Mathematics | Chris Aldrich

🔖 Introduction to Category Theory | UCLA Continuing Education

Bookmarked Introduction to Category Theory (UCLA Continuing Education)

This course is an introduction to the basic tenets of category theory, as formulated and illustrated through examples drawn from algebra, calculus, geometry, set theory, topology, number theory, and linear algebra.

Category theory, since its development in the 1940s, has assumed an increasingly center-stage role in formalizing mathematics and providing tools to diverse scientific disciplines, most notably computer science. A category is fundamentally a family of mathematical obejcts (e.g., numbers, vector spaces, groups, topological spaces) along with “mappings” (so-called morphisms) between these objects that, in some defined sense, preserve structure. Taking it one step further, one can consider morphisms (so-called functors) between categories. This course is an introduction to the basic tenets of category theory, as formulated and illustrated through examples drawn from algebra, calculus, geometry, set theory, topology, number theory, and linear algebra. Topics to be discussed include: isomorphism; products and coproducts; dual categories; covariant, contravariant, and adjoint functors; abelian and additive categories; and the Yoneda Lemma. The course should appeal to devotees of mathematical reasoning, computer scientists, and those wishing to gain basic insights into a hot area of mathematics.

January 8, 2019 - March 19, 2019
Tuesday 7:00PM - 10:00PM
Location: UCLA
Instructor: Michael Miller
Fee: $453.00

The new catalog is out today and Mike Miller’s Winter class in Category Theory has been officially announced.

Oddly, it wasn’t listed in the published physical catalog, but it’s available online. I hope that those interested in mathematics will register as well as those who are interested in computer science.

🎧 Episode 48: Ain’t No Amoeba (MEN, Part 2) | Scene on Radio

Listened to Episode 48: Ain’t No Amoeba (MEN, Part 2) by John Biewen and Celeste Headlee from Scene on Radio

For millennia, Western culture (and most other cultures) declared that men and women were different sorts of humans—and, by the way, men were better. Is that claim not only wrong but straight-up backwards?

Co-hosts Celeste Headlee and John Biewen explore the current state of the nature-nurture gender debate, with help from Lisa Wade of Occidental College and Mel Konner of Emory University.

Music by Alex Weston, and by Evgueni and Sacha Galperine. Music and production help from Joe Augustine at Narrative Music.

🔖 Surreal number | Wikipedia

Bookmarked Surreal numbers (Wikipedia)

In mathematics, the surreal number system is a totally ordered proper class containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number. The surreals share many properties with the reals, including the usual arithmetic operations (addition, subtraction, multiplication, and division); as such, they form an ordered field.[a] If formulated in Von Neumann–Bernays–Gödel set theory, the surreal numbers are a universal ordered field in the sense that all other ordered fields, such as the rationals, the reals, the rational functions, the Levi-Civita field, the superreal numbers, and the hyperreal numbers, can be realized as subfields of the surreals.[1] The surreals also contain all transfinite ordinal numbers; the arithmetic on them is given by the natural operations. It has also been shown (in Von Neumann–Bernays–Gödel set theory) that the maximal class hyperreal field is isomorphic to the maximal class surreal field; in theories without the axiom of global choice, this need not be the case, and in such theories it is not necessarily true that the surreals are a universal ordered field.

👓 Sci-Fi Writer Greg Egan and Anonymous Math Whiz Advance Permutation Problem | Quanta Magazine

Read Sci-Fi Writer Greg Egan and Anonymous Math Whiz Advance Permutation Problem (Quanta Magazine)

A new proof from the Australian science fiction writer Greg Egan and a 2011 proof anonymously posted online are now being hailed as significant advances on a puzzle mathematicians have been studying for at least 25 years.

I wonder what happens when the reverse process is run on numbers like pi? This could be an interesting thing to take a look at in my current math class.

👓 Learning How to Learn Math | Math3ma | Tai-Danae Bradley

Read Learning How to Learn Math by

Tai-Danae Bradley (math3ma.com)

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.

A Szilassi polyhedron in math class

Instagram filter used: unknown

Photo taken at: UCLA Math Sciences Building

🔖 Collaborative Workshop for Women in Mathematical Biology | IPAM

Bookmarked Collaborative Workshop for Women in Mathematical Biology (IPAM)

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)

👓 Mathematics matters | Bits of DNA

Read Mathematics matters by

Lior Pachter (Bits of DNA)

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.

👓 Atiyah Riemann Hypothesis proof: final thoughts | The Aperiodical

Read Atiyah Riemann Hypothesis proof: final thoughts by Katie Steckles and Christian Lawson-Perfect (The Aperiodical)

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.