It’s “a definitive study for all time, like writing the final book,” says one researcher who’s mapping out new classes of geometric structures.
Category: Mathematics
🔖 Elementary Algebraic Geometry by Klaus Hulek
This is a genuine introduction to algebraic geometry. The author makes no assumption that readers know more than can be expected of a good undergraduate. He introduces fundamental concepts in a way that enables students to move on to a more advanced book or course that relies more heavily on commutative algebra. The language is purposefully kept on an elementary level, avoiding sheaf theory and cohomology theory. The introduction of new algebraic concepts is always motivated by a discussion of the corresponding geometric ideas. The main point of the book is to illustrate the interplay between abstract theory and specific examples. The book contains numerous problems that illustrate the general theory. The text is suitable for advanced undergraduates and beginning graduate students. It contains sufficient material for a one-semester course. The reader should be familiar with the basic concepts of modern algebra. A course in one complex variable would be helpful, but is not necessary. It is also an excellent text for those working in neighboring fields (algebraic topology, algebra, Lie groups, etc.) who need to know the basics of algebraic geometry.
Dr. Miller emailed me yesterday to confirm that the textbook for his Fall UCLA Extension course Introduction to Algebraic Geometry will be Elementary Algebraic Geometry by Klaus Hulek (AMS, 2003) ISBN: 0-8218-2952-1.
Sadly, I totally blew the prediction of which text he’d use. I was so far off that this book wasn’t even on my list to review! I must be slipping…
🔖 Homotopy Type Theory: Univalent Foundations of Mathematics
Homotopy type theory is a new branch of mathematics that combines aspects of several different fields in a surprising way. It is based on a recently discovered connection between homotopy theory and type theory. It touches on topics as seemingly distant as the homotopy groups of spheres, the algorithms for type checking, and the definition of weak ∞-groupoids. Homotopy type theory offers a new “univalent” foundation of mathematics, in which a central role is played by Voevodsky’s univalence axiom and higher inductive types. The present book is intended as a first systematic exposition of the basics of univalent foundations, and a collection of examples of this new style of reasoning — but without requiring the reader to know or learn any formal logic, or to use any computer proof assistant. We believe that univalent foundations will eventually become a viable alternative to set theory as the “implicit foundation” for the unformalized mathematics done by most mathematicians.
Introduction to Algebraic Geometry | UCLA Extension in Fall 2017
Algebraic geometry is the study, using algebraic tools, of geometric objects defined as the solution sets to systems of polynomial equations in several variables. This introductory course, the first in a two-quarter sequence, develops the basic theory of the subject, beginning with seminal theorems—the Hilbert Basis Theorem and Hilbert’s Nullstellensatz—that establish the dual relationship between so-called varieties—both affine and projective—and certain ideals of the polynomial ring in some number of variables. Topics covered in this first quarter include: algebraic sets, projective spaces, Zariski topology, coordinate rings, the Grassmannian, irreducibility and dimension, morphisms, sheaves, and prevarieties. The theoretical discussion will be supported by a large number of examples and exercises. The course should appeal to those with an interest in gaining a deeper understanding of the mathematical interplay among algebra, geometry, and topology. Prerequisites: Some exposure to advanced mathematical methods, particularly those pertaining to ring theory, fields extensions, and point-set topology.
Dr. Michael Miller has announced the topic for his Fall math class at UCLA Extension: Algebraic Geometry!!
Yes math fans, as previously hinted at in prior conversations, we’ll be taking a deep dive into the overlap of algebra and geometry. Be sure to line up expeditiously as registration for the class won’t happen until July 31, 2017.
While it’s not yet confirmed, some sources have indicated that this may be the first part of a two quarter sequence on the topic. As soon as we have more details, we’ll post them here first. As of this writing, there is no officially announced textbook for the course, but we’ve got some initial guesses and the best are as follows (roughly in decreasing order):
- Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra (Undergraduate Texts in Mathematics) 4th ed. by David A. Cox, John Little, and Donal O’Shea
- Algebraic Geometry: An Introduction (Universitext) by Daniel Perrin
- An Invitation to Algebraic Geometry (Universitext) by Karen E. Smith, Lauri Kahanpää, Pekka Kekäläinen, William Traves
- Algebraic Geometry (Dover Books on Mathematics) by Solomon Lefschetz (Less likely based on level and age, but Dr. Miller does love inexpensive Dover editions)
For those who are new to Dr. Miller’s awesome lectures, I’ve written some hints and tips on what to expect.
Most of his classes range from about 20-30 people, many of them lifelong regulars. (Yes, there are dozens of people like me who will take almost everything he teaches–he’s that good. This class, my 22nd, will be the start of my second decade of math with him.)
🔖 A Course in Game Theory by Martin J. Osborne, Ariel Rubinstein | MIT Press
A Course in Game Theory presents the main ideas of game theory at a level suitable for graduate students and advanced undergraduates, emphasizing the theory's foundations and interpretations of its basic concepts. The authors provide precise definitions and full proofs of results, sacrificing generalities and limiting the scope of the material in order to do so. The text is organized in four parts: strategic games, extensive games with perfect information, extensive games with imperfect information, and coalitional games. It includes over 100 exercises.
Tangentially suggested after reading In Game Theory, No Clear Path to Equilibrium by Erica Klarreich (Quanta Magazine)
Free, personal copy is downloadable in .pdf format with registration here.
🔖 Subjectivity and Correlation in Randomized Strategies by Robert J. Aumann | Journal of Mathematical Economics
(.pdf download) Subjectivity and correlation, though formally related, are conceptually distinct and independent issues. We start by discussing subjectivity. A mixed strategy in a game involves the selection of a pure strategy by means of a random device. It has usually been assumed that the random device is a coin flip, the spin of a roulette wheel, or something similar; in brief, an ‘objective’ device, one for which everybody agrees on the numerical values of the probabilities involved. Rather oddly, in spite of the long history of the theory of subjective probability, nobody seems to have examined the consequences of basing mixed strategies on ‘subjective’ random devices, i.e. devices on the probabilities of whose outcomes people may disagree (such as horse races, elections, etc.).
Suggested by In Game Theory, No Clear Path to Equilibrium by Erica Klarreich (Quanta Magazine)
🔖 Communication complexity of approximate Nash equilibria | arXiv
For a constant ϵ, we prove a poly(N) lower bound on the (randomized) communication complexity of ϵ-Nash equilibrium in two-player NxN games. For n-player binary-action games we prove an exp(n) lower bound for the (randomized) communication complexity of (ϵ,ϵ)-weak approximate Nash equilibrium, which is a profile of mixed actions such that at least (1−ϵ)-fraction of the players are ϵ-best replying.
Suggested by In Game Theory, No Clear Path to Equilibrium by Erica Klarreich (Quanta Magazine)
👓 Kaisa Matomäki Dreams of Primes | Quanta Magazine
Kaisa Matomäki has proved that properties of prime numbers over long intervals hold over short intervals as well. The techniques she uses have transformed the study of these elusive numbers.
👓 In Game Theory, No Clear Path to Equilibrium | Quanta Magazine
John Nash’s notion of equilibrium is ubiquitous in economic theory, but a new study shows that it is often impossible to reach efficiently.
There’s a couple of interesting sounding papers in here that I want to dig up and read. There are some great results that sound like they are crying out for better generalization and classification. Perhaps some overlap with information theory and complexity?
To some extent I also find myself wondering about repeated play as a possible random walk versus larger “jumps” in potential game play and the effects this may have on the “evolution” of a solution by play instead of a simpler closed mathematical solution.
Cotton twill hat features a full color embroidered Johns Hopkins lacrosse design showcasing our shielded blue jay. Unstructured low profile fit. Just the right wash; renowned perfect fit. Fabric strap closure with brass slide buckle. 100% cotton twill. Adjustable. Black. By Legacy.
I’d love to have a Johns Hopkins hat just like this with “Math” instead of “Lacrosse”. Surely the department has them made occasionally?
An Information Theory Playlist on Spotify
In honor of tomorrow’s release of Jimmy Soni and Rob Goodman’s book A Mind at Play: How Claude Shannon Invented the Information Age, I’ve created an Information Theory playlist on Spotify.
Songs about communication, telephones, conversation, satellites, love, auto-tune and even one about a typewriter! They all relate at least tangentially to the topic at hand. To up the ante, everyone should realize that digital music would be impossible without Shannon’s seminal work.
Let me know in the comments or by replying to one of the syndicated copies listed below if there are any great tunes that the list is missing.
Enjoy the list and the book!
👓 Maryam Mirzakhani | What’s New
I am totally stunned to learn that Maryam Mirzakhani died today, aged 40, after a severe recurrence of the cancer she had been fighting for several years. I had planned to email her some wishes for a speedy recovery after learning about the relapse yesterday; I still can’t fully believe that she didn’t make it.
A nice obituary about a fantastic mathematician from a fellow Fields Prize winner.
The quintessential poolside summer reading: A Mind at Play
Instagram filter used: Clarendon
Photo taken at: Gerrish Swim & Tennis Club
👓 Abel Prize 2017: Yves Meyer wins ‘maths Nobel’ for work on wavelets
Frenchman wins prestigious prize for theory that links maths, information technology and computer science
A Long-Sought Proof, Found and Almost Lost | Quanta Magazine
When a German retiree proved a famous long-standing mathematical conjecture, the response was underwhelming.