If Creativity were anything but Random, Someone would have figured out the Algorithm by now. -- Dilbert

There are ten kinds of people in the world. Those who think in binary and those who don't. Joel Spencer

# Joel Spencer

## Email: {lowercaselastname}@cims.nyu.edu

Department of Computer Science and Department of Mathematics
Courant Institute, New York University

## Contact

Room 829, 251 Mercer St.
New York, NY 10012, U.S.A.
212-998-3219 (voice) 212-995-4124 (fax)-- but email is MUCH better

## CV/Papers/Talks

• Vita--Selected Work (short form)
• Vita--Publication List (long form)
• Papers Description of selected papers. Vita and those papers in postscript.
• Talks Slides for various talks.

## Course Information

In Spring 2015 I am teaching Algebra Honors II and Fundamental Algorithms. Students may email with queries at any time.

In Fall 2016 I am teaching Algebra Honors I. Click here for website with very tentative information.

## Asymptopia

This book is aimed at strong undergraduates, though it is also suitable for particularly good high school students or for graduates wanting to learn some basic techniques. From the back cover: Asymptotics in one form or another are part of the landscape for every mathematician. The objective of this book is to present the ideas of how to approach asymptotic problems that arise in discrete mathematics, analysis of algorithms, and number theory. A broad range of topics is covered, including distribution of prime integers, Erdos Magic, random graphs, Ramsey numbers, and asymptotic geometry. The author is a disciple of Paul Erdos, who taught him about Asymptopia. Primes less than n, graphs with v vertices, random walks of t steps--Erdos was fascinated by the limiting behavior as the variables approached, but never reached, infinity. Asymptotics is very much an art. The various functions nlnn, n^2, (ln n)/n, \sqrt{\ln n}, ln(ln n) all have distinct personalities. Erdos knew these functions as personal friends. It is the author's hope that these insights may be passed on, that the reader may similarly feel which function has the right temperament for a given task.

Asymptopia is a beautiful world. Enjoy!

Click (Material will be slightly different from the text here and below) for preface

Chapters: An Infinity of Primes; Stirling's Formula; Big Oh, Little Oh and All That; Integration in Asymptopia; From Integrals to Sums; Asymptotics of Binomial Coefficients; Unicyclic Graphs; Ramsey Numbers; Large Deviations; Primes; Asymptotic Geometry; Algorithms; Potpourri; Really Big Numbers!

## The Probabilistic Method -- Third Edition

Click here for the frontcover front cover with a great picture of Uncle Paul

Click here for a phasetransition new chapter on the Erdos-Renyi Phase Transition, with particular emphasis on the Critical Window.

Click for more, available from archives here

Hannah1 Hannah2 Hannah3 Hannah4 for our most beautiful and most attentive reader. She's a fox! And, in strong competition: johanna and jazmin

SPECIAL OFFER: Yes, your child (grandchild, niece, nephew, sibling) can have an Erdos Number of 1+\sqrt{-1}! Just send a photo of him/her absorbed in reading/playing/teething our book and I'll post it!

## The Erdos-Selberg Controversy

Over 60 years ago a controversy erupted between Paul Erdos and Atle Selberg over the elementary proof of the Prime Number Theorem. Ernst Straus -- then a young mathematician at the scene -- wrote a description of the events. These have been published in Math Intelligencer, along with commentary by Ron Graham and myself, and a mathematical postscript by Carl Pomerance.

Photo by MA

## Math News !

Alantha Newman and Aleksandar Nikolov have solved the Beck Three Permutaion Problem. I had offered a prize of 100 USD for its resolution. Here are Alantha and Sasha with the prize jpg. They found three permutations of 1,...,n=3^t so that for any coloring of 1,...,n with +1,-1 in one of the permutations there will be a prefix whose sum has absolute value at least ct. The conjecture had been that there was an absolute constant K so that for any three permutations of any size n there would be such a coloring where in each permutation all prefixes would have sum in [-K,+K], and so the conjecture was disproved.

Robin Moser, a student of Emo Welzl (ETH), has given an algorithmic implementation of the Lovasz Local Lemma. A rough explanation:

Nikhil Bansal (at TU Eindhoven (Holland)) has given an algorithmic implementation of my result that given any n sets on n points there is a 2-coloring with all discrepencies less than 6 \sqrt{n}. I had long conjectured that no algorithm would exist. Semidefinite Programming is a key ingredient. His preprint, Constructive Algorithms for Discrepancy Minimization, is available on arXiv. Click for Bansal's paper . or here
postscript LaTeX pdf for my own rough explanation. Shachar Lovett and Raghu Meka ArXiV link . have come up with another argument, using a restricted Brownian Motion.

## High School MathCamps

This is a topic dear to my heart. I am former chair of an AMS committee that gives grants to High School Math Camps. Click for camp information or for information about donations to a worthy cause!