Once upon a time, while in college, I decided to take my first intro-to-proofs class. I was so excited. "This is it!" I thought, "now I get to learn how to think like a mathematician." You see, for the longest time, my mathematical upbringing was very... not mathematical. As a student in high school and well into college, I was very good at being a robot. Memorize this formula? No problem. Plug in these numbers? You got it. Think critically and deeply about the ideas being conveyed by the mathematics? Nope. It wasn't because I didn't want to think deeply. I just wasn't aware there was anything to think about. I thought math was the art of symbol-manipulation and speedy arithmetic computations. I'm not good at either of those things, and I never understood why people did them anyway. But I was excellent at following directions. So when teachers would say "Do this computation," I would do it, and I would do it well. I just didn't know what I was doing. By the time I signed up for that intro-to-proofs class, though, I was fully aware of the robot-symptoms and their harmful side effects. By then, I knew that math not just fancy hieroglyphics and that even people who aren't super-computers can still be mathematicians because—would you believe it?—"mathematician" is not synonymous with "human calculator." There are even—get this—ideas in mathematics, which is something I could relate to. ("I know how to have ideas," I surmised one day, "so maybe I can do math, too!") One of my instructors in college was instrumental in helping to rid me of robot-syndrome. One day he told me, "To fully understand a piece of mathematics, you have to grapple with it. You have to work hard to fully understand every aspect of it." Then he pulled out his cell phone, started rotating it about, and said, "It's like this phone. If you want to understand everything about it, you have to analyze it from all angles. You have to know where each button is, where each ridge is, where each port is. You have to open it up and see how it the circuitry works. You have to study it—really study it—to develop a deep understanding." "And that" he went on to say, "is what studying math is like."

A nice little essay on mathematics for old and young alike–and particularly for those who think they don’t understand or “get” math. It’s ultimately not what you think it is, there’s something beautiful lurking underneath.

In fact, I might say that unless you can honestly describe mathematics as “beautiful”, you should read this essay and delve a bit deeper until you get the understanding that’s typically not taught in mathematics until far too late in most people’s academic lives.

]]>]]>June 17-21, 2019

This workshop will tackle a variety of biological and medical questions using mathematical models to understand complex system dynamics. Working in collaborative teams of 6, each with a senior research mentor, participants will spend a week making significant progress with a research project and foster innovation in the application of mathematical, statistical, and computational methods in the resolution of problems in the biosciences. By matching senior research mentors with junior mathematicians, the workshop will expand and support the community of scholars in mathematical biosciences. In addition to the modeling goals, an aim of this workshop is to foster research collaboration among women in mathematical biology. Results from the workshop will be published in a peer-reviewed volume, highlighting the contributions of the newly-formed groups. Previous workshops in this series have occurred at IMA, NIMBioS, and MBI.

This workshop will have a special format designed to facilitate effective collaborations.

- Each senior group leader will present a problem and lead a research group.
- Group leaders will work with a more junior co-leader, someone with whom they do not have a long-standing collaboration, but who has enough experience to take on a leadership role.
- Additional team members will be chosen from applicants and invitees. We anticipate a total of five or six people per group.
It is expected that each group will continue to work on their project together after the workshop, and that they will submit results to the Proceedings volume for the workshop.

The benefit of such a structured program with leaders, projects and working groups planned in advance is based on the successful WIN, Women In Numbers, conferences and is intended to provide vertically integrated mentoring: senior women will meet, mentor, and collaborate with the brightest young women in their field on a part of their research agenda of their choosing, and junior women and graduate students will develop their network of colleagues and supporters and encounter important new research areas to work in, thereby fostering a successful research career. This workshop is partially supported by NSF-HRD 1500481 – AWM ADVANCE grant.

## ORGANIZING COMMITTEE

Rebecca Segal (Virginia Commonwealth University)

Blerta Shtylla (Pomona College)

Suzanne Sindi (University of California, Merced)

In number theory, a]]>SierpinskiorSierpiński numberis an odd natural numberksuch that {\displaystyle k\times 2^{n}+1} is composite, for all natural numbersn. In 1960, Wacław Sierpiński proved that there are infinitely many odd integerskwhich have this property. In other words, whenkis a Sierpiński number, all members of the following set are composite:

- {\displaystyle \left\{\,k\cdot {}2^{n}+1:n\in \mathbb {N} \,\right\}.}

Six years ago I received an email from a colleague in the mathematics department at UC Berkeley asking me whether he should participate in a study that involved “collecting DNA from the brigh…

Not sure how I had missed this in the brouhaha a few weeks back, but it’s one of the more sober accounts from someone who’s actually got some math background and some reasonable idea about the evolutionary theory involved. It had struck me quite significantly that both Gowers and Tao weighed in as they did given their areas of expertise (or not). Perhaps it was worthwhile simply for the attention they brought? Gowers did specifically at least call out his lack of experience and asked for corrections, though I didn’t have the fortitude to wade through his hundreds of comments–perhaps this stands in part because there was little, if any indication of the background and direct identity of any of the respondents within the thread. As an simple example, while reading the comments on Dr. Pachter’s site, I’m surprised there is very little indication of Nicholas Bray’s standing there as he’s one of Pachter’s students. It would be much nicer if, in fact, Bray had a more fully formed and fleshed out identity there or on his linked Gravatar page which has no detail at all, much less an actual avatar!

This post, Gowers’, and Tao’s are all excellent reasons for a more IndieWeb philosophical approach in academic blogging (and other scientific communication). Many of the respondents/commenters have little, if any, indication of their identities or backgrounds which makes it imminently harder to judge or trust their bonafides within the discussion. Some even chose to remain anonymous and throw bombs. If each of the respondents were commenting (preferably using their real names) on their own websites and using the Webmention protocol, I suspect the discussion would have been richer and more worthwhile by an order of magnitude. Rivin at least had a linked Twitter account with an avatar, though I find it less than useful that his Twitter account is protected, a fact that makes me wonder if he’s only done so recently as a result of fallout from this incident? I do note that it at least appears his Twitter account links to his university website and vice-versa, so there’s a high likelihood that they’re at least the same person.

I’ll also note that a commenter noted that they felt that their reply had been moderated out of existence, something which Lior Pachter certainly has the ability and right to do on his own website, but which could have been mitigated had the commenter posted their reply on their own website and syndicated it to Pachter’s.

Hiding in the comments, which are generally civil and even-tempered, there’s an interesting discussion about academic publishing that could have been its own standalone post. Beyond the science involved (or not) in this entire saga, a lot of the background for the real story is one of process, so this comment was one of my favorite parts.

]]>The Lathisms podcast shares the varied stories of Hispanic and Latinx mathematicians]]>

After Sir Michael Atiyah’s presentation of a claimed proof of the Riemann Hypothesis earlier this week at the Heidelberg Laureate Forum, we’ve shared some of the immediate discussion in the aftermath, and now here’s a round-up of what we’ve learned.

I’m not sure I agree wholly with some of the viewpoint taken here, but I will admit that I was reading some of the earlier reports and not as much of the popular press coverage. Most reports I heard specifically mentioned the proof hadn’t been seen or gone over by others and suggested caution both as a result of that as well as the fact that Atiyah had had some recent false starts in the past several years. Some went as far as to mention that senior mathematicians in the related areas had not commented at all on the purported proof and hinted that this was a sign that they didn’t think the proof held water but also as a sign of respect for Atiyah so as not to besmirch his reputation either. In some sense, the quiet was kind of a kiss of death.

]]>The Riemann hypothesis, a formula related to the distribution of prime numbers, has remained unsolved for more than a century

One of the lesser articles I’ve seen on the topic thus far…

]]>]]>Sylvester's line problem, known as the Sylvester-Gallai theorem in proved form, states that it is not possible to arrange a finite number of points so that a line through every two of them passes through a third unless they are all on a single line. This problem was proposed by Sylvester (1893), who asked readers to "Prove that it is not possible to arrange any finite number of real points so that a right line through every two of them shall pass through a third, unless they all lie in the same right line."

Woodall (1893) published a four-line "solution," but an editorial comment following his result pointed out two holes in the argument and sketched another line of enquiry, which is characterized as "equally incomplete, but may be worth notice." However, no correct proof was published at the time (Croft

et al.1991, p. 159), but the problem was revived by Erdős (1943) and correctly solved by Grünwald (1944). Coxeter (1948, 1969) transformed the problem into an elementary form, and a very short proof using the notion of Euclidean distance was given by Kelly (Coxeter 1948, 1969; Chvátal 2004). The theorem also follows using projective duality from a result of Melchior (1940) proved by a simple application of Euler's polyhedral formula (Chvátal 2004).Additional information on the theorem can be found in Borwein and Moser (1990), Erdős and Purdy (1991), Pach and Agarwal (1995), and Chvátal (2003).

In September 2003, X. Chen proved a conjecture of Chvátal that, with a certain definition of a line, the Sylvester-Gallai theorem extends to arbitrary finite metric spaces.

T. Gallai's proof has been outlined by P. Erdös in his submission of the problem to The American Mathematical Monthly in 1943. Solution Given the set Π of noncollinear points, consider the set of lines Σ that pass through at least two points of Π. Such lines are said to be connecting. Among the connecting lines, those that pass through exactly two points of Π are called ordinary.]]>

]]>The Sylvester–Gallai theorem in geometry states that, given a finite number of points in the Euclidean plane, either

* all the points lie on a single line; or

* there is a line which contains exactly two of the points.

It is named after James Joseph Sylvester, who posed it as a problem in 1893, and Tibor Gallai, who published one of the first proofs of this theorem in 1944.A line that contains exactly two of a set of points is known as an ordinary line. According to a strengthening of the theorem, every finite point set (not all on a line) has at least a linear number of ordinary lines. There is an algorithm that finds an ordinary line in a set of n points in time proportional to n log n in the worst case.

]]>The discrepancy of a sequence f(1), f(2), ... of numbers is defined to be the largest value of |f(d) + f(2d) + ... + f(nd)| as n and d range over the natural numbers. In the 1930s, Erdős posed the question of whether any sequence consisting only of +1 and -1 could have bounded discrepancy. In 2010, the collaborative Polymath5 project showed (among other things) that the problem could be effectively reduced to a problem involving completely multiplicative sequences. Finally, using recent breakthroughs in the asymptotics of completely multiplicative sequences by Matomaki and Radziwiłł, as well as a surprising application of the Shannon entropy inequalities, the Erdős discrepancy problem was solved in 2015. In his talk TT will discuss this solution and its connection to the Chowla and Elliott conjectures in number theory.

]]>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

]]>

]]>Tuesday, April 11th, 2017 9:30 am – 10:30 am

Structure vs. Randomness

Speaker: Terry Tao, UCLAWe 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.

]]>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)
- Ramsey Theory
- Erdős (1943)
- Gallai (1944)
- Steinberg’s alternate (1944)
- DeBruijn and Erdős (1948)
- Motzkin (1951)
- Dirac (1951)
- Kelly & Moser (1958)
- Tao-Green Proof

- Homework 1 (homeworks are generally not graded)

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.

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.

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’ve written some general thoughts, hints, and tips on these courses in the past.

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.

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!

]]>In the last week or so there has been some discussion on the internet about a paper (initially authored by Hill and Tabachnikov) that was initially accepted for publication in the Mathematical Inte…

I wish there were more on the math here or at least some solid discussion of the actual science. The huge number of comments make me just think that this is gasoline, however well intentioned it may be.

]]>This morning Sir Michael Atiyah gave a presentation at the Heidelberg Laureate Forum with a claimed proof of the Riemann hypothesis. The Riemann hypothesis (RH) is the most famous open problem in mathematics, and yet Atiyah claims to have a simple proof.

Based on the update that the whole thing may fall apart, but the fact that it’s based on the Todd function as it reaches a limit for the fine structure constant might provide an answer to Sean Carroll’s issues? We’ll see what comes of it.

]]>

]]>

Mathematician Michael Atiyah has presented his claimed proof of one of the most famous unsolved problems in maths, but others remain cautiously sceptical]]>

Sir Michael Atiyah, one of the world’s greatest living mathematicians, has proposed a derivation of α, the fine-structure constant of quantum electrodynamics. A preprint is here. The math her…]]>

]]>## "Category theory is a universal modeling language."

## Background.

Success is founded on information. A tight connection between success (in anything) and information. It follows that we should (if we want to be more successful) study what information is.

Grant proposals. These are several grant proposals, some funded, some in the pipeline, others not funded, that explain various facets of my research project.

Introductory talk (video, slides).

Blog post, on John Baez's blog Azimuth, about my motivations for studying this subject. (Here's a .pdf version.)

There have been a couple news stories regarding proofs of major theorems. First, an update on Shinichi Mochizuki’s proof of the abc conjecture, then an announcement that Sir Michael Atiyah claims to have proven the Riemann hypothesis.]]>

Michael Atiyah, a famed UK mathematician, claims that he has a "simple proof" of the Riemann hypothesis, a key unsolved question about the nature of prime numbers]]>

Two mathematicians have found what they say is a hole at the heart of a proof that has convulsed the mathematics community for nearly six years.

This break in the story of the ABC conjecture is sure to make that portion of Mike Miller’s upcoming math class on Gems And Astonishments of Mathematics: Past and Present at UCLA much more interesting.

]]>Cauchy-Lorentz: "Something alarmingly mathematical is happening, and you should probably pause to Google my name and check what field I originally worked in."

I love that it’s all the exact same data points…

]]>First, I’d like to thank the large number of commenters on my previous post for keeping the discussion surprisingly calm and respectful given the topic discussed. In that spirit, and to try t…

The analysis here makes me think there might be some useful tidbits hiding in the 300+ comments of his prior article. I wish I had the time to dig back into it.

Our prehistoric ancestors were not doing higher mathematics, so we would need to think of some way that being on the spectrum could have caused a man at that time to become highly attractive to women. ❧

One needs to remember that it isn’t always the men that themselves need to propagate the genes directly (ie, they don’t mate with someone to hand their genes down to their progeny directly). Perhaps a man on the autism spectrum, while not necessarily attractive himself, has traits which improve the lives and fitness of the offspring of his sister’s children? Then it’s not his specific genes which are passed on as a result, but those of his sister’s which have a proportion of his genes since they both share their parent’s genes in common.

September 19, 2018 at 03:35PM

variability amongst males ❧

Does it need to be a mate-related thing? Why not an environmental one. I seem to recall that external temperature had a marked effect on the sexual selection within alligator populations such that a several degree change during gestation would swing the sex proportion one way or another. Could these effects of environment have caused a greater variability?

Further, what other factors may be at play? What about in sea horse populations where males carry the young? Does this make a difference?

September 19, 2018 at 03:41PM

Update to post, added 11th September. As expected, there is another side to the story discussed below. See this statement about the decision by the Mathematical Intelligencer and this one about the…

I agree in large part with his assessment, and do so in part based on Ted Hill’s Quillette article and not having read the actual paper yet.

I will say that far more people have now either heard about or read Hill’s paper than would have ever otherwise been aware of it had it actually gone ahead and actually been published and kept up. This is definitely an academic case of the Barbara Streisand effect, though done somewhat in reverse.

]]>I provide a (very) brief introduction to game theory. I have developed these notes to provide quick access to some of the basics of game theory; mainly as an aid for students in courses in which I assumed familiarity with game theory but did not require it as a prerequisite.]]>

This is a collection of introductory, expository notes on applied category theory, inspired by the 2018 Applied Category Theory Workshop, and in these notes we take a leisurely stroll through two themes (functorial semantics and compositionality), two constructions (monoidal categories and decorated cospans) and two examples (chemical reaction networks and natural language processing) within the field. [PDF]

hat tip:

Friends! I am so happy to share that my little booklet “What is Applied Category Theory?” is now available on the arXiv. It’s a collection of introductory, expository notes inspired by the ACT workshop that took place earlier this year. Enjoy! https://t.co/EPYP19z14x pic.twitter.com/O4uVhj401s

— Tai-Danae Bradley (@math3ma) September 18, 2018

See also Notes on Applied Category Theory

]]>"Crucial life lessons from the end of hockey games, Idris Elba, and some Wall Street guys with a lot of time on their hands."

Revisionist History wades into the crowded self-help marketplace, with some help with from a band of math whizzes and Hollywood screenwriters. It's late in a hockey game, and you're losing. When should you pull your goalie? And what if you used that same logic when a bad guy breaks into your house and holds your entire family hostage? We think the unthinkable, so you don’t have to.

Why one should be a bit more disagreeable and “pull the goalie”.

Pulling the Goalie: Hockey and Investment Implications on SSRN.

]]>Welcome to our Election Update for Thursday, Sept. 13! The biggest update: We now have a Senate forecast to go with our House forecast! The “Classic” version of the Senate forecast currently gives Democrats a 1 in 3 chance of flipping the upper chamber. Meanwhile, the “Classic” version of our House forecast hasn’t really changed much since yesterday: Democrats still have a 5 in 6 chance of winning control. Across thousands of simulations, Democrats’ average gain was 39 seats.]]>

Last weekend, Steelers running back Le’Veon Bell sat out the first game of the regular season rather than play under the NFL franchise tag. Slated to earn $14.5 million in guaranteed money in 2018, Bell loses out on $855,529 each week he fails to report. The franchise tag would make Bell the third highest paid running back in the NFL this season — but only if he actually plays. Around the league, there is a wide range of speculation on how long Bell’s holdout will last. ESPN’s Adam Schefter reports that his sources believe Bell could be back by the end of September, while others note his holdout could conceivably last through Week 10.]]>

This statement addresses some unfounded allegations about my personal involvement with the publishing of Ted Hill's preprint "An evolutionary theory for the variability hypothesis" (and the earlier version of this paper co-authored with Sergei Tabachnikov). As a number of erroneous statements have been made, I think it's important to state formally what transpired and my beliefs overall about academic freedom and integrity. I first saw the publicly-available paper of Hill and Tabachnikov on 9/6/17, listed to appear in The Mathematical Intelligencer. While the original link has been taken down, the version of the paper that was publicly available on the arxiv at that time is here. I sent an email, on 9/7/17, to the Editor-in-Chief of The Mathematical Intelligencer, about the paper of Hill and Tabachnikov. In it, I criticized the scientific merits of the paper and the decision to accept it for publication, but I never made the suggestion that the decision to publish it be reversed. Instead, I suggested that the journal publish a response rebuttal article by experts in the field to accompany the article. One day later, on 9/8/17, the editor wrote to me that she had decided not to publish the paper. I had no involvement in any editorial decisions concerning Hill's revised version of this paper in The New York Journal of Mathematics. Any indications or commentary otherwise are completely unfounded. I would like to make clear my own views on academic freedom and the integrity of the editorial process. I believe that discussion of scientific merits of research should never be stifled. This is consistent with my original suggestion to bring in outside experts to rebut the Hill-Tabachnikov paper. Invoking purely mathematical arguments to explain scientific phenomena without serious engagement with science and data is an offense against both mathematics and science.

A response to an article I read the other day in Quillette.

]]>