Undergraduate Seminars I (Number Theory & Statistics), Fall 2019


Time: Tuesday 7-9 pm
Location: Math Room 622
Section Leader: Chung-Hang (Kevin) Kwan

0. Announcement

1. Introduction:

Many good stories start with "once upon a time" and this include number theory! Number theory has been a source of fascination for millennia and is described by many as the "queen of mathematics" for its beauty in the purest form. Classically, it refers to the study of the interesting qualitative properties of numbers, but in this seminar the audience will be introduced to the global and quantitative perspectives of number theory.

To illustrate this, let's consider the sequence of prime numbers: 2,3,5,7,11, ... , 2^13466917-1, 143332^786432-143332^393216+1, ... The sequence of prime numbers seem to have no pattern and prime numbers become rare when they get large.

In terms of qualitative perspective, one hopes for a prime-producing formula. There has been a lot of failed attempts (say Euler & Fermat). Finally, mathematicians did found one in 1976 but it is extremely complicated! All prime numbers can be captured by a system of 14 equations in 26 variables (just enough to be expressed in terms of English alphabets!), which can be be seen in this Wiki page . Of course, such formula has no practical use in the investigation of prime numbers.

In terms of quantitative perspective, we may ask for the rarity/ abundance of prime numbers. Essentially, we want to count how many primes are there (say from 1 to x). Amazingly, there is a very simple formula for this --- there are roughly (x/log x)-many primes from 1 to x, as x goes to infinity. This is known as the Prime Number Theorem and undoubtedly it is one of the major triumph in the modern number theory! As a side note, it is quite remarkable that Gauss conjected this formula simply by staring at the table for prime numbers (probably for a long time)!

2. Structures & Learning Outcomes

This will be a student-oriented seminar. Each student enrolled in this section is expected to read through some short mathematical articles which are relevant to the scopes of this seminar, and then give a talk, share your passion with fellow classmates. There are two possibilities for choosing a topic:

Both options are equally good. In any case, feel free to share your thoughts, interests and backgrounds with me before you choose a topic. I am sure we can come up with something fun together! Students with similar interests may form a group to study a single topic (with a couple of sub-topics so that every member has different things to present on). It is always a good idea to discuss with your classmates when you have any problem. It is beneficial to learn the perspectives from others that are different from yours. The maximum number of students allowed in a group depends on the chosen topic. (In any case no more than 4.)

I hope this seminar will provide students the experience of learning mathematics actively, independently, and being able to explain the mathematical concepts he/she has learnt to others effectively. Another important goal is to follow the footsteps of mathematicians in the past and appreciate their brilliant yet simple ideas.

Please email me if you would like to join this seminar.

3. Pre-requisite

Most importantly, passion in numbers! Also, calculus (up to infinite series and integration) and some knowledge of probability (e.g., expected value, variance, addition/multiplication rules of probability, normal distribution).

4. Suggested Topics (More to come!)

Supplementary materials



The schedule is tentative and subject to change as the seminar moves along.

  Date  Speaker  Topic  References 
Lecture 1  9/10  Kevin   Introduction    
Lecture 2  9/17  Kevin   Introduction    
Lecture 3  9/24  Amara   The Primes that Euclid Forgot   
Lecture 4  10/1  Maya & Cecilia   Randomness, Benford's Law & Diophantine Approximation    
Lecture 5  10/8  Catherine & Maggie   The Multiplication Table and the American Flag    
Lecture 6  10/15  Gia & Sung-Jun  Ramanujan and partition function    
Lecture 7  10/22  Pallavi & Jarek   Probabilistic Methods in Constructive Number Theory    
Lecture 8  10/29  Helen, Mariya   Monte Carlo algorithm and primality testing, Euler's polynomial and unique factorization    
Lecture 9  11/12  Alec & Joon   Turan's sieve & Counting Irreducible Polynomials    
Lecture 10  11/19  Rohan & Abdoulaye   Lattice Point Counting Inside Circles    
Lecture 11  11/26  Kexin, Akiva   Square Sieve, Anatomy of Middle Binomial Coefficients    
Lecture 12  12/3  Ben, William  TBC   


Last updated: September 2, 2019.

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