University of Michigan Student Arithmetic Seminar

Wednesday 3:10 - 4:00

3088 East Hall

If you would like to be added to the email list, give a talk, or hear a talk on a particular topic, email Jonathan Bober, or Ben Weiss.

Date: Speaker: Title:
9/24/2008 Ben Weiss Continued Fractions
10/1/08 Ben Weiss An Improvised Talk on P-adic Hilbert Irreducibility
10/8/08 Hester Graves The Fibonacci-Adics
10/15/08 Ben Weiss Measure Theory Of Continued Fractions
10/22/08 Chris Hall Covers and Permutations
10/29/08 Hester Graves The Continued Fraction of p/q - 1/Nq^2
11/5/08 Jonathan Bober The Beurling-Nyman criterion for the Riemann Hypothesis
11/12/08 TBA TBA
11/19/08 Leo Goldmakher Large character sums and the Polya-Vinogradov inequality
11/26/08 Mr. Turkey Gobble Gobble
12/3/08 Chris Hall Twin-prime pairs of polynomials
12/10/08 Not Meeting Not Meeting
1/7/09 Not Meeting Not Meeting
1/14/09 Organizational Meeting Arrange speakers
1/21/09 Johnson Jia Modular Forms for Number Theorists
1/28/09 Julian Rosen On Generalizations of the Zeta Function
2/4/09 Leo Goldmakher Sumset and Productset Inequalities
2/11/09 Hester Graves Harper's Construction and Recursive Sequences
2/18/09 Ben Weiss An Introduction to Kissing Numbers
2/25/09 Not Meeting Spring Break
3/4/09 Zach Scherr On the size of Kakeya sets in finite fields
3/11/09 Ben Weiss An Introduction to transcendental number theory
3/18/09 Alex Mueller Analysis as it was intended: A brief tour of non-standard analysis
3/25/09 TBA TBA
4/1/09 Julian Rosen TBA
4/8/09 No Meeting TBA
4/22/09 Johnson Jia Uniform distribution of Heegner points


9/24/08 Ben Weiss An Introduction to Continued Fractions
We will discuss the preliminary definitions, and some intuition as to what these are, and why they are useful, including why they are best approximations. We may also mention (although there will be no time to discuss at length) some of the results concerning random numbers and their continued fraction elements.


10/8/08 Hester Graves The Fibonacci-Adics
We will define, and explain, the Fibonacci-Adics.


10/15/08 Chris Hall Covers and Permutations
We'll describe a concrete combinatorial problem involving a symmetric group $S_n$ and $b$-tuples of elements in $S_n$ satisfying certain properties. We'll also describe a correspondence between $b$-tuples and covers of the projective line. For fixed $n$ and $b$, there is a natural permutation action of a braid group on the $b$-tuples, and we'll describe the action and state some (mostly open) questions that arise.


10/8/08 Hester Graves The Continued Fraction of p/q - 1/Nq^2
A surprising and pretty result by Mark Sheingorn, which is shocking in its novelty.


10/8/08 Jonathan Bober The Beurling-Nyman criterion for the Riemann Hypothesis
The Beurling-Nyman criterion is a real-variable reformulation of the Riemann Hypothesis. More specifically, the Riemann Hypothesis is true if and only if a certain set of functions is dense in L^2((0,1)). I'll describe this, with at least partial proof, and I'll give some related examples.


11/19/08 Leo Goldmakher Large character sums and the Polya-Vinogradov inequality
Because of their close connection with L-functions, character sums have become one of the central objects of study in analytic number theory. In 1918, Polya and Vinogradov independently proved an upper bound on character sums which is non-trivial for N > q^{1/2 + \epsilon}, where N is the number of terms in the sum and q is the conductor. Despite some significant progress since then on shorter character sums, the Polya-Vinogradov inequality remains the strongest result in the full range N < q. Quite recently, Granville and Soundararajan broke this barrier for the first time in nearly 90 years, by showing that for characters of odd order one can improve Polya-Vinogradov. I'll talk about their work, as well as my recent results improving Polya-Vinogradov for characters whose conductor is smooth (i.e. has only small prime factors).


10/8/08 Chris Hall Twin-prime pairs of polynomials
The well-known twin-prime problem asks whether there are infinitely many rational primes p so that p+2 is also prime. While the empirical evidence supports an affirmative answer, the problem remains open. One can pose a similar question for polynomials over a finite field: are there infinitely many pairs f,f+1 in F_q[t] such that f,f+1 are prime? We will show that the answer is affirmative if there is an odd prime ell dividing q-1, and we will explain how to prove analogous results for n-tuples of polynomials when n>2. If time permits, we will also show how one can obtain results when q-1 is a power of 2.


2/4/09 Leo Goldmakher Sumset and Productset Inequalities
Given a finite set A of integers, let A+A denote the set consisting of all possible sums a+b and A*A the set of all products ab, where a and b are any two elements of A. It is conjectured that the size of A+A is significantly larger than that of A unless A "looks like" an arithmetic progression, and similarly, that the only way A*A can be small is if A looks like a geometric progression. (In fact, a deep theorem of Freiman shows that a form of the first conjecture holds; to my knowledge, not much is known about the second conjecture.) Since a set cannot simultaneously look like both an arithmetic and a geometric progression, one might be led to conjecture that at least one of the two sets A+A, A*A must be large. A quantitative form of this conjecture was made by Erdos and Szemeredi. In a beautiful paper posted on the arXiv in mid-2008, Solymosi improved previous results on this problem using a surprisingly elementary argument. In this talk I'll describe his proof in full. No knowledge beyond the classical Cauchy-Schwarz inequality will be assumed.


2/11/09 Hester Graves Harper's Construction and Recursive Sequences
This will be a short talk about a minor new result. Unlike most talks, where everyone is fine the first five minutes and lost the rest of the time, non-number theorists may be confused the first five minutes and able to understand everything after that, as the techniques used are from calculus, undergrad analysis, and a first course in abstract algebra. Delicious food will be served.


2/18/09 Ben Weiss An Introduction to Kissing Numbers
This will be a survey talk on spheres, sphere packing, and kissing numbers. We'll go over definitions, conjectures, basic results, and a few constuctions and proofs. This talk should be accessible to all.


3/4/09 Zachary Scherr On the size of Kakeya sets in finite fields
A Kakeya set was originally defined as the space through which a needle passes while turning around. Over finite fields, it is defined as a subset of affine n-space which contains a line in every direction. In this talk I will prove a recent result of Zeev Dvir, an Israeli graduate student, on lover bounds for the size of Kakeya sets. In 2008, Dvir managed to improve the best known lover bound from Omega(q^(4n/7)) to the near optimal Omega(q^n).

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