MA3105 Advanced real analysis, Spring 2012

13.03 One more schedule change, there will be a lecture on Tuesday, March 20th, but no lecture on Friday March 30th.
The last lecture is on April 20th, the oral exam will be arranged before June 8th.

27.02 Extra lectures on Tuesday will be given on 6th and 13th of March, room 734 as usual, 14.15-16.00. No lectures next two Thursdays March 1st and 8th.

23.02 There will be no lecture next Thursday, 01.03.
On Friday 02.03 we will discuss dynamical systems (problems from lecture note 6).
Drafts of the last lecture notes are below.

12.01 First lecture is today at 14.15 in room 734.
Our topic for this week is foundation of probability theory from analytical point of view.
We will learn the zero-one law and prepare the ground for the law of large numbers scheduled for the next week.

Lectures

Thursday 14.15-16.00, room 734, SB II
Friday 14.15-16.00, room 734, SB II

Lecturer: Eugenia Malinnikova
room 948, Sentralbygg II
phone 73550257
e-mail eugenia [at] math [dot] ntnu [dot] no

Preliminary Curriculum

Signed measures and the Radon-Nykodym theorem.
Riesz-Markov theorem, the dual of C(X).
Mathematical model of probability, The Law of Large Numbers.
Dynamical systems and Ergodic Theorem.
Hausdorff measures and dimension
Fourier transform and applications.

We will mostly use the textbook:
McDonald, Weiss, A course in Real Analysis, Academic press.

More detailed plan of the lectures: Preliminary plan

Lecture notes

week 2 LN 1 : Foundations of the probability theory, zero-one law
week 3 LN 2: Laws of large numbers, Shannon's theorem
week 4 LN 3 : Signed measures, Radon-Nikodym theorem and conditional expectations
week 5 LN 4 : Decomposition of measures.
week 6 LN 5: Measurable Dynamical Systems, Pointwise Ergodic theorem
week 7 LN 6 : Entropy of dynamical system
week 8 LN 7 : Hausdorff measures
week 9 Examples of ergodic systems (presentations)
week 10 Riesz theorem in Hausdorff locally compact spaces (Rudin, Real and Complex analysis, ch. 2)
week 11 LN 8 : Central Limit Theorem
week 11 LN 9 : Central Limit Theorem and Law of the Iterated Logarithm
week 12 LN 10 : The Law of the Iterated Logarithm for dyadic martingales
week 13 Lattice random walk
week 14 Easter break
week 15 Probabilistic number theory: Hardy-Ramanujan and Erdös-Kac theorems

2012-04-12, Eugenia Malinnikova