Joshua Zahl

Teaching

I am currently the TA for Ma245b. Dicussion sections are T 13:00-13:50 in MS 6229. Office hours are TF 14:00-14:40 in MS6154. More information can be found here.

Research

I am interested in harmonic analysis and problems relating to the Kakeya conjecture in particular. As an undergrad I did much of my work in combinatorics, which is why the papers listed below are focused on combinatorics/discrete mathematics rather than analysis.

Papers

Bounds on degrees of p-adic separating polynomials. (with D. Katz). J. Comb. Theory Ser. A, 115:1310-1319, 2008.

Abstract: We study a discrete optimization problem introduced by Babai, Frankl, Kutin, and Stefankovic (2001), which provides bounds on degrees of polynomials with p-adically controlled behavior. Such polynomials are of particular interest because they furnish bounds on the size of set systems satisfying Frankl�Wilson-type conditions modulo prime powers, with lower degree polynomials providing better bounds. We elucidate the asymptotic structure of solutions to the optimization problem, and we also provide an improved method for finding solutions in certain circumstances.
On universal cycles for multisets. (with G. Hurlbert and T. Johnson). Discrete Math., 309::5321-5327

Abstract: A Universal Cycle for t-multisets of [n] = {1,..., n} is a cyclic sequence of n+t-1 choose t integers from [n] with the property that each t-multiset of [n] appears exactly once consecutively in the sequence. For such a sequence to exist it is necessary that n divides n+t-1 choose t and it is reasonable to conjecture that this condition is sufficient for large enough n in terms of t. We prove the conjecture completely for t = 2,3 and partially for t = 4,6.

Unpublished Notes and Expository Articles

Fefferman's proof that the Fourier multiplier for the disk is unbounded in $L^p$, $p \neq 2$ [Expository].
There are hundreds of expositions on this result, and each one tries to stike a balance between berevity and being self contained. Hopefully the balance I have chosen will be useful for some readers. This article is based heavily on the exposition given in Krantz's A panorama of harmonic analysis, as well as (of course) Fefferman's original paper.
A generalized De Bruijn cycle algorithm.

Abstract: Let B be a collection of sets of strings over an alphabet X where each string has length n. A generalized De Bruijn cycle S is a cyclic string of letters over an alphabet X such that the collection of substrings of S of length n (with multiplicity) contains precisely one element from each of the sets in the collection B. Given a collection B we present an algorithm to find strings S which satisfy the generalized De Bruijn cycle condition. Several practical considerations are addressed such as optimizing the algorithm to be run in parallel across many computers.

Talks

On the expected height of t-ary trees under random edge compression. AMS-MAA Joint Meeting, New Orleans LA, January 2007.
Universal cycles on multisets. (with T. Johnson), MathFest 2006, Knoxville TN, August 2006.
Properties of random restricted minors. MathFest 2006, Knoxville TN, August 2006.
Determining bounds for minimum separating polynomial degree. Caltech summer undergraduate research seminar, October 2005.

CV

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