Freshmen seminar, Fall 2016 - N. Reshetikhin

office: 917 Evans Hall
email: reshetik@math.berkeley.edu
Seminars: Wednesday 10-12pm, 891 Evans Hall
Office Hours: Friday 11:30am-12:30pm,

Description

The goal of this course is an introduction to probability and its applications. Flipping a coin and estimating how many times it will land on one side and how many times it will land on the other side is a good illustration to how determinism enters into randomness. We will start with this example (after a short recollection of basic principles of probability). We will compute the probability of a coin landing n times on one side after N flipping for large n and N. Then we will discuss random processes and an important class them known as Markov processes. We will also discuss the question known in probability theory as large deviations and will see that some times there is an element of determinism in randomness. We will consider some simple combinatorial examples such as pile of squares to illustrate this phenomenon. The seminar will start with a series of introductory lectures, and then, towards the end of the seminar, students will give presentations. Knowledge of elements of probability theory is desirable but not required. It is expected that during the seminar every participating student will give a short presentation on the subject of the seminar (in a broad sense). Suggested references may look advanced. This is done to show where the subject is evolving from the introductory level of the seminar.

Suggested References

Sheldon Ross, "A First Course in Probability", Prentice Hall, 1998

More references will be posted later.

Tentative syllabus

We will start with the study of probabilities of flipping coin. The goal is to understand the probabilities for large number of flips. This will follow the discussion of central limit theorems in probability theory. Then we will discuss random processes and how to sample large systems. All these notions will be illustrated with examples. Below is the tentative schedule of seminars. It may change depending on how fast the discussions will go.

Seminar 1

This is an introductory meeting. We will start the detailed discussion of flipping coins: how the outcome is distributed for large n and N, where n is the number of heads and N is the number of flips.

Seminar 2

We will finish the coins and will start discussing random processes.

Seminar 3

The discussion of simplest random processes: Markov processes.

Seminar 4

Random walks

Seminar 5

to be posted

Seminar 6

to be posted

Seminar 7

Presentations

Seminar 8

Presentations
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