Category: Mathematics

The L-functions and modular forms database

Bookmarked The L-functions and modular forms database by Tim Gowers (Gowers's Weblog)
...I rejoice that a major new database was launched today. It’s not in my area, so I won’t be using it, but I am nevertheless very excited that it exists. It is called the L-functions and modular forms database. The thinking behind the site is that lots of number theorists have privately done lots of difficult calculations concerning L-functions, modular forms, and related objects. Presumably up to now there has been a great deal of duplication, because by no means all these calculations make it into papers, and even if they do it may be hard to find the right paper. But now there is a big database of these objects, with a large amount of information about each one, as well as a great big graph of connections between them. I will be very curious to know whether it speeds up research in number theory: I hope it will become a completely standard tool in the area and inspire people in other areas to create databases of their own.

…I rejoice that a major new database was launched today. It’s not in my area, so I won’t be using it, but I am nevertheless very excited that it exists. It is called the L-functions and modular forms database. The thinking behind the site is that lots of number theorists have privately done lots of difficult calculations concerning L-functions, modular forms, and related objects. Presumably up to now there has been a great deal of duplication, because by no means all these calculations make it into papers, and even if they do it may be hard to find the right paper. But now there is a big database of these objects, with a large amount of information about each one, as well as a great big graph of connections between them. I will be very curious to know whether it speeds up research in number theory: I hope it will become a completely standard tool in the area and inspire people in other areas to create databases of their own.

–Tim Gowers

Tom M. Apostol, 1923–2016

Bookmarked Tom M. Apostol, 1923–2016 (CalTech Division of Physics, Mathematics and Astronomy)
Tom M. Apostol, professor of mathematics, emeritus at California Institute of Technology passed away on May 8, 2016. He was 92.
My proverbial mathematical great-grandfather passed away yesterday.

As many know, for over a decade, I’ve been studying a variety of areas of advanced abstract mathematics with Michael Miller. Mike Miller received his Ph.D. in 1974 (UCLA) under the supervision of Basil Gordon who in turn received his Ph.D. in 1956 (CalTech) under the supervision of Tom M. Apostol.

Incidentally going back directly three generations is Markov and before that Chebyshev and two generations before that Lobachevsky.

Sadly, I never got to have Tom as a teacher directly myself, though I did get to meet him several times in (what mathematicians might call) social situations. I did have the advantage of delving into his two volumes of Calculus as well as referring to his book on Analytic Number Theory. If it’s been a while since you’ve looked at calculus, I highly recommend an evening or two by the fire with a glass of wine while you revel in Calculus, Vol 1 or Calculus, Vol 2.

It’s useful to take a moment to remember our intellectual antecedents, so in honor of Tom’s passing, I recommend the bookmarked very short obituary (I’m sure more will follow), this obituary of Basil, and this issue of the Notices of the AMS celebrating Basil as well. I also came across a copy of Fascinating Mathematical People which has a great section on Tom and incidentally includes some rare younger photos of Sol Golomb who suddenly passed away last Sunday. (It’s obviously been a tough week for me and math in Southern California this week.)

Mathematician Basil Gordon (wearing a brown sweater) sitting near Chris Aldrich
Somehow Basil Gordon (brown sweater) and I were recognized on the same day at an event for Johns Hopkins University. Later we discovered that he had overseen the Ph.D. of one of my long-time mathematics professors and he was a lifelong friend of my friend/mentor Solomon Golomb. — at Loews Santa Monica Beach Hotel on March 13, 2010

Michael Miller making a "handwaving argument" during a lecture on Algebraic Number Theory at UCLA on November 15, 2015. I've taken over a dozen courses from Mike in areas including Group Theory, Field Theory, Galois Theory, Group Representations, Algebraic Number Theory, Complex Analysis, Measure Theory, Functional Analysis, Calculus on Manifolds, Differential Geometry, Lie Groups and Lie Algebras, Set Theory, Differential Geometry, Algebraic Topology, Number Theory, Integer Partitions, and p-Adic Analysis.
Michael Miller making a “handwaving argument” during a lecture on Algebraic Number Theory at UCLA on November 15, 2015. I’ve taken over a dozen courses from Mike in areas including Group Theory, Field Theory, Galois Theory, Group Representations, Algebraic Number Theory, Complex Analysis, Measure Theory, Functional Analysis, Calculus on Manifolds, Differential Geometry, Lie Groups and Lie Algebras, Set Theory, Differential Geometry, Algebraic Topology, Number Theory, Integer Partitions, and p-Adic Analysis.

Devastating News: Sol Golomb has apparently passed away on Sunday

I was getting concerned that I hadn’t heard back from Sol for a while, particularly after emailing him late last week, and then I ran across this notice through ITSOC & the IEEE:

Solomon W. Golomb (May 30, 1932 – May 1, 2016)

Shannon Award winner and long-time ITSOC member Solomon W. Golomb passed away on May 1, 2016.
Solomon W. Golomb was the Andrew Viterbi Chair in Electrical Engineering at the University of Southern California (USC) and was at USC since 1963, rising to the rank of University and Distinguished Professor. He was a member of the National Academies of Engineering and Science, and was awarded the National Medal of Science, the Shannon Award, the Hamming Medal, and numerous other accolades. As USC Dean Yiannis C. Yortsos wrote, “With unparalleled scholarly contributions and distinction to the field of engineering and mathematics, Sol’s impact has been extraordinary, transformative and impossible to measure. His academic and scholarly work on the theory of communications built the pillars upon which our modern technological life rests.”

In addition to his many contributions to coding and information theory, Professor Golomb was one of the great innovators in recreational mathematics, contributing many articles to Scientific American and other publications. More recent Information Theory Society members may be most familiar with his mathematics puzzles that appeared in the Society Newsletter, which will publish a full remembrance later.

A quick search a moment later revealed this sad confirmation along with some great photos from an award Sol received just a week ago:

As is common in academia, I’m sure it will take a few days for the news to drip out, but the world has certainly lost one of its greatest thinkers, and many of us have lost a dear friend, colleague, and mentor.

I’ll try touch base with his family and pass along what information sniff I can. I’ll post forthcoming obituaries as I see them, and will surely post some additional thoughts and reminiscences of my own in the coming days.

Golomb and national medal of science
President Barack Obama presents Solomon Golomb with the National Medal of Science at an awards ceremony held at the White House in 2013.

Online Lectures in Information Theory

Replied to Where can I find good online lectures in information theory? (quora.com)
There aren’t a lot of available online lectures on the subject of information theory, but here are the ones I’m currently aware of:

Introductory

Advanced

Fortunately, most are pretty reasonable, though vary in their coverage of topics. The introductory lectures don’t require as much mathematics and can probably be understood by those at the high school level with just a small amount of basic probability theory and an understanding of the logarithm.

The top three in the advanced section (they generally presume a prior undergraduate level class in probability theory and some amount of mathematical sophistication) are from professors who’ve written some of the most commonly used college textbooks on the subject. If I recall a first edition of the Yeung text was available via download through his course interface. MacKay’s text is available for free download from his site as well.

Feel free to post other video lectures or resources you may be aware of in the comments below.

Editor’s Update: With sadness, I’ll note that David MacKay died just days after this was originally posted.

“ALOHA to the Web”: Dr. Norm Abramson to give 2016 Viterbi Lecture at USC

Bookmarked USC - Viterbi School of Engineering - Dr. Norm Abramson (viterbi.usc.edu)

“ALOHA to the Web”

Dr. Norman Abramson, Professor Emeritus, University of Hawaii

Lecture Information

Thursday, April 14, 2016
Hughes Electrical Engineering Center (EEB)
Reception 3:00pm (EEB Courtyard)
Lecture 4:00pm (EEB 132)

Abstract

Wireless access to the Internet today is provided predominantly by random access ALOHA channels connecting a wide variety of user devices. ALOHA channels were first analyzed, implemented and demonstrated in the ALOHA network at the University of Hawaii in June, 1971. Information Theory has provided a constant guide for the design of more efficient channels and network architectures for ALOHA access to the web.

In this talk we examine the architecture of networks using ALOHA channels and the statistics of traffic within these channels. That traffic is composed of user and app oriented information augmented by protocol information inserted for the benefit of network operation. A simple application of basic Information Theory can provide a surprising guide to the amount of protocol information required for typical web applications.

We contrast this theoretical guide of the amount of protocol information required with measurements of protocol generated information taken on real network traffic. Wireless access to the web is not as efficient as you might guess.

Biography

Norman Abramson received an A.B. in physics from Harvard College in 1953, an M.A. in physics from UCLA in 1955, and a Ph.D. in Electrical Engineering from Stanford in 1958.

He was an assistant professor and associate professor of electrical engineering at Stanford from 1958 to 1965. From 1967 to 1995 he was Professor of Electrical Engineering, Professor of Information and Computer Science, Chairman of the Department of Information and Computer Science, and Director of the ALOHA System at the University of Hawaii in Honolulu. He is now Professor Emeritus of Electrical Engineering at the University of Hawaii. He has held visiting appointments at Berkeley (1965), Harvard (1966) and MIT (1980).

Abramson is the recipient of several major awards for his work on random access channels and the ALOHA Network, the first wireless data network. The ALOHA Network went into operation in Hawaii in June, 1971. Among these awards are the Eduard Rhein Foundation Technology Award (Munich, 2000), the IEEE Alexander Graham Bell Medal (Philadelphia, 2007) and the NEC C&C Foundation Award (Tokyo, 2011).

2016 North-American School of Information Theory, June 21-23

Bookmarked 2016 North-American School of Information Theory, June 21-23, 2016 (itsoc.org)

The 2016 School of information will be hosted at Duke University, June 21-23. It is sponsored by the IEEE Information Theory Society, Duke University, the Center for Science of Information, and the National Science Foundation. The school provides a venue where doctoral and postdoctoral students can learn from distinguished professors in information theory, meet with fellow researchers, and form collaborations.

Program and Lectures

The daily schedule will consist of morning and afternoon lectures separated by a lunch break with poster sessions. Students from all research areas are welcome to attend and present their own research via a poster during the school.  The school will host lectures on core areas of information theory and interdisciplinary topics. The following lecturers are confirmed:

  • Helmut Bölcskei (ETH Zurich): The Mathematics of Deep Learning
  • Natasha Devroye (University of Illinois, Chicago): The Interference Channel
  • René Vidal (Johns Hopkins University): Global Optimality in Deep Learning and Beyond
  • Tsachy Weissman (Stanford University): Information Processing under Logarithmic Loss
  • Aylin Yener (Pennsylvania State University): Information-Theoretic Security

Logistics

Applications will be available on March 15 and will be evaluated starting April 1.  Accepted students must register by May 15, 2016.  The registration fee of $200 will include food and 3 nights accommodation in a single-occupancy room.  We suggest that attendees fly into the Raleigh-Durham (RDU) airport located about 30 minutes from the Duke campus. Housing will be available for check-in on the afternoon of June 20th.  The main part of the program will conclude after lunch on June 23rd so that attendees can fly home that evening.

To Apply: click “register” here (fee will accepted later after acceptance)

Administrative Contact: Kathy Peterson, itschool2016@gmail.com

Organizing Committee

Henry Pfister (chair) (Duke University), Dror Baron (North Carolina State University), Matthieu Bloch (Georgia Tech), Rob Calderbank (Duke University), Galen Reeves (Duke University). Advisors: Gerhard Kramer (Technical University of Munich) and Andrea Goldsmith (Stanford)

Sponsors

Introduction to Information Theory | SFI’s Complexity Explorer

Many readers often ask me for resources for delving into the basics of information theory. I hadn’t posted it before, but the Santa Fe Institute recently had an online course Introduction to Information Theory through their Complexity Explorer, which has some other excellent offerings. It included videos, fora, and other resources and was taught by the esteemed physicist and professor Seth Lloyd. There are a number of currently active students still learning and posting there.

Introduction to Information Theory

About the Tutorial:

This tutorial introduces fundamental concepts in information theory. Information theory has made considerable impact in complex systems, and has in part co-evolved with complexity science. Research areas ranging from ecology and biology to aerospace and information technology have all seen benefits from the growth of information theory.

In this tutorial, students will follow the development of information theory from bits to modern application in computing and communication. Along the way Seth Lloyd introduces valuable topics in information theory such as mutual information, boolean logic, channel capacity, and the natural relationship between information and entropy.

Lloyd coherently covers a substantial amount of material while limiting discussion of the mathematics involved. When formulas or derivations are considered, Lloyd describes the mathematics such that less advanced math students will find the tutorial accessible. Prerequisites for this tutorial are an understanding of logarithms, and at least a year of high-school algebra.

About the Instructor(s):

Professor Seth Lloyd is a principal investigator in the Research Laboratory of Electronics (RLE) at the Massachusetts Institute of Technology (MIT). He received his A.B. from Harvard College in 1982, the Certificate of Advanced Study in Mathematics (Part III) and an M. Phil. in Philosophy of Science from Cambridge University in 1983 and 1984 under a Marshall Fellowship, and a Ph.D. in Physics in 1988 from Rockefeller University under the supervision of Professor Heinz Pagels.

From 1988 to 1991, Professor Lloyd was a postdoctoral fellow in the High Energy Physics Department at the California Institute of Technology, where he worked with Professor Murray Gell-Mann on applications of information to quantum-mechanical systems. From 1991 to 1994, he was a postdoctoral fellow at Los Alamos National Laboratory, where he worked at the Center for Nonlinear Systems on quantum computation. In 1994, he joined the faculty of the Department of Mechanical Engineering at MIT. Since 1988, Professor Lloyd has also been an adjunct faculty member at the Sante Fe Institute.

Professor Lloyd has performed seminal work in the fields of quantum computation and quantum communications, including proposing the first technologically feasible design for a quantum computer, demonstrating the viability of quantum analog computation, proving quantum analogs of Shannon’s noisy channel theorem, and designing novel methods for quantum error correction and noise reduction.

Professor Lloyd is a member of the American Physical Society and the Amercian Society of Mechanical Engineers.

Tutorial Team:

Yoav Kallus is an Omidyar Fellow at the Santa Fe Institute. His research at the boundary of statistical physics and geometry looks at how and when simple interactions lead to the formation of complex order in materials and when preferred local order leads to system-wide disorder. Yoav holds a B.Sc. in physics from Rice University and a Ph.D. in physics from Cornell University. Before joining the Santa Fe Institute, Yoav was a postdoctoral fellow at the Princeton Center for Theoretical Science in Princeton University.

How to use Complexity Explorer: How to use Complexity Explore
Prerequisites: At least one year of high-school algebra
Like this tutorial? 

Syllabus

  1. Introduction
  2. Forms of Information
  3. Information and Probability
  4. Fundamental Formula of Information
  5. Computation and Logic: Information Processing
  6. Mutual Information
  7. Communication Capacity
  8. Shannon’s Coding Theorem
  9. The Manifold Things Information Measures
  10. Homework

Marvin Minsky, Pioneer in Artificial Intelligence, Dies at 88 | The New York Times

Professor Minsky laid the foundation for the field by demonstrating the possibilities of imparting common-sense reasoning to computers.

Source: Marvin Minsky, Pioneer in Artificial Intelligence, Dies at 88 – The New York Times

Einstein’s Equations From Entanglement

Brian Swingle Colloquium at Caltech

From the Physics Research Conference 2015-2016
on Thursday, November 19, 2015 at 4:00 pm
at the California Institute of Technology, East Bridge 201 – Norman Bridge Laboratory of Physics, East

All talks are intended for a broad audience, and everyone is encouraged to attend. A list of future conferences can be found here.
Sponsored by Division of Physics, Mathematics and Astronomy

In recent years we have learned that the physics of quantum information plays a crucial role in the emergence of spacetime from microscopic degrees of freedom.

I will review the idea that entanglement is the glue which holds spacetime together and show how Einstein’s equations plausibly emerge from this perspective. One ubiquitous feature of these dynamical equations is the formation of black holes, so I will conclude by discussing some new ideas about the nature of spacetime inside a black hole.

Brian Swingle, postdoctoral fellow at the Stanford Institute for Theoretical Physics and physicist focusing on quantum matter, quantum information, and quantum gravity
in Physics Research Conference | Caltech

Click here for full screen presentation.

Why math? JHU mathematician on teaching, theory, and the value of math in a modern world | Hub

Bookmarked Why math? JHU mathematician on teaching, theory, and the value of math in a modern world (The Hub)
Great to see this interview with my friend and mathematician Richard Brown from Johns Hopkins Unviersity.  Psst: He’s got an interesting little blog, or you can follow some of his work on Facebook and Twitter.

Click through for the full interview: Q+A with Richard Brown, director of undergraduate studies in Johns Hopkins University’s Department of Mathematics

 

https://youtu.be/kg2mOl042ng

Algebraic Number Theory: The Sequel | UCLA Extension

I know you’ve all been waiting for the announcement with bated breath! We’ve known for a while that Mike Miller’s Winter course would be a follow-on course to his Algebraic Number Theory course this Fall, but it’s been officially posted, so now you can register for it: Algebraic Number Theory: The Sequel.

I’m sure, as always, that there are a few who are interested, but who couldn’t make the Fall lectures. Never fear, there’s a group of us that can help you get up to speed to keep pace with us during the second quarter. Just drop us a note and we’ll see what we can do.

Algebraic Number Theory: The Sequel

In no field of mathematics is there such an irresistible fascination as in the theory of numbers. This course, the second in a two-quarter sequence, is an introductory, yet rigorous, survey of algebraic number theory, which evolved historically through attempts to prove Fermat’s Last Theorem. Beginning with a quick review of the previous quarter’s work, the course continues discussions on the structure of algebraic number fields, focusing particular attention on primes, units, and roots of unity in quadratic, cubic, and cyclotomic fields. Topics to be discussed include: norms and traces; the ideal class group; Minkowski’s Translate, Convex Body, and Linear Forms theorems; and Dirichlet’s Unit Theorem.

UCLA: 5137 Math Sciences
Tuesday, 7-10pm,
January 5 – March 15
11 meetings total

MATH X450.9
3.00 units

Recommended Textbook

We’ll be using Introductory Algebraic Number Theory by Saban Alaca and Kenneth S. Williams (Cambridge University Press, 2003, ISBN: 978-0521183048).

Introductory Algebraic Number Theory

Eugenia Cheng, author of How to Bake Pi, on Colbert Tonight

Earlier this year, I read Eugenia Cheng’s brilliant book How to Bake Pi: An Edible Exploration of the Mathematics of Mathematics. Tonight she’s appearing (along with Daniel Craig apparently) on the The Late Show with Stephen Colbert. I encourage everyone to watch it and read her book when they get the chance.

How-to-bake-pi

You can also read more about her appearance from Category Theorist John Carlos Baez here: Cakes, Custard, Categories and Colbert | The n-Category Café

My brief review of her book on GoodReads.com:

How to Bake Pi: An Edible Exploration of the Mathematics of MathematicsHow to Bake Pi: An Edible Exploration of the Mathematics of Mathematics by Eugenia Cheng
My rating: 4 of 5 stars

While most of the book is material I’ve known for a long time, it’s very well structured and presented in a clean and clear manner. Though a small portion is about category theory and gives some of the “flavor” of the subject, the majority is about how abstract mathematics works in general.

I’d recommend this to anyone who wants to have a clear picture of what mathematics really is or how it should be properly thought about and practiced (hint: it’s not the pablum you memorized in high school or even in calculus or linear algebra). Many books talk about the beauty of math, while this one actually makes steps towards actually showing the reader how to appreciate that beauty.

Like many popular books about math, this one actually has very little that goes beyond the 5th grade level, but in examples that are very helpfully illuminating given their elementary nature. The extended food metaphors and recipes throughout the book fit in wonderfully with the abstract nature of math – perhaps this is why I love cooking so much myself.

I wish I’d read this book in high school to have a better picture of the forest of mathematics.

More thoughts to come…

What An Actual Handwaving Argument in Mathematics Looks Like

I’m sure we’ve all heard them many times, but this is what an actual handwaving argument looks like in a mathematical setting.

Handwaving during Algebraic Number Theory

Instagram filter used: Normal

Photo taken at: UCLA Math Sciences Building

Handwaving during Algebraic Number Theory

A photo posted by Chris Aldrich (@chrisaldrich) on

Winter Q-BIO Quantitative Biology Meeting February 15-18, 2016

Bookmarked Winter Q-BIO Quantitative Biology Meeting February 15-18, 2016 (w-qbio.org)
The Winter Q-BIO Quantitative Biology Meeting is coming up at the Sheraton Waikiki in Oahu, HI, USA

A predictive understanding of living systems is a prerequisite for designed manipulation in bioengineering and informed intervention in medicine. Such an understanding requires quantitative measurements, mathematical analysis, and theoretical abstraction. The advent of powerful measurement technologies and computing capacity has positioned biology to drive the next scientific revolution. A defining goal of Quantitative Biology (qBIO) is the development of general principles that arise from networks of interacting elements that initially defy conceptual reasoning. The use of model organisms for the discovery of general principles has a rich tradition in biology, and at a fundamental level the philosophy of qBIO resonates with most molecular and cell biologists. New challenges arise from the complexity inherent in networks, which require mathematical modeling and computational simulation to develop conceptual “guideposts” that can be used to generate testable hypotheses, guide analyses, and organize “big data.”

The Winter q-bio meeting welcomes scientists and engineers who are interested in all areas of q-bio. For 2016, the meeting will be hosted at the Sheraton Waikiki, which is located in Honolulu, on the island of Oahu. The resort is known for its breathtaking oceanfront views, a first-of-its-kind recently opened “Superpool” and many award-winning dining venues. Registration and accommodation information can be found via the links at the top of the page.

Source: Winter Q-BIO Quantitative Biology Meeting

Shinichi Mochizuki and the impenetrable proof of the abc conjecture

Read The biggest mystery in mathematics: Shinichi Mochizuki and the impenetrable proof of the ABC Conjecture (Nature News & Comment)
A Japanese mathematician claims to have solved one of the most important problems in his field. The trouble is, hardly anyone can work out whether he's right.

The biggest mystery in mathematics

This article in Nature is just wonderful. Everyone will find it interesting, but those in the Algebraic Number Theory class this fall will be particularly interested in the topic – by the way, it’s not too late to join the class. After spending some time over the summer looking at Category Theory, I’m tempted to tackle Mochizuki’s proof as I’m intrigued at new methods in mathematical thinking (and explaining.)

The abc conjecture refers to numerical expressions of the type a + b = c. The statement, which comes in several slightly different versions, concerns the prime numbers that divide each of the quantities a, b and c. Every whole number, or integer, can be expressed in an essentially unique way as a product of prime numbers — those that cannot be further factored out into smaller whole numbers: for example, 15 = 3 × 5 or 84 = 2 × 2 × 3 × 7. In principle, the prime factors of a and b have no connection to those of their sum, c. But the abc conjecture links them together. It presumes, roughly, that if a lot of small primes divide a and b then only a few, large ones divide c.

Thanks to Rama for bringing this to my attention!