I'm a biomedical and electrical engineer with interests in information theory, complexity, evolution, genetics, signal processing, IndieWeb, theoretical mathematics, and big history.
I'm also a talent manager-producer-publisher in the entertainment industry with expertise in representation, distribution, finance, production, content delivery, and new media.
This article gives a simplified account of some of the ideas behind Tao’s resolution of the Erdős discrepancy problem. http://dx.doi.org/10.1090/bull/1598 | PDF
We discuss a variant of the density and energy increment arguments that we call an "entropy decrement method", which can be used to locate a scale in which two relevant random variables share very little mutual information, and thus behave somewhat like independent random variables. We were able to use this method to obtain a new correlation estimate for multiplicative functions, which in turn was used to establish the Erdos discrepancy conjecture that any sequence taking values in {-1,+1} had unbounded sums on homogeneous arithmetic progressions.
We show that for any sequence f:N→{−1,+1} taking values in {−1,+1}, the discrepancy
supn,d∈N∣∣∣∣∑j=1nf(jd)∣∣∣∣
of f is infinite. This answers a question of Erdős. In fact the argument also applies to sequences f taking values in the unit sphere of a real or complex Hilbert space.
The argument uses three ingredients. The first is a Fourier-analytic reduction, obtained as part of the Polymath5 project on this problem, which reduces the problem to the case when f is replaced by a (stochastic) completely multiplicative function g. The second is a logarithmically averaged version of the Elliott conjecture, established recently by the author, which effectively reduces to the case when g usually pretends to be a modulated Dirichlet character. The final ingredient is (an extension of) a further argument obtained by the Polymath5 project which shows unbounded discrepancy in this case.
Kaisa Matomäki, Maksym Radziwiłł, and I have just uploaded to the arXiv our paper “Sign patterns of the Liouville and Möbius functions”. This paper is somewhat similar to our previous p…
We introduce a general result relating "short averages" of a multiplicative function to "long averages" which are well understood. This result has several consequences. First, for the M\"obius function we show that there are cancellations in the sum of μ(n) in almost all intervals of the form [x,x+ψ(x)] with ψ(x)→∞ arbitrarily slowly. This goes beyond what was previously known conditionally on the Density Hypothesis or the stronger Riemann Hypothesis. Second, we settle the long-standing conjecture on the existence of xϵ-smooth numbers in intervals of the form [x,x+c(ε)x−−√], recovering unconditionally a conditional (on the Riemann Hypothesis) result of Soundararajan. Third, we show that the mean-value of λ(n)λ(n+1), with λ(n) Liouville's function, is non-trivially bounded in absolute value by 1−δ for some δ>0. This settles an old folklore conjecture and constitutes progress towards Chowla's conjecture. Fourth, we show that a (general) real-valued multiplicative function f has a positive proportion of sign changes if and only if f is negative on at least one integer and non-zero on a positive proportion of the integers. This improves on many previous works, and is new already in the case of the M\"obius function. We also obtain some additional results on smooth numbers in almost all intervals, and sign changes of multiplicative functions in all intervals of square-root length.
Using crowd-sourced and traditional mathematics research, Terence Tao has devised a solution to a long-standing problem posed by the legendary Paul Erdős.
In the middle of the lecture last night, I was thinking to myself that this problem seems like a mixture of combinatorics, integer partitions and coding theory. Something about this article reminds me of that fact again. Most of the references I’m seeing however are directly to number theory and don’t relate to the integer partition piece–perhaps worth delving into to see what shakes out.
The article does a reasonable job of laying out some of the problem and Tao’s solution to it. I was a bit bothered by the idea of “magical” in the title, but it turns out it’s a different reference than the one I was expecting.
Last night was the first lecture of Dr. Miller’s Gems And Astonishments of Mathematics: Past and Present class at UCLA Extension. There are a good 15 or so people in the class, so there’s still room (and time) to register if you’re interested. While Dr. Miller typically lectures on one broad topic for a quarter (or sometimes two) in which the treatment continually builds heavy complexity over time, this class will cover 1-2 much smaller particular mathematical problems each week. Thus week 11 won’t rely on knowing all the material from the prior weeks, which may make things easier for some who are overly busy. If you have the time on Tuesday nights and are interested in math or love solving problems, this is an excellent class to consider. If you’re unsure, stop by one of the first lectures on Tuesday nights from 7-10 to check them out before registering.
Lecture notes
For those who may have missed last night’s first lecture, I’m linking to a Livescribe PDF document which includes the written notes as well as the accompanying audio from the lecture. If you view it in Acrobat Reader version X (or higher), you should be able to access the audio portion of the lecture and experience it in real time almost as if you had been present in person. (Instructions for using Livescribe PDF documents.)
We’ve covered the following topics:
Class Introduction
Erdős Discrepancy Problem
n-cubes
Hilbert’s Cube Lemma (1892)
Schur (1916)
Van der Waerden (1927)
Sylvester’s Line Problem (partial coverage to be finished in the next lecture)
Over the coming days and months, I’ll likely bookmark some related papers and research on these and other topics in the class using the class identifier MATHX451.44 as a tag in addition to topic specific tags.
Course Description
Mathematics has evolved over the centuries not only by building on the work of past generations, but also through unforeseen discoveries or conjectures that continue to tantalize, bewilder, and engage academics and the public alike. This course, the first in a two-quarter sequence, is a survey of about two dozen problems—some dating back 400 years, but all readily stated and understood—that either remain unsolved or have been settled in fairly recent times. Each of them, aside from presenting its own intrigue, has led to the development of novel mathematical approaches to problem solving. Topics to be discussed include (Google away!): Conway’s Look and Say Sequences, Kepler’s Conjecture, Szilassi’s Polyhedron, the ABC Conjecture, Benford’s Law, Hadamard’s Conjecture, Parrondo’s Paradox, and the Collatz Conjecture. The course should appeal to devotees of mathematical reasoning and those wishing to keep abreast of recent and continuing mathematical developments.
Suggested Prerequisites
Some exposure to advanced mathematical methods, particularly those pertaining to number theory and matrix theory. Most in the class are taking the course for “fun” and the enjoyment of learning, so there is a huge breadth of mathematical abilities represented–don’t not take the course because you feel you’ll get lost.
I’d complained to the UCLA administration before about how dirty the windows were in the Math Sciences Building, but they went even further than I expected in fixing the problem. Not only did they clean the windows they put in new flooring, brand new modern chairs, wood paneling on the walls, new projection, and new white boards! I particularly love the new swivel chairs, and it’s nice to have such a lovely new environment in which to study math.
The newly renovated classroom space in UCLA’s Math Sciences Building
Category Theory for Winter 2019
As I mentioned the other day, Dr. Miller has also announced (and reiterated last night) that he’ll be teaching a course on the topic of Category Theory for the Winter quarter coming up. Thus if you’re interested in abstract mathematics or areas of computer programming that use it, start getting ready!
Remembering the veteran news media hoaxer Alan Abel.
This past week’s coverage of Hurricane Florence has had all the trappings of a terrible storm: the satellite images, the sandbags and empty grocery stores, the newscasters dressed in flood gear. One recurring side character we seem to have avoided this time around, though, is the doctored image of a shark swimming on a flooded highway.
It’s a preposterous hoax that succeeds, occasionally, on the merits of some kernel of truth; for instance, whole swathes of interstate highway in North Carolina are, as of this moment, covered with water. That kernel of truth is what hoaxers and jokers build their handiwork upon — as did the veteran hoaxer Alan Abel, who passed away late last week at the age of 94.
Abel made a name for himself inventing characters and causes and turning the joke on the media; in 1980 he staged his own death and got himself an obituary in the New York Times.
Brooke spoke to Abel — and his daughter, Jenny Abel, the director of the documentary, "Abel Raises Cain" — in 2008.
Listened toFEMA Time from On the Media | WNYC Studios
As Hurricane Florence bears down, we discover that FEMA has $10 million less in its budget. The money was siphoned off to pay for detention and removal of immigrants.
On Wednesday, as Florence swirled ominously off the coast of the Carolinas, and states prepared for imminent disaster, Senator Jeff Merkley (D-OR) thought it would be a good time to draw everyone’s attention to the shifting priorities of this administration. Specifically, he released a budget that showed the Department of Homeland Security had transferred nearly 10 million dollars from the Federal Emergency Management Agency to Immigration and Customs Enforcement to pay for detention and removal operations.
FEMA officials maintain that the smaller budget won’t hinder their operations, but as wildfires rage and hurricanes make landfall, they have a lot on their plate. We don't think about FEMA much, until that's all we think about. Historian Garrett Graff says the agency’s, quote, “under-the-radar nature” was originally a feature, not a bug. Graff wrote about "The Secret History of FEMA" for Wired last September and he spoke to Bob about the agency's Cold War origins as civil defense in the event of a nuclear attack and how it transitioned to "natural" disaster response. Plus, they discuss the limitations to FEMA's capabilities and why it has such a spotty record. Graff is also author of Raven Rock: The Story of the U.S. Government's Secret Plan to Save Itself -- While The Rest of Us Die.
The anonymous op-ed, the Kavanaugh hearings, decorum, civility, and the freedom to speak.
Between the Kavanaugh confirmation hearings on Capitol Hill and an anonymous op-ed from within the Trump White House, a wave of rule-bending and -breaking has crashed on Washington. This week, we explore how political decorum and popular dissent have evolved since the early days of our republic — and how the legal protections for those core freedoms could transform our future.
2. Geoffrey Stone, professor of law at University of Chicago, on our evolving — and occasionally faulty — interpretations of the first amendment. And, Laura Weinrib, professor of law at University of Chicago, on how early-20th century labor struggles gave birth to our modern ideas about freedom of speech. Listen.
3. Tim Wu [@superwuster], professor of law at Columbia University, on how the first amendment could inform new regulations for Silicon Valley. Listen.
Terence Tao (YouTube)
Erica Klarreich (Quanta Magazine)
