gaussian processes are used extensively in mathematics, mathematical statistics, and also in applications such as statistical regression for machine learning or the design of compressed sensing matrices. We will not consider applications in this course. We will focus on mathematics.

The course will be organised around 4-hour lectures and a 2-hour exercise class.

Literatur

Evarist Gine and Richard Nickl, "Mathematical foundations of infinite-dimensional statistical models", Cambridge Series in Statistical and Probabilistic Mathematics (2016)

Bemerkung

Students who would like to participate in the course must inform the lecturer by e-mail that they would like to participate in the course.

Students who can register on PULS must register for the course on PULS and inform the lecturer by e-mail.

 

Voraussetzungen

This course requires that participants know the following:

1) real analysis, for example epsilon-delta proofs, series and sequences, measure theory, Lebesgue integration, Fatou's lemma, monotone convergence theorem, dominated convergence theorem, Fubini-Tonelli theorem, differentiation and integration, totally bounded sets, precompact sets, compact sets, Bolzano-Weierstrass theorem, limit inferior and limit superior of sequences, Taylor expansions, uniformly continuous functions;

2) functional analysis, including linear spaces, Hilbert spaces, Banach spaces, dual spaces, topological spaces, metric spaces, linear operators;

3) measure-theoretic probability theory, including sigma-algebras, probability spaces, random variables, conditional expectations, different types of convergence of random variables, limit superior and limit inferior of events, Borel-Cantelli lemmas, Chebyshev's inequality.

Students who have read through ”Real Analysis” by Gerald Folland (Wiley, 1999) and ”Real analysis and probability” by Richard Dudley (Cambridge University Press, 2002) will be sufficiently prepared for this course.

Below are some titles of books that cover some of the topics above and that are available through the University of Potsdam library.

Christian Clason, "Einführung in die Funktionalanalysis", Cham: Birkhäuser (2019)

Achim Klenke, "Probability theory: a comprehensive course", Springer (2014)

Achim Klenke, "Wahrscheinlichkeitstheorie", Springer (2013)

Norbert Henze, "Stochastik: Eine Einführung mit Grundzügen der Maßtheorie", Springer Spektrum (2019)

Christopher Heil, "Introduction to Real Analysis" Cham: Springer (2019)

Sergei Ovchinnikov, "Functional Analysis: An introductory course", Cham: Springer (2018)

Satish Shirali, "Measure and Integration", Cham: Springer (2019)

Leistungsnachweis

To receive credit for this course, students must

  1. Obtain at least 50% of the total homework points, and
  2. Give at least one successful presentation of their solutions to the homework assignments, and
  3. Pass an oral examination of 30 minutes.
Lerninhalte

The planned course content is to present some parts of the following topics in Gaussian processes:

  • Definitions and basic concepts
  • Isoperimetric inequalities and concentration
  • Metric entropy bounds for sub-Gaussian processes
  • Comparison theorems and Sudakov’s lower bound
  • Reproducing kernel Hilbert spaces

The plan may change during the lectures.

 

Zielgruppe

This course is targeted at students who

  1. have very strong mathematical background and satisfy the prerequisites / Voraussetzungen, and
  2. have very good English and/or German language skills, and
  3. enjoy rigorous proof-based mathematics, and
  4. are interested in analysis and probability theory.

Strukturbaum
Keine Einordnung ins Vorlesungsverzeichnis vorhanden. Veranstaltung ist aus dem Semester SoSe 2020 , Aktuelles Semester: WiSe 2020/21