Topics in Math and Physics

A First Course in Loop Quantum Gravity

2012-01-08T12:28:00.000-05:00

Same idea as Zwiebach's A First Course in String Theory. I highly recommend both books, whichever side of the intellectual divide you're on, or even if (particularly if?) you are agnostic on the question of the correct way to quantize gravity.

Graduate Physics Education ... Free!

2011-12-31T13:11:00.005-05:00

Perimeter Scholars International
The 2010-2011 PSI course series is a condensed program of first and second year graduate physics study including (but not limited to) quantum mechanics, quantum field theory, quantum information, general relativity, cosmology, the standard model, mathematical physics, condensed matter physics, string theory, quantum gravity and scientific computation.

Perimeter Institute Recorded Seminar Archive
The PIRSA website contains the usual collection of advanced graduate courses:
PIRSA:C11003 - Higher Spin Theories
PIRSA:C10018 - Space-time, Quantum Mechanics and Scattering Amplitudes
PIRSA:C10008 - Cosmology Mini Course
PIRSA:C10003 - Quantum Field Theory for Cosmology
PIRSA:C10002 - Foundations and Interpretation of Quantum Theory
PIRSA:C09020 - Introduction to Effective Field Theory
PIRSA:C09019 - General Relativity for Cosmology
PIRSA:C08019 - Quantum Field Theory
PIRSA:C08018 - Astrophysics & Cosmology through Problems
PIRSA:C08002 - Advanced General Relativity
PIRSA:C08001 - New Horizons In Fundamental Physics
PIRSA:C07007 - Advanced Topics in Cosmology
PIRSA:C07002 - Introduction to Quantum Groups
PIRSA:C06014 - Introduction to Quantum Information and Computation
PIRSA:C06012 - The Standard Model and Quantum Field Theory
PIRSA:C06001 - Introduction to Quantum Gravity
PIRSA:C05001 - Interpretation of Quantum Theory
as well as a slew of specialized courses on narrower topics.
(This list includes all courses of interest through 12/31/2011.)

Funny

2011-10-11T10:41:00.002-04:00

2011 Nobel Prize in Physics

2011-10-04T23:44:00.000-04:00

The Nobel Prize in Physics 2011 was divided, one half awarded to Saul Perlmutter, the other half jointly to Brian P. Schmidt and Adam G. Riess "for the discovery of the accelerating expansion of the Universe through observations of distant supernovae."

http://www.nobelprize.org/nobel_prizes/physics/laureates/2011/

University of Colorado at Colorado Springs Math Video Archive

2010-12-31T16:44:00.024-05:00

http://www.uccs.edu/~math/vidarchive.html

Spring Semester 2010
Math 448- Mathematical Modeling - Dr. Radu Cascaval

Spring Semester 2009
Math 443- Ordinary Differential Equations - Dr. Radu Cascaval

Fall Semester 2008
Math 447- Applied Mathematics - Dr. Radu Cascaval

Summer Semester 2008
Math 442 - Optimization - Dr. Radu Cascaval

Fall Semester 2007
Math 414 - Modern Algebra - Dr. Gene Abrams
Math 533 - Real Analysis - Dr. Rinaldo Schinazi

Summer Semester 2007
Math 425 - Chaotic Dynamical Systems - Dr. Greg Morrow

Spring Semester 2007
Math 432 - Modern Analysis II - Dr. Bob Carlson

Fall Semester 2006
Math 431 - Modern Analysis I - Dr. Rinaldo Schinazi

Summer Semester 2006
Math 483 - Linear Statistical Models - Dr. Greg Morrow

Spring Semester 2006
Math 535 - Applied Functional Analysis - Dr. Greg Morrow

ICTP Diploma Programs in Math & Physics

2010-12-31T15:33:00.006-05:00

High Energy Physics

Relativistic Quantum Mechanics

Quantum Electrodynamics

Quantum Field Theory

Lie Groups and Lie Algebras

Introduction to Particle Physics

General Relativity

The Standard Model

Susy Field Theory

String Theory

Condensed Matter Physics

Advanced Statistical Mechanics

Advanced Quantum Mechanics

Many Body Physics

Solid State Physics

Graduate Mathematics

Partial Differential Equations

Topology

Algebraic Topology

Abstract Algebra

Complex Analysis

Real Analysis I and Real Analysis II

Functional Analysis I and Functional Analysis II

Differential Equations and Dynamical Systems

Ergodic Theory

Differential Geometry

Algebraic Geometry

Probability Theory

Online Physics Video Courses

2010-12-31T14:22:00.031-05:00

Everybody knows about MIT OCW and Stanford on iTunes, but here are a few less well known video lectures on advanced physics topics.

Classical Mechanics at McGill University

Computational Physics at Oregon State University

Foundations of Theoretical Physics at USC

General Relativity at McGill University

Mathematical Physics I & II at University of New Mexico

Quantum Mechanics I & II at University of New Mexico

Quantum Physics by NPTEL on YouTube

Quantum Physics A, B & C at UCSD

Quantum Field Theory I & II at University of New Mexico

Here's someone else who shouldn't be trying to do math

2010-12-22T14:54:00.003-05:00

http://www.thebigquestions.com/2010/12/22/a-big-answer-2/

The correct answer is 1/2. In statistical notation, he is asking us to calculate E[G]/E[B+G], the expected proportion of females in the total population. However, he turns it into the different question E[G/(B+G)], the expected proportion of females in an average family, which is not generally equal to the first expression (since families are of different sizes) and which in this case gives the incorrect answer of 30.6%. The guy is impervious to all the good arguments that have been posted to his blog pointing out his error.

His argument is exactly the same as if headed down to the roulette tables in Vegas and placed bets on black, just making sure that at each session I stop when black hits. According to his "math", that strategy should provide a 69.4% win rate (slightly less once we account for 0 and 00, but still well above 50%). A sure-fire way to beat the house!

String Theory Tested, Fails Black Hole Predictions

2010-11-30T22:10:00.000-05:00

Back in 2006 there was a lot of talk of testing String Theory. Well, today CERN released a statement for the Compact Muon Solenoid Experiment. The short of it is simply that as far as they could tell, 'No experimental evidence for microscopic black holes has been found.' The long statement indicates that since the highly precise CMS detector found no spray of sub-atomic particles of normal matter while LHC smashed particles together, the hypothesis by string theory that micro black holes would be formed and quickly evaporated in this experiment was incorrect. These tests have given the team confidence to say that they can exclude a 'variety of theoretical models' for the cases of black holes with a mass of 3.5-4.5 TeV. While you may not be able to run around claiming that string theory is dead and disproved, evidently there are some adjustments that need to be made. (Source: Slashdot)

2010 Nobel in Physics

2010-10-09T17:05:00.000-04:00

http://nobelprize.org/nobel_prizes/physics/laureates/2010/
The Nobel Prize in Physics 2010 was awarded jointly to Andre Geim and Konstantin Novoselov "for groundbreaking experiments regarding the two-dimensional material graphene."

World's Worst Physicist?

2010-09-30T13:56:00.001-04:00

University of Central Florida physics professor Costas Efthimiou's work debunks pseudoscientific ideas, such as vampires and zombies, in an attempt to enhance public literacy. Legend has it that vampires feed on human blood and once bitten a person turns into a vampire and starts feasting on the blood of others. Efthimiou's debunking logic: On Jan 1, 1600, the human population was 536,870,911. If the first vampire came into existence that day and bit one person a month, there would have been two vampires by Feb. 1, 1600. A month later there would have been four, and so on. In just two-and-a-half years the original human population would all have become vampires with nobody left to feed on. If mortality rates were taken into consideration, the population would disappear much faster. Even an unrealistically high reproduction rate couldn't counteract this effect.

Wow! Leaving aside that his work debunking pseudoscience is NOT what we as a society expect our well-paid physicists to be working on, let's look at his own pseudoscientific work...

536,870,911 - really? and he knows this how? Has he ever heard of significant digits?

"once bitten a person turns into a vampire" - false; anybody with even a passing understanding of the vampire myth knows this to be wrong; dude needed to do some basic research

He's essentially assuming (a) vampires are immortal and (b) vampires must feed at least once a month on a human. These assumptions are mutually contradictory. If vampires were immortal then they would not have to feed every month to stay alive. If they must feed every month to stay alive then the well-known predator-prey model (based on the Lotka-Volterra differential equation) can predict all sorts of end-states, including extinction of humans, extinction of vampires, equilibrium, oscillation, or chaotic swings in the vampire and human populations, depending on the various parameters that can be used in the equations. Seriously, what on earth was this guy thinking when he tackled this problem? I wonder how long he worked on his "solution"? I've seen better thought-out logic on freshman physics papers.

Actually, on second thought, maybe it's better that this guy isn't doing physics. That could be dangerous!

Peer review

2010-08-11T10:22:00.001-04:00

Breakthroughs from the second tier
Peer review isn't perfect - meet 5 high-impact papers that should have ended up in bigger journals.
by the Scientist staff

I hate your paper
Many say the peer review system is broken. Here's how some journals are trying to fix it.
by Jef Akst

Peer review and the age of aquarius
It's time to reinvent the system that validates scientific discovery.
by Sarah Greene

Peer pressure
What should we do with peer review? Tell us in our new poll.
by Richard Grant

Chess & Physics

2010-04-08T15:13:00.001-04:00

My nominee for the decade's worst scientific reporting

2010-04-06T14:38:00.002-04:00

http://www.foxnews.com/scitech/2010/04/05/freaky-physics-proves-parallel-universes/?test=faces

Wolf says that time - at least in quantum mechanics - doesn't move straight like an arrow. It zig-zags, and he thinks it may be possible to build a machine that lets you bend time. Consider Sergei Krikalev, the Russian astronaut who flew six space missions. Richard Gott, a physicist at Princeton University, says Krikalev aged 1/48th of a second less than the rest of us because he orbited at very high speeds. And to age less than someone means you've jumped into the future - you did not experience the same present. In a sense, he says, Krikalev time-traveled to the future - and back again!

1) Writer switches from QM to relativity in a completely unwarranted manner.
2) The "time travel to the future and back again" idea is completely idiotic. I find it hard to believe that Richard Gott actually said that. My bet is the reporter got it wrong. (If not, then shame on Richard Gott.)
3) None of this has anything to do with the actual discovery, which is that researchers at UCSB appear to have found a macro-system that exhibits quantum superposition.
4) The headline is that this proves parallel universes exist. I'm still trying to figure out wtf that even means in the context of this story.

LHC breaks record

2010-03-30T10:21:00.000-04:00

The world's biggest atom smasher - the $10 billion Large Hadron Collider that straddles the Swiss-French border - set another record today by colliding protons at three times the previous record for energy created.

Get your geek on!

2010-02-11T12:28:00.002-05:00

For those (like myself) who wanted to play with Mathematica but couldn't justify the price ... there's now Mathematica Home Edition!!

http://www.wolfram.com/products/mathematicahomeedition/qa.html

(The functionality is exactly the same as in the full edition; it's just the licensing that is more restricted.)

One of my favorite comics

2010-01-04T16:04:00.001-05:00

(From the college geometry class I took over 20 years ago - God I'm old!)

Undergraduate Physics Education ... Free!

2009-10-10T11:23:00.005-04:00

iTunes U has a free downloadable course by Stanford's Leonard Susskind called The Theoretical Minimum. It consists of 57 video lectures covering almost an entire undergraduate physics education - Classical Mechanics, Quantum Mechanics, Statistical Mechanics, Special Relativity & Electromagnetism, General Relativity, even Cosmology.

2009 Nobel in Physics

2009-10-06T13:32:00.000-04:00

This year's Nobel Prize in Physics is awarded for two scientific achievements that have helped to shape the foundations of today's networked societies. They have created many practical innovations for everyday life and provided new tools for scientific exploration.

In 1966, Charles K. Kao made a discovery that led to a breakthrough in fiber optics. He carefully calculated how to transmit light over long distances via optical glass fibers. With a fiber of purest glass it would be possible to transmit light signals over 100 kilometers, compared to only 20 meters for the fibers available in the 1960s.

In 1969 Willard S. Boyle and George E. Smith invented the first successful imaging technology using a digital sensor, a Charge-Coupled Device. Digital photography has become an irreplaceable tool in many fields of research. The CCD has provided new possibilities to visualize the previously unseen. It has given us crystal clear images of distant places in our universe as well as the depths of the oceans.

Loop Quantum Gravity

2009-08-26T16:00:00.001-04:00

I attended a lecture today by Professor Abhay Ashtekar (of the Institute for Gravitation at Penn State University) about Loop Quantum Gravity at Georgia Tech. Apparently recent work has made considerable progress towards solving the infrared problem (i.e., proving agreement of LQG with classical General Relativity at low densities). Also, a density cutoff (of approximately 41% of the Planck density) has appeared naturally out of the formalism, instead of being imposed arbitrarily as an additional condition.

Slides covering material very similar to his Georgia Tech lecture can be found at
http://universe2009.obspm.fr/fichiers/Recherche/Friday-3/ashtekar.pdf

Possible ramifications of this work:
1) Help resolve the well-known horizon problem
2) Provide a mechanism for propagation of "seeds" for creating structure in the universe
3) Predicts a period of super-inflation, which could have implications for gravitational waves
4) Cause quantum corrections to the cosmic microwave backround radiation

#3 and #4 might provide testable predictions unlike superstring theory

Pure v Applied

2009-08-01T13:22:00.000-04:00

Lyman-Alpha Discovered Near Dawn of Time

2009-04-23T13:39:00.000-04:00

A newly found primordial blob may represent the most massive object ever discovered in the early universe, researchers announced today. The gas cloud, spotted from 12.9 billion light-years away, could signal the earliest stages of galaxy formation back when the universe was just 800 million years old.

12 yr old pursuing math/physics double major at my alma mater

2009-04-21T15:43:00.001-04:00

http://news.fiu.edu/?p=2985

Relativistic Quantum Mechanics Solutions

2008-12-15T00:25:00.001-05:00

Problem 4-2
Problem 4-4
Problem 4-7
Problem 7-7
Problem 7-8
Chapter 8 Problems (8-1, 8-2, 8-3, 8-5, 8-7)

Baez's Classical Mechanics Website

2008-12-11T23:45:00.000-05:00

As long as I'm posting these Goldstein solutions, I should definitely point out one of the best classical mechanics websites out there.

http://math.ucr.edu/home/baez/classical/

Goldstein 2nd Edition Chapter 2 Solutions

2008-12-10T00:32:00.000-05:00

· Problem 2-4
· Problem 2-5
· Problem 2-6
· Problem 2-9
· Problem 2-13
· Problem 2-15
· Problem 2-16
· Problem 2-20
· Problem 2-21

2008 Nobel Prize in Physics

2008-10-07T18:25:00.002-04:00

Yoichiro Nambu, who discovered the mechanism of spontaneous symmetry breaking, and Makoto Kobayashi and Toshihide Maskawa, who discovered the origin of the broken symmetry that predicts the existence of at least three families of quarks in nature, are announced as winners of the 2008 Nobel Prize in Physics.

LHC

2008-09-10T18:02:00.001-04:00

The Large Hadron Collider goes online today at CERN.

Google conmemorates this in their banner.

The Monty Hall Problem

2008-05-17T17:35:00.009-04:00

Let's start with a very standard probability question (the Monty Hall problem) that was put to Marilyn vos Savant.

Q: Suppose you're on a game show, and you're given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say #1, and the host, who knows what's behind the doors, opens another door, say #3, which has a goat. He says to you, "Do you want to pick door #2?" Is it to your advantage to switch your choice of doors?

A: Yes; you should switch. The first door has a 1/3 chance of winning, but the second door has a 2/3 chance. Here's a good way to visualize what happened. Suppose there are a million doors, and you pick door #1. Then the host, who knows what's behind the doors and will always avoid the one with the prize, opens them all except door #777,777. You'd switch to that door pretty fast, wouldn't you?

Now, I understand that the immediate instinct is to think that door #1 and door #2 each have a 50-50 chance of having the car. However, if you think about it for a few minutes (and I would certainly expect you to do so before writing Ms. vos Savant to tell her she's wrong), you should pretty quickly realize that the host will always open the door from 2,3 that does NOT have a car, and therefore the probability of the remaining door is 2/3 as opposed to 1/3 for the original door #1. I would certainly expect a PhD in math to come to this conclusion in short order. Even if they couldn't do it on their own, this is a standard problem covered in dozens of introductory statistics textbooks (hell, it's so well understood it's got a freakin' name), so I would at least expect a PhD in math to LOOK IT UP! However, I guess my expectations would be sadly shattered, not once but nine times (the names have been bolded to call out the guilty)...

Since you seem to enjoy coming straight to the point, I'll do the same. You blew it! Let me explain. If one door is shown to be a loser, that information changes the probability of either remaining choice, neither of which has any reason to be more likely, to 1/2. As a professional mathematician, I'm very concerned with the general public's lack of mathematical skills. Please help by confessing your error and in the future being more careful. -- Robert Sachs, Ph.D., George Mason University

DUH, we're not talking about quantum tunnelling.

You blew it, and you blew it big! Since you seem to have difficulty grasping the basic principle at work here, I'll explain. After the host reveals a goat, you now have a one-in-two chance of being correct. Whether you change your selection or not, the odds are the same. There is enough mathematical illiteracy in this country, and we don't need the world's highest IQ propagating more. Shame! -- Scott Smith, Ph.D., University of Florida

And you have a 0% chance of being correct with this reasoning. I didn't know so much of the mathematical illiteracy in this country resided in mathematics faculties.

May I suggest that you obtain and refer to a standard textbook on probability before you try to answer a question of this type again? -- Charles Reid, Ph.D., University of Florida

Preferably not the one that he uses to teach.

I am sure you will receive many letters on this topic from high school and college students. Perhaps you should keep a few addresses for help with future columns. -- W. Robert Smith, Ph.D., Georgia State University

And with future lectures at GSU.

You are utterly incorrect about the game show question, and I hope this controversy will call some public attention to the serious national crisis in mathematical education. If you can admit your error, you will have contributed constructively towards the solution of a deplorable situation. How many irate mathematicians are needed to get you to change your mind? -- E. Ray Bobo, Ph.D., Georgetown University

Apparently the necessary number of IRATE mathematicians is n where n>9. To get her to change her mind, she needed to hear from exactly 1 CORRECT mathematician.

Your answer to the question is in error. But if it is any consolation, many of my academic colleagues have also been stumped by this problem. -- Barry Pasternack, Ph.D., California Faculty Association

You're in error, but Albert Einstein earned a dearer place in the hearts of people after he admitted his errors. -- Frank Rose, Ph.D., University of Michigan

I have been a faithful reader of your column, and I have not, until now, had any reason to doubt you. However, in this matter (for which I do have expertise), your answer is clearly at odds with the truth. -- James Rauff, Ph.D., Millikin University

You made a mistake, but look at the positive side. If all those Ph.D.'s were wrong, the country would be in some very serious trouble. -- Everett Harman, Ph.D., U.S. Army Research Institute

Serious trouble indeed! What a sad state of affairs that not only can't these nine PhD's not get the correct answer on their own, but they can't even recognize the correct answer when it is put before them in excruciating detail. I am so sad for their students, more so than for the country!

[Thanks to WM for the link]

Another one for the DUH files

2008-02-25T23:24:00.002-05:00

The expected introduction of plug-in hybrid electric vehicles could cut gasoline use but could increase deadly air pollution in some areas, two reports say. That's because a plug-in's lower tailpipe emissions may be offset by smokestack emissions from the utility generating plants supplying electricity to recharge the big batteries that allow plug-ins to run up to 40 miles without kicking on their gasoline engines.

I don't understand why this would be a big surprise to folks. Anybody who stayed awake in high school physics grasps this idea. Using electricity instead of gasoline is not a free ride. That power is still produced somewhere. And the laws of thermodynamics tells us that will produce emissions, unless we use nuclear power and that brings a whole other price tag with it (not that the green freaks want us using nuclear power anyway).

2007 Nobel Prize in Physics

2007-10-09T15:18:00.001-04:00

Goes to Albert Fert and Peter Grünberg, for the discovery of Giant Magnetoresistance.

http://nobelprize.org/nobel_prizes/physics/laureates/2007/

This discovery made the iPod possible.

http://www.nytimes.com/2007/10/09/health/09iht-nobel.4.7820918.html?_r=1&emc=eta1

Schrodinger's Cat

2007-10-01T22:57:00.003-04:00

Queen's Brian May

2007-08-23T23:03:00.000-04:00

Just earned his PhD in astrophysics from Imperial College by completing his dissertation on interstellar dust, 35 years after he dropped out to concentrate on his music.

Diffusion with a source on a semi-infinite line

2007-04-21T10:55:00.000-04:00

(Budak/Samarskii/Tikhonov, problem 3.7)

u_t = a²u_xx + f(x,t)
u(x,0) = φ(x)
u(0,t) = μ(t)
0 ≤ x < ∞ and 0 ≤ t < ∞

The solution of the problem can be represented as a sum
u(x,t) = u₁(x,t) + u₂(x,t) + u₃(x,t)
where u₁(x,t) represents the effect of the initial condition u(x,o) = φ(x), u₂(x,t) represents the effect of the boundary condition u(0,t) = μ(t), and u₃(x,t) represents the effect of the nonhomogeneous term f(x,t).

Solution for u₁

u_t = a²u_xx
u₁(x,0) = φ(x)
u₁(0,t) = 0

u₁(x,t) = ½ (1/√π) ∫₀^∞ 1/√(a²t) [ e ^{-(x-x')²/4a²t} - e ^{-(x+x')²/4a²t} ] φ(x') dx'

Solution for u₂

u_t = a²u_xx
u₂(x,0) = 0
u₂(0,t) = μ(t)

u₂(x,t) = ½ (a²/√π) ∫₀^t x/[a²(t-t')]^3/2 e^{-x²/4a²(t-t')} μ(t') dt'

Solution for u₃u₃(x,t) = ½ (1/√π) ∫₀^∞ dx' ∫₀^t dt' { 1/√[a²(t-t')] [ e ^{-(x-x')²/4a²(t-t')} - e ^{-(x+y)²/4a²(t-t')} ] f(x',t') }

Financial Mathematics

2007-04-14T23:51:00.000-04:00

Since finishing my ASA late last year, I've started rediscovering my interest in financial mathematics. I will be posting assorted theorems, problems, observations, etc, from my readings in my finance and investments blog from time to time.

The E8 Lie Group Successfully Mapped

2007-03-19T18:47:00.000-04:00

Computer scientists and mathematicians from the United States and Europe said they had mapped the Lie group E8, a problem that was discovered in 1887.

[Thanks to LL for sending me the link.]

How Ticked Off Physicists Write Checks

2007-02-10T11:10:00.000-05:00

[Thanks to NE for sending me this.]

Disappointed

2006-11-30T21:57:00.000-05:00

I just finished auditing a graduate mathematics class at a university that shall remain nameless, but I did not enjoy the experience at all. I have never taken a math class where everything was so pre-digested for me. At the beginning of each section, the professor assigned problems from the book as exercises, standard operating procedure. A few lectures later, he would select a couple of the more challenging ones (translation: the only interesting ones) and provide "hints" so comprehensive that 90% of the proof had been worked out. Before the exam, he handed out a graded homework assignment that contained the same proof again (not a similar proof, the same proof). And then on exam day, the same proof would show up a third time. A shame.

Fraleigh Exercise 4.39

2006-10-09T00:18:00.000-04:00

Prove that a nonempty set G together with a binary operation * on G such that ax=b and ya=b have [unique] solutions for all a and b in G is a group.

(1) The binary operation is associative.

(2) Fix a in G. Then the equation ax=a has a unique solution, call it e. Then ae=a. Now pick an arbitrary b in G. Then the equation ya=b has a unique solution, call it c. Then ca=b. Multiplying ae=a by c on the left, cae=ca. Thus, be=b. Therefore, e is a right identity on G.

(3) Again pick an arbitrary b in G. The equation bx=e has a unique solution, call it b'. Therefore, every element b as a right inverse in G.

This suffices to show that G is a group.

2006 Nobel Prize in Physics

2006-10-03T00:20:00.000-04:00

For their discovery of the blackbody form and anisotropy of the cosmic microwave background radiation

John C. Mather
Goddard Space Flight Center

George F. Smoot
UC-Berkeley

http://nobelprize.org/nobel_prizes/physics/laureates/2006/

What They Don’t Teach You in Graduate School

2006-09-12T10:21:00.001-04:00

http://insidehighered.com/layout/set/print/workplace/2005/11/30/tips

The Bogdanov Affair

2006-09-12T08:00:00.000-04:00

To think that all these years I have tried to master the field of theoretical physics and possibly make a contribution. It turns out that I didn't need to learn anything and could have just made up my contribution. Has the field of physics been reduced to the level of "social criticism" where any idea no matter how nonsensical can be published? How can this happen?

What, you may ask, am I talking about. I'm speaking about the Bogdanov affair. First, let's start with just the facts (from Wikipedia)...

The Bogdanov Affair is an academic dispute regarding a series of theoretical physics papers written by French twin brothers Igor and Grichka Bogdanov (or Bogdanoff). These papers were published in reputable scientific journals, and culminated in a proposed theory for describing what occurred at the Big Bang. While the Bogdanovs defend the veracity of their work, many physicists have alleged that the papers are nonsense, emphasizing the fallibility of the peer review system used to approve papers. The Bogdanovs obtained Ph.D. degrees from the University of Bourgogne; Grichka Bogdanov received his degree in mathematics in 1999, and Igor Bogdanov received his in theoretical physics in 2002. Both were given the low but passing grade of "honorable", in Igor's case only after publishing in respected physics journals to establish the acceptability of his work.

Before moving on to other topics, let's consider briefly the appalling intellectual laziness of allowing journals to establish the "acceptability" of the work. If you don't understand the paper, you have no business recommending the awarding of a doctorate.

Inside Higher Education said it best: "A PhD is a license to reproduce and an obligation to maintain the quality of your intellectual descendants. [...] If you vote to pass someone who is marginal or worse they, in turn, have the same privilege. If they are not up to standard, it is likely that some of their descendants will also not be."

Leaving this shameful aspect of the affair, let's now look at some actual text from the papers in question, courtesy of John Baez's blog...

We consider inertia as a topological field, linked to the topological charge Q = 1 of the "singular zero size gravitational instanton" which can be identified with the initial singularity of space-time in the standard model.

Huh?

Baez: "The paper goes on to discuss the supposed connection between N = 2 supergravity, Donaldson theory, KMS states and the Foucault pendulum experiment, which he claims 'cannot be explained satisfactorily in either classical or relativistic mechanics.' If you know some physics you'll find this statement odd. The Foucault pendulum behaves exactly the way classical mechanics predicts: it is a standard textbook exercise."

We draw from the above that whatever the orientation, the plane of oscillation of Foucault's pendulum is necessarily aligned with the initial singularity marking the origin of physical space S3, that of Euclidean space E4, and, finally, that of Lorentzian space-time M4.

Baez: "How in the world could the plane of oscillation of a Foucault's pendulum be 'aligned with the initial singularity', i.e. the big bang?"

We simply suggest that at 0 scale, the observables must be replaced by the homology cycles in the moduli space of gravitational instantons. We then get a deep correspondence - a symmetry of duality - between physical theory and topological field theory.

Baez: "(1) Could you please define "at 0 scale"? (2) "Observables" in which theory? (3) You say "must be replaced". Why? How? (4) You speak of a "deep correspondence" between some unspecified physical theory and some unspecified topological field theory. Which theories are you talking about here? How does the correspondence go?"

Lee Smolin: "They were at best wrong, and most likely just throwing around words with no calculations or proofs to back them up."

Jacques Distler: "The papers consist of buzzwords from various fields of mathematical physics, string theory and quantum gravity, strung together into syntactically correct, but semantically meaningless, prose."

Eli Hawkins: "This paper is built around the idea that 'at the Planck scale, the space-time system is in a themodynamical equilibrium state.' It is not quite clear what the author means by this. He may mean that when the matter is at the Planck temperature, it is in thermodymanic equilibrium with the geometry. He does not explain why there should not be thermal equilibrium at all temperatures. It may be simply that the author does not know what he is talking about. The main result of this paper is that this thermodynamic equilibrium should be a KMS state. This almost goes without saying; for a quantum system, the KMS condition is just the concrete definition of thermodynamic equilibrium. The hard part is identifying the quantum system to which the condition should be applied. It is difficult to describe what is wrong in Section 4, since almost nothing is right. The author seems to believe that just because an analytic continuation of a function exists, the argument must be considered a complex number. He also makes the rather obvious claims that complex numbers should be the sums of real and imaginary parts. The remainder of the paper is a jumble of misquoted results from math and physics. It would take up too much space to enumerate all the mistakes: indeed it is difficult to say where one error ends and the next begins."

So the papers are clearly recognizable as nonsense to anybody with a basic understanding of physics. Now we get to the more depressing elements of the story. Not only did these con artists get their theses by ignorant committee members, but they got them published in some of the most prestigious physics journals (Classical and Quantum Gravity, Nuovo Cimento, Annals of Physics). Here are some of the comments regarding the quality of the papers by what I would have thought were strong physicists...

Roman Jackiw (MIT) (According to most sources, the thesis had "many" things Dr. Jackiw didn't understand, but I have been unable to find a direct quote to this effect.): "The author proposes a novel, speculative solution to the problem of the pre-Big-Bang initial singularity ... the thesis and the published papers provide an excellent introduction to these ideas, and can serve as a useful springboard for further research in this area. All these were ideas that could possibly make sense. It showed some originality and some familiarity with the jargon. That's all I ask."

That's all you ask?!?! WTF?!?! I have some familiarity with the jargon. I'm sure I can write something original (especially if these Bogdanov papers are setting the bar for "originality"). Where the hell is my Ph.D.?!?!

Jack Morava (Johns Hopkins University): "The thesis work is of great interest, dominated by new ideas with fundamental physical implications in cosmology and in many other fields connected with gravitation."

Lubos Motl (Harvard): "They are proposing something that has, speculatively, the potential to be an alternative story about quantum gravity. What they are proposing is a potential new calculational framework for gravity."

So, there you have it. Physicists at some of the best universities in the world have been taken in by this meaningless pile of crap. Amazing! One almost hopes that these folks were bribed or have some other ulterior motive in stating that these papers constitute good physics. Because the alternative explanation - that they are fools - is too terrible to contemplate. However, deep down I have become convinced that this is in fact the case. I have to accept that the field that I have loved for over 20 years is turning into a pathetic joke. I will finish this post with two quotes that most clearly explain the situation...

Jackiw: "One person looks at a piece of art and says it is gibberish; another person looks and says it's wonderful."

When did physics become comparable to modern art in deciding what was of value? I seem to recall learning about something called "the scientific method" in high school (oh, so many years ago). I will give the last word to Frank Wilczek (MIT) who provides the best explanation of why we find ourselves in this sorry state.

"This says something profound about what happens to theoretical physics in the absence of the discipline of experiment."

Dark Matter, Yes?

2006-09-04T22:30:00.000-04:00

Experimental evidence for the existence of dark matter explained by Sean Carroll in his blog.

Here is a picture of the distribution of the matter based on gravitational lensing. (Of course, labeling it Dark Matter assumes the very thing we are trying to prove, so that is a little disappointing.)

Here is a picture of the distribution of matter based on X-ray observations of the hot gas.

The previous two images superimposed:

The gravitational field, as reconstructed from lensing observations, is not pointing toward the ordinary matter. That’s exactly what you’d expect if you believed in dark matter.

There is no denying the conclusion from the third picture that there is definitely "dark matter" there in the sense that the hot gas does not constitute the bulk of the gravitational mass. However, the conclusion that there is Dark Matter (meaning nonbaryonic cold dark matter, in the sense that the term is usually used in cosmology) depends on the assumption that the hot gas is the bulk of the regular matter and there cannot be nonvisible regular matter causing this effect.

Whatever matter is in the blue areas in the pictures above cannot interact heavily with regular matter (else, it would have been dragged along with the pink areas). This tends to support the hypothesis that it is Dark Matter. However, it is my opinion that our imperfect understanding of the universe simply does not allow us at this time the luxury of treating it as the only possible hypothesis to the exclusion of all others. True Dark Matter (Higgs, Superpartners, WIMPs, etc)? Baryons? Neutrinos? Who knows!

One response to Sean Carroll's post seems worth reproducing here: The history of the Vulcan hypothesis shows the way: the strange movement of Mercury was not caused by something which was huge and strangely impossible to see; it was caused by a small relativistic effect which was hidden in normal situations by the very fact of its weakness. When people start saying that an effect is caused by something three times the mass of the visible universe but which has escaped notice for the whole of history, is it really unreasonable to doubt them? I don’t think so. Dark matter is a fudge; a placeholder until we come up with something sensible.

No Dark Matter?

2006-09-04T12:09:00.001-04:00

First they invented dark matter. However, so far there is no evidence that such a thing exists, although the Higgs particle does turn out to have precisely the right properties to be the missing dark matter. Then they invented dark energy. And as far as I know, there's not even a candidate for what this mysterious substance might actually be.

This is all very radical, although I suppose that Planck's quantum hypothesis and the Bohr hypothesis about electronic orbits were just a radical in their day. But now there's another radical theory, although perhaps somewhat less radical than dark matter and dark energy.

According to Discover, Mordehai Milgrom proposed an alternative to dark matter and dark energy named MOND (MOdified Newtonian Dynamics), which consists of the idea that when the accelerations become very small the well-known formula F=ma changes to F=ma^2/a_0. Of course, there is no more theoretical justification for this formula than for the introduction of dark matter. Turns out Milgrom wrote a Scientific American article back in 2002 titled Does Dark Matter Really Exist?

The story is back in the pages of the popular science media is because in 2004 Jacob Bekenstein published a relativistic extension of MOND called TeVeS (Tensor, Vector, Scalar) which can correctly explain gravitational lensing and should be applicable to the problem of galaxy formation.

Also, here's a link to a webpage with many resources on Modified Newtonian Dynamics: http://www.astro.umd.edu/~ssm/mond/litsub.html.

This is what a critique of Woit should look like

2006-09-04T11:51:00.000-04:00

http://zippy.ph.utexas.edu/~abergman/Review.pdf

I will focus on one passage for a minute...

There is a seemingly robust description of zillions and zillions of vacua, each of which seems to describe consistent physics completely unlike our world. Faced with this proliferation, one immediately begins to worry about the predictivity of the theory. With so many vacua it might be possible to explain any experimental result one can imagine. If this were true, it would not mean that string theory was wrong, but it would mean that it would be completely useless as it could never make a prediction. To avoid this unhappy state of affairs, a fair number of senior people have decided to take a radically new approach to predictivity. The idea is that one should determine all the possible vacua that are consistent with the existence of intelligent life and imagine that it is equally likely that we could be in any of them. Then, while we could not precisely predict anything about our world, we could assign probabilities to the results of future experiments.

Philosophically, the bolded phrase is indicative of the major problem that critics have with supersymmetry and string theory.

Supersymmetry solves the Higgs problem by introducing a supersymmetric partner to each known particle - never mind that nobody has ever observed a supersymmetric particle
Superstring theory requires more than four dimensions - no problem, compactify the superfluous ones
M Theory has no predictability - no problem, redefine predictability

I can see where this last one is the straw that broke the camel's back for some scientists. Nevertheless, Bergman makes a good even-handed case for the merits of superstring theory while making a detached and solid review of Woit's book in the process. A must read.

Another article on Smolin and Woit

2006-09-04T00:52:00.000-04:00

Scientific American joined the fray...

[Critics claim] string theory is not just untested but untestable, incapable of ever making predictions that can be experimentally checked. [...] There are, in fact, some 10^500 perfectly good M theories, each describing a different physics. The theory of everything, as Smolin puts it, has become a theory of anything.

I have no idea which side of this debate is correct. However, I would prefer to see reasonable debate on the questions as befits proper science. Instead I see the following (to be fair, these comments are all from Motl, but I do not see anything to the contrary from his string theory colleagues) ...

Sean Carroll has essentially joined Peter Woit and others in their irrational attack against existing high-energy theoretical physics (Carroll actually wrote that he did not agree with Woit, but apparently in Motl's world even allowing that Woit's book has a right to exist is an attack not on him personally but on the entire enterprise of theoretical physics)
semi-official unholy alliance between Cosmic Variance and the anti-science activists (another reference to Carroll)
They're orders of magnitude and decades of education from being able to do something like [presenting an alternative to superstrings]. The only thing that Woit and others offer and share is their hatred against science and a very poor knowledge of the actual problems that are being discussed.
[Greene's victory in the debate was] because of Brian Greene's virtues, based not only on his superior communication skills and psychological balance but also on his active knowledge of the topics and clean conscience of a real contributor to science. (So anyone with whom Motl disagrees is not only not "a real contributor to science" but is psychologically unbalanced? It is also worth noting that Motl's description of the debate bears VERY little resemblance to the audio.)
a scientific microbe (referring to Woit)
the attack is led by people with poor quantitative skills who dislike a careful and detailed technical analysis of the scientific issues [...] This description is primarily about Peter Woit
Another postmodern diatribe against modern physics and scientific method (about Smolin's book) and bitter emotions and obsolete understanding of high-energy physics (about Woit)

I find this unseemly. It's as if Bohr and Einstein had tried to settle their differences on the meaning of quantum physics by brawling in a bar.

Exchange with Professor Motl

2006-09-03T22:39:00.000-04:00

I posted the following response to Motl's comments about the Time magazine article on superstring theory.

The article hardly seems to warrant your "physics is a sin" response. Other than engaging in ad hominem attacks, you haven't really explained why you disagree with the article. That hardly seems like science to me. What's up?

He gave me the following reply, which I thought I'd reproduce here.

Dear Alberto,

I won't be explaining why I disagree with the article because this is not a matter of agreement or disagreement. The article is a pile of vitriolic nonsense and conspiratory theories written by rather incompetent people and, in my opinion, there is nothing in the article that merits a scientific discussion. I apologize but if you need someone who will be commenting every sentence of every vitriolic author with high-school physics education, you will be disappointed.

It is up to you whether you believe that I/we know what we're saying or not. Be aware that if you choose to take the Time magazine more seriously than my fast comment, I won't think that you are an intelligent person.

Best
Lubos

The Unraveling of String Theory?

2006-09-03T19:30:00.001-04:00

Ever since I was an undergraduate, superstring theory has made me uneasy philosophically. Green,Schwartz,Witten came out in 1987 (I was a junior), and I remember discussing with classmates and professors my concerns that "physics was still an experimental science." The results of the theory are probably mathematically valid (although the lack of rigor with which some results are obtained would probably bring tears to the eyes of most mathematicians). However, I question the validity (as physics). The field has always seemed to me to be an exercise in mathematics, pursued with little or no concern regarding its possible relation to experimental proof. As the field currently stands, it isn't even experimentalists or the experimental equipment that are the limiting factor. It is the theory itself that cannot even be formulated in a way that allows it to be applied to the universe around us.

With the recent publication of Not Even Wrong by Peter Woit and The Trouble with Physics by Lee Smolin, this point of view is starting to be debated among top physicists (sometimes rudely so as in the blog wars between Woit and Harvard's Lubos Motl) and this month even made the pages of Time magazine.

Now, it seems [to critics], at least some superstring advocates are ready to abandon the essential definition of science itself on the basis that string theory is too important to be hampered by old-fashioned notions of experimental proof.

The story does seem a little one-sided in that the opinions of physicists who support string theory are never solicited (it's sad that the media screw up basic things like this even when political bias is not a consideration). However Motl's hysterical response describing the article as a "dirty, immoral, dishonest, and anti-scientific enterprise" seems extreme. What ever happened to logical scientific debate? I find his review of Smolin's book on Amazon particularly incoherent, as he calls it "a post-modern diatribe" and illogically claims that Smolin posits that "it is wrong for mathematics to play a crucial role in theoretical physics." He isn't helping his side in the debate with his idiocy.

[I attempted to add a trackback to this post from Motl's blog, but because I just transferred my own blog to the beta version of Blogger I am unable to do so at this time.]

The Sokal Affair

2006-08-31T19:28:00.000-04:00

I won't bore the reader by describing the well-known hoax, but I did want to provide a link to a website that has collected a lot of documentation related to it.

http://www.physics.nyu.edu/faculty/sokal/

Set Theory Proof

2006-08-29T13:29:00.001-04:00

Here's a proof that I have always found fascinating.

Theorem: Given any set A, the power set of A has strictly larger cardinality than A.

For two sets to have the same cardinality, there must exist a bijection between the two sets. So, let us assume that there exists such a bijection between A and P(A), denoted f:A->P(A).

For each x in A, either x is in f(x) or x is not in f(x). Consider the subset S of A defined as
S = {x in A | x is not in f(x)}.

Since we have assumed that f:A->P(A) is a bijection and S is in P(A), then there exists some t in A such that f(t) = S.

Either t is in S or t is not in S. Suppose t is in S. Then t satisfies the condition that t is not in f(t) which is S. Therefore t is not in S. Suppose t is not in S. Then t does not satisfy the condition, and t is in f(t)=S. Therefore t is in S.

So in either case, we arrive at the contradiction that t is in S and simultaneously t is not in S. The source of the contradiction is the assumption that there exists a bijection f:A->P(A). Therefore, there exists no such bijection and the cardinality of P(A) is not the same as the cardinality of A. Since the cardinality of P(A) cannot be smaller than the cardinality of A, the conclusion is that P(A) has strictly larger cardinality than A. (Q.E.D.)

Finite Simple Group (of Order Two)

2006-08-26T11:39:00.000-04:00

The lyrics of this song can be found here. More information on the group (from Northwestern University) can be found at http://www.kleinfour.com/.

[Thanks to TMB for sending me this video.]

Fields Medals 2006

2006-08-25T10:34:00.000-04:00

Awarded once every 4 years by the International Mathematical Union, the Fields Medal recognizes outstanding contributions to mathematics.

This year's winners...

Andrei Okounkov “for his contributions briding probability; representation theory and algebraic geometry.”

Grigori Perelman “for his contributions to geometry and his revolutionary insights into the analytical and geometric structure of the Ricci flow.” (Perelman's work may prove the more general Thurston Geometrization Conjecture from which the Poincare Conjecture would then follow. Perelman is quite the recluse and actually turned down the Fields Medal.)

Terence Tao “for his contributions to partial differential equations, combinatorics, harmonic analysis and additive number theory.” (At 31, Tao is one of the youngest mathematicians to be awarded the Fields Medal.)

Wendelin Werner “for his contributions to the development of stochastic Loewner evolution, the geometry of two-dimensional Brownian motion, and conformal field theory.”

Busy year for Pluto

2006-08-24T23:34:00.000-04:00

The International Astronomical Union defined the term "planet" for the first time. This definition excluded Pluto from planethood, so Pluto was reclassified under the new category of "dwarf planet" along with and Ceres and Eris.

At Berkeley

2006-08-15T23:41:00.000-04:00

On a bulletin board in the mathematics department at UC Berkeley, somebody had left a flier offering a $300 reward for a solution to the equation x^5 + ax^4 + bx^3 + cx^2 + dx = e in terms of a, b, c, d and e. If you studied college level math, you know this problem is unsolvable. The insolubility of the quintic is proved as part of Galois theory in the second semester of undergraduate abstract algebra. I am not sure if this was a hoax or exactly what the purpose of this flier might have been, but this being Berkeley several people had scribbled sarcastic offers to square the circle (another problem that has been proven to be unsolvable) or to prove that pi is a natural number.

Nix and Hydra

2006-06-21T23:25:00.000-04:00

Two additional moons of Pluto (in addition to Charon) were imaged by astronomers working with the Hubble Space Telescope on 5/15/2005. The International Astronomical Union officially named the moons Nix and Hydra. These moons orbit Pluto at approximately two and three times the distance of Charon: Nix at 48,700 kilometres and Hydra at 64,800 kilometres from the barycenter of the system.

Geeky Math Humor

2006-05-21T22:09:00.000-04:00

Q: What's the value of a contour integral around Western Europe?
A: Zero, because all the Poles are in Eastern Europe.

Itzykson & Zuber is back in print

2006-05-14T21:15:00.000-04:00

Dover rocks!

http://www.amazon.com/gp/product/0486445682/

Ridiculous

2006-02-22T22:03:00.000-05:00

On today's Law & Order Criminal Intent, I learned that elementary age kids without any training in advanced mathematics or supervision by a knowledgeable mathematician nevertheless have a reasonable chance of solving the Riemann Hypothesis just because they are geniuses.

A couple of weeks ago, I learned that in theoretical physics "theories" are formulated without any appeal to mathematics, and then mathematics is used to prove those theories.

Ridiculous

2006-01-23T18:33:00.000-05:00

Having a bad day? No wonder. According to one scientist, today, January 23, is the gloomiest day of 2006. Cliff Arnall, a health psychologist at the University of Cardiff in Wales, has devised a formula that combines personal and seasonal factors to calculate the year's emotional low point.

Caught this story in today's paper. The same story made the rounds last year. It's utterly ridiculous. According to this "scientist," the variables are weather (W), debt (D), monthly salary (S), time since Christmas (T), time since failure to quit a bad habit (Q), low motivational levels (M) and the need to take action (N). Sounds reasonable enough. But now let's look at his "formula."

Gloominess = [W + (D-S)xTxQ] / [M x N]

So we take time (since Christmas) times time (since failure) times money. So we have a quantity that is measured in $ x days squared. Then we add this to a weather variable, which is measured in . . . ?

Then we take this already meaningless concoction and further divide it by "low motivational levels" and "the need to take action." I'm unclear on how either of these can be measured well enough to allow any mathematical comparisons to be made between January 23 and January 20 or January 26 ... or July or November or any other random date I pull out of a hat. Furthermore, why do we divide by them; why not multiply?

I am appalled that some idiot can get this "research" published. What a freakin' joke.

Ridiculous

2005-10-31T23:23:00.000-05:00

I'm watching an episode of Law & Order: Criminal Intent, and in just a five minute interval, I've learned the following:

"After three tries, most password programs give you a hint." Funny, most password programs I've ever seen after three tries LOCK YOU OUT!
Senior math majors take multivariable calculus. (I took it first semester of sophomore year, and I was far from the most advanced student in my cohort.)
Seniors who have not completed multivariable calculus are given research grants by the Department of Defense.
Random number generator programs increase your odds of winning the lottery.

2005 Nobel Prize in Physics

2005-10-04T22:13:00.001-04:00

Roy J. Glauber
"For his contribution to the quantum theory of optical coherence."

John L. Hall
Theodor W. Hänsch
"For their contributions to the development of laser-based precision spectroscopy, including the optical frequency comb technique."

http://nobelprize.org/physics/laureates/2005/

MIT Open CourseWare

2005-09-24T11:41:00.000-04:00

MIT has a website, Open Course Ware, where professors present a lot of the teaching materials from their various courses. When the site first started, the math and physics offerings were pretty thing. However, over the last year, many of the courses have been substantially enhanced, to the point where they are now quite useful resources, including extensive lecture notes and many solutions to homework assignments. I am also providing the link to the MIT math and physics departments, since those sites still have the home pages of professors with resources for courses being offered in the current academic years.

Physics OCW

Physics Department Website

Mathematics OCW

Mathematics Department Website

Dr. Serge Lang, R.I.P.

2005-09-12T11:33:00.001-04:00

Serge Lang died today. Rather than create a lengthy post that would still not do this great mathematician justice, I'll just link to his wikipedia entry.

Methods of Theoretical Physics

2005-09-01T11:45:00.000-04:00

The famous mathematical physics textbook by Morse & Feshbach has finally been republished. I've long wanted to get my hands on a copy, but not at this price ($329 for the set, up from the original price of $250).

Not sure why the price is going up or what is he thinking, but I don't think his company will never be successful at this price. It's a shame.

Thirteen things that do not make sense

2005-03-21T13:51:00.000-05:00

New Scientist has an article on thirteen things that do not make sense in science. Historically, such paradoxes have provided the impetus for new discoveries. The most notable examples are the Michelson-Morley null experiment (leading to special relativity), the unexpected blackbody radiation spectrum (leading to the quantum hypothesis), and the anomalous precession of the perihelion of Mercury (leading to general relativity).

1. The placebo effect
2. The cosmological horizon problem
3. Ultra-energetic cosmic rays
4. Belfast homeopathy results
5. Dark matter
6. Viking's discovery methane on Mars
7. Tetraneutrons
8. The Pioneer anomaly
9. Dark energy
10. The Kuiper cliff
11. The "Wow" signal
12. Not-so-constant constants
13. Cold fusion

[Thanks to DVD for the link]

Superstrings <> Cosmic Strings

2005-03-03T22:45:00.000-05:00

I have seen this article twice today, so I thought I'd make a quick post about it.

Could two lookalike galaxies, barely a whisker apart in the night sky, herald a revolution in our understanding of fundamental physics? Some physicists believe that the two galaxies are the same - its image has been split into two, they maintain, by a "cosmic string." If this interpretation is correct, then CSL-1 - the name of the curious double galaxy - is the first concrete evidence for "superstring theory": the best candidate for a "theory of everything", which attempts to encapsulate all the phenomena of nature in one neat set of equations.

Not sure why the author of this article switched from "cosmic strings" to "superstrings" without a word of explanation, but these two concepts are completely unrelated. Superstrings are the extension of Quantum Field Theory of point particles to one-dimensional "strings," a theory which leads to highly speculative predictions about a 10 dimensional universe and may succeed in unifying quantum field theory with classical General Relativity. Cosmic strings are astronomical objects described by the rules of classical general relativity. The two things have absolutely nothing to do with each other.

http://www.astronomycafe.net/qadir/ask/a11828.html

Astronomers find star-less (dark matter?) galaxy

2005-02-23T17:08:00.000-05:00

Astronomers say they have discovered an object that appears to be an invisible galaxy made almost entirely of dark matter. The team, led by Cardiff University, claimed it is the first to be detected. It was found 50 million light years away using radio telescopes in Cheshire and Puerto Rico.

[Thanks to GB for the link.]

No Evidence of Life on Mars

2005-02-18T17:02:00.000-05:00

News reports on February 16, 2005, that NASA scientists from Ames Research Center have found strong evidence that life may exist on Mars are incorrect. NASA does not have any observational data from any current Mars missions that supports this claim. The work by the scientists mentioned in the reports cannot be used to directly infer anything about life on Mars, but may help formulate the strategy for how to search for martian life. Their research concerns extreme environments on Earth as analogs of possible environments on Mars. No research paper has been submitted by them to any scientific journal asserting martian life.

Life on Mars? Never mind!

2005-02-17T15:47:00.000-05:00

A pair of NASA scientists told a group of space officials at a private meeting Sunday that they have found strong evidence that life may exist today on Mars, hidden away in caves and sustained by pockets of water. The scientists, Carol Stoker and Larry Lemke of NASA's Ames Research Center in Silicon Valley, told the group that they have submitted their findings to the journal Nature for publication in May, and their paper currently is being peer reviewed. What Stoker and Lemke have found, according to several attendees of the private meeting, is not direct proof of life on Mars, but methane signatures and other signs of possible biological activity remarkably similar to those recently discovered in caves here on Earth.

Dr. Herbert Goldstein, R.I.P.

2005-01-12T15:49:00.001-05:00

Herbert Goldstein, Professor Emeritus of Nuclear Science and Engineering at Columbia, died today. He was 82. Long recognized for his scholarship in classical mechanics and reactor shielding, he was the author of the classic (pun intended) graduate textbook, Classical Mechanics. The book has been a standard text since it first appeared 55 years ago and has been translated into nine languages.

Goldstein's contributions to nuclear energy were honored by the U.S. Department of Energy, which awarded him the E.O. Lawrence Memorial Award in 1962. In 1977, he was the recipient of the Distinguished Service Award from the shielding division of the American Nuclear Society. He was a professor of nuclear science and engineering at the Fu Foundation School of Engineering and Applied Science since 1961. He received the Great Teacher Award, given by the Society of Columbia Graduates, in 1976. In 1984, he was the first to hold the Thomas Alva Edison Professorship at the University. In addition to research, he devoted time to promoting scientific literacy by teaching undergraduate courses. In 1977, he taught a course he designed to increase scientific understanding of energy issues "Nuclear Energy: A Semi-technical View for the Non-scientist." He was also one of the faculty members instrumental in developing an innovative science course for non-scientists, "The Theory and Practice of Science," at the College. He was a consultant for Oak Ridge National Laboratory and Brookhaven National Laboratory.

Goldstein was a fellow of the American Physical Society, the American Nuclear Society, the New York Academy of Sciences and the American Association for the Advancement of Science. He also was a member of the American Association of Physics Teachers and was a founding member and president of the Association of Orthodox Jewish Scientists. He received a B.S. from City College of New York in 1940 and a Ph.D. from Massachusetts Institute of Technology in 1943.

Goldstein is survived by his wife, Channa; his children, Penina, Aaron Meir and Shoshanna; and 10 grandchildren.

2005

2005-01-01T11:48:00.002-05:00

is the 100th anniversary of Einstein's miracle year (Annus Mirabilis)!

Putnam Competition

2004-12-04T08:00:00.001-05:00

The William Lowell Putnam Mathematics Competition is today.

The Putnam Competition is a North American math contest for college students offered each year on the first Saturday in December. Over 3000 students will spend 6 hours (in two sittings) trying to solve 12 problems (at 10 points a problem, the maximum score is 120). Individual and team winners get some money and a few minutes of fame. High scorers almost invariably go on to earn their Ph.D. and generally end up as top-tier mathematicians in their particular research areas.

I took this exam once in my undergraduate career and scored either 11 or 13 (I don't remember). A most humbling experience, but also not a bad result, relatively speaking. The median score is usually 2 (yes, 2 out of 120, that's not a typo).

For past exams and their solutions see this website.

Also, the wikipedia article has a list of past winners.

Progress on Two Unsolved Problems

2004-11-19T16:06:00.000-05:00

Some news on two of the unsolved problems previously mentioned in this blog. See Seven Mathematical Tidbits by Eric W. Weisstein and Ed Pegg Jr. for more details.

Conjecture: There are no odd perfect numbers.

It is still not known if any odd perfect numbers exist, although it is known that any such number would have to be greater than 10^300. Kevin Hare has been investigating the existence of odd perfect numbers and has proven that any odd perfect number must have at least 47 prime factors (including repetitions), and that improving this result depends upon finding factors of three large numbers. In particular, a higher bound can be computed if a certain composite 301-digit number can be factored.

References:
Odd Perfect Numbers by K. G. Hare
Some Factorizations That I Want by K. G. Hare

Riemann Hypothesis: All of the non-trivial zeros of the Zeta Function are located on the line Re(s) = ½.

On October 13, Xavier Gourdon and Patrick Demichel announced that they had used an efficient technique due to Odlyzko and Schönhage to find the first ten trillion nontrivial zeroes of the Riemann zeta function, more than ever before computed. Every single one of these zeros lies along the critical line, which is a necessary requirement for the Riemann hypothesis to hold but not sufficient.

References:
Computation of Zeros of the Zeta Function by Xavier Gourdon and Patrick Demichel
The 10^13 First Zeros of the Zeta Function, and Zeros Computation at Very Large Height by Xavier Gourdon

Second black hole found at the centre of our Galaxy

2004-11-10T13:47:00.000-05:00

Just three years ago, astronomers confirmed that the Milky Way revolves around a supermassive black hole, called Sagittarius A*, which is about 2.6 million times more massive than the Sun. But now a much smaller black hole, just 1,300 times our Sun's mass, has been found orbiting about three light years away from its supermassive cousin.

"This is the first intermediate-mass black hole found in our Galaxy." - Jean-Pierre Maillard, an astronomer from the Institute of Astrophysics in Paris, France.

[Thanks to DVD for providing this link.]

An interesting discussion of black holes with special emphasis on the famous Cygnus X-1 black hole candidate can be found here.

Differential Equations & Linear Algebra

2004-11-02T21:32:00.000-05:00

With all the questionable innovations which are accepted in mathematics education whole-heartedly, there is one innovation in undergraduate mathematics which seems natural and extremely useful but has never caught on ... the integration of differential equations and linear algebra for college freshmen and sophomores after the completion of the introductory calculus sequence.

While there are many excellent textbooks for both of these subjects, there are obvious synergies between the two topics which can only be exploited by combining their teaching into one course. In fact, many professors believe (and I agree) that these two subjects must be integrated if they are to be properly understood since their intricate interaction is where all of the action is.

Some of the excellent texts which take this integrated approach are:
Introduction to Linear Algebra and Differential Equations by John Dettman
Differential Equations and Linear Algebra by Edwards & Penney (includes a lot of Matlab stuff)
Differential Equations and Linear Algebra by Jerry Farlow et al
Linear Algebra and Differential Equations by Peterson & Sochacki
Linear algebra & differential equations: An integrated approach by Charles Cullen

General Relativity's Frame Dragging Confirmed

2004-10-21T11:03:00.000-04:00

Earth's spin warps space around the planet, according to a new study that confirms a key prediction of Einstein's general theory of relativity. After 11 years of watching the movements of two Earth-orbiting satellites, researchers found each is dragged by about 6 feet (2 meters) every year because the very fabric of space is twisted by our whirling world. The results, announced today, are much more precise than preliminary findings published by the same group in the late 1990s.

[Thanks to DVD for bringing this link to my attention.]

The Ph.D. gap

2004-10-16T13:36:00.000-04:00

http://discusseducation.blogspot.com/2004/10/more-on-phd-gap.html

On the black hole information paradox

2004-10-13T13:16:00.000-04:00

After nearly 30 years of arguing that a black hole destroys everything that falls into it, Stephen Hawking is saying he was wrong. It seems that black holes may after all allow information within them to escape. Hawking will present his latest finding at a conference in Ireland next week.

Other articles:
http://www.newscientist.com/news/news.jsp?id=ns99996193
http://www.spacedaily.com/news/blackhole-04c.html

I didn't see this story back when it was published. The conference was on July 21st, and I'm just now learning about it through the American Physical Society's coverage. It's not yet clear that this new explanation is correct. John Preskill has not yet accepted the encyclopedia in part because he says "I didn't understand the talk." Kip Thorne (Hawking's betting partner) is apparently also unconvinced.

A very accessible (but still technical) discussion of the information paradox can be found on Baez's website.

Field Quantization

2004-10-07T16:33:00.000-04:00

I'm currently refreshing my understanding of quantum field theory by reading Field Quantization by Greiner & Reinhardt. I hope to follow this up by tackling those parts of Relativistic Quantum Fields (Bjorken & Drell) and Quantum Field Theory (Itzykson & Zuber) that I did not adequately master during my graduate training. Field Quantization covers things in much the same level of detail as Greiner's Relativistic Quantum Mechanics and Quantum Electrodynamics, although some steps are skipped in derivations, as befits a book meant for advanced graduate students. Still, the book covers the mathematical details of relativistic quantum field theory in a simple way, contains more details than most other standard treatments and includes many instructive examples that most other authors just gloss over or exclude altogether.

One thing worth noting about this book is that although it has far too many typos, it has nowhere near the ridiculous number contained in Greiner's RQM or QED books. The publisher should be completely ashamed of himself for marring such an excellent pedagogical exposition in those other books with typographic blunders so numerous that the inexpert reader is thoroughly distracted from the clarity of the exposition in those two books. Field Quantization at least keeps the number of errors down low enough where the text is still readable and maybe even usable in a classroom environment.

So what is the Poincaré Conjecture?

2004-10-05T13:04:00.000-04:00

Wolfram's site explains it quite well.

In its original form, the Poincaré conjecture stated that every simply connected closed three-manifold is homeomorphic to the three-sphere. More colloquially, the conjecture says that the three-sphere is the only type of bounded three-dimensional space possible that contains no holes. This conjecture was first proposed in 1904 by Poincaré.

Soon thereafter this specialized conjecture for the case n=3 was generalized to the conjecture that every compact n-manifold is homotopy-equivalent to the n-sphere if and only if it is homeomorphic to the n-sphere, which is logically equivalent to the earlier weaker conjecture in the case n=3. The n=1 case is trivial, the n=2 case was known to 19th century mathematicians, n=4 was proved by Freedman in 1982 for which he was awarded the 1986 Fields medal, n=5 was proved by Zeeman in 1961, n=6 was proved by Stallings in 1962, and n>6 was proved by Smale in 1961 (although Smale subsequently extended his proof to include all n>4).

The n=3 case has baffled mathematicians since it was first proposed, spawning a multitude of incorrect proofs by many mathematicians, including Poincare and Whitehead. However, it now appears that this remaining case has finally been proved by Perelman although the proof has not yet been fully verified. In fact, Perelman's work may prove the more general Thurston Geometrization Conjecture from which the Poincare Conjecture would then follow.

Perelman's two papers are The entropy formula for the Ricci flow and its geometric applications and Ricci flow with surgery on three-manifolds. For the non Fields Medal candidates among us, a more accessible (but still highly technical) discussion can be found on the AMS site.

2004 Nobel Prize in Physics

2004-10-05T11:25:00.000-04:00

The winners of the 2004 Nobel Prize in Phyiscs have just been announced.

"for the discovery of asymptotic freedom in the theory of the strong interaction"

David J. Gross - Kavli Institute for Theoretical Physics, University of California at Santa Barbara
H. David Politzer - California Institute of Technology
Frank Wilczek - Massachusetts Institute of Technology

Most Interesting Open Math Questions

2004-10-02T17:40:00.000-04:00

1. Twin Prime Conjecture: There are an infinite number of twin primes. (This problem was known to the ancient Greeks; it’s related to Hilbert’s Eighth Problem.)

2. Goldbach Conjecture: Every even integer larger than 2 is the sum of two primes. (This problem was known to the ancient Greeks; it’s related to Hilbert’s Eighth Problem.)

3. Riemann Hypothesis/Conjecture: All of the non-trivial zeros of the Zeta Function are located on the line Re(s) = ½. (This is the central point of Hilbert’s Eighth Problem; also one of the Millennium Problems.)

4. Birch & Swinnerton-Dyer Conjecture: If a given elliptic curve has an infinite number of solutions, then the associated L-series has value 0 at a certain fixed point. (One of the Millennium Problems)

5. Hodge Conjecture: For projective algebraic varieties, Hodge cycles are actually rational linear combinations of algebraic cycles. (One of the Millennium Problems)

6. Jacobian Conjecture: If det[F’(x)]=1 for a polynomial map F, then F is bijective with polynomial inverse. (One of Steve Smale’s 1998 problems)

7. Under what conditions does a solution exist to the Navier-Stokes non-linear partial differential equation? Do these equations have solutions that last for all time, given arbitrary sufficiently nice initial data, or do singularities develop in the fluid flow that prevent the solution from continuing? (One of the Millennium Problems)

8. How many limit cycles are possible for a given ODE? (Half of Hilbert’s Sixteenth Problem)

9. Describe the shapes possible for the graphs of algebraic functions with only real numbers allowed as solutions. (Half of Hilbert’s Sixteenth Problem)

10. Does P = NP? (One of the Millennium Problems)

The Poincare Conjecture is not included in this list because based on what I've read (not that I'm qualified to evaluate a lot of what's being written about it) it seems that the solution by Perelman is correct.

A plethora of additional unsolved mathematics problems can be found at mathworld.wolfram.com.

Most Interesting Open Physics Questions

2004-10-02T17:02:00.003-04:00

I make no pretense of originality in the following. This list of what I consider the most interesting and important unanswered physics questions borrows heavily from Warren Siegel, John Baez and Lee Smolin. However, this compilation does introduce one or two new ideas not contained in any of their writings. It is worth noting that nine questions (with multiple subparts) is almost certainly excessive. A greater mind than mine could probably collapse these into fewer questions, and it is likely that identifying the right relations between these questions would in and of itself lead to a breakthrough.

Question 1 – The Standard Model
(a) What is the mechanism of CP violation? Is it explicable entirely within the Standard Model, or is some new force or mechanism required? Related to this, is there more matter than antimatter? If so, why?
(b) Are the measurable dimensionless parameters that characterize Nature calculable in principle or are they merely determined by historical or quantum mechanical accident and uncalculable?
(c) Are there exactly three generations of leptons and quarks? Why do the generations have the structure they do? Can the various particles be explained as manifestations of a more fundamental entity? Do quarks or leptons have any substructure, or are they truly elementary?
(d) Why do these 12 particles have the precise masses they do? Or is this an unanswerable question? How do we understand neutrino mass?
(e) Is there really a Higgs boson? If so, what is its mass, can this mass be calculated, and what is the expectation value of the Higgs field? If not, what breaks the symmetry between the electromagnetic and weak forces and gives all the elementary particles their masses?
(f) Is it possible to calculate masses of hadrons from QCD?

Question 2 – String Theory
(a) Does string theory actually work? Do problems fixed at the perturbative level return with the non-perturbative 11th dimension described by non-renormalizable membrane theory? Is a 10-dimensional perturbation expansion reasonable for an 11-dimensional theory?
(b) Are there any other strings than the D=10(11) and 26 ones, not counting dual theories?
(c) Do the four forces really become unified at sufficiently high energy?
(d) What are the fundamental degrees of freedom of M-theory? Does M-theory describe Nature? Does M-theory give specific predictions about elementary particles? If so, are they correct?

Question 3 - Gravity
(a) What is gravity? Can we merge quantum theory and general relativity to create a quantum gravity? Can quantum gravity help explain the origin of the universe?
(b) Can we detect gravity waves? What will they teach us about the universe?
(c) Does the graviton exist? If so, is it fundamental?

Question 4 – Cosmology
(a) What happened at or before the Big Bang? Was there really an initial singularity? Does the history of the Universe go back in time forever or only a finite amount? Will the future of the Universe go on forever or not?
(b) Is the Universe infinite in spatial extent? More generally, what is the topology of space?
(c) Why is there an arrow of time? Why is the future so much different from the past?
(d) How can we understand the cosmological horizon problem? Why is the Universe almost, but not quite, homogeneous, on the very largest distance scales? Is this the result of an inflationary epoch? If so, what caused this inflation?
(e) Why does the cosmological constant have the value that it has? Is it zero? How do we reconcile the requirements of cosmology with the predictions of quantum field theory or string theory? Is the cosmological constant actually constant?
(f) What is the real solution to the "dark matter" and "dark energy" problems? Do they actually exist? If so, what are they? If not, how and why is gravity modified on large scales?

Question 5 - Confinement & The Mass Gap
(a) Does confinement work? Can we calculate the observed linear Regge trajectories and see what happens to the bound states as their excitation energy increases?
(b) Is there a mathematically rigorous formulation of a relativistic quantum field theory describing interacting fields in four dimensions?
(c) Can we rigorously solve the SU(2) Yang-Mills theory in four dimensions so that we can quantitatively predict quark and gluon confinement as well as the existence of a mass gap between the Planck scale and the electroweak unification scale? (This is actually one of the seven Millenium mathematics problems.)
(d) Does QCD predict that quarks and gluons become deconfined and form plasma at high temperature? If so, what is the nature of the deconfinement phase transition? Does this really happen in Nature?

Question 6 – Black Holes
(a) Do black holes exist? What do you do with the singularities? Doesn't a singularity signify a breakdown of the theory? Is the Cosmic Censorship Hypothesis true?
(b) Do black holes evaporate through Hawking radiation? If so, what happens when they radiate completely away?
(c) Has the information paradox really been resolved? Was Hawking right in 1975, or was Hawking right in 2005?

Question 7 – Nature’s Dimensionality
(a) Is Nature four-dimensional? If not, then why does Nature appear to have one time and three space dimensions?
(b) Does compactification work? What forces the extra dimensions to hide and prevents them from reappearing? Does compactification destroy predictability? Do the extra dimensions really do anything we couldn't reproduce without them?

Question 8 - Supersymmetry
(a) Is Nature supersymmetric? If so, how is supersymmetry broken? If not, is supersymmetry still useful?
(b) Is fine tuning really that much worse than any other kind of tuning? Are superpartners any worse than non-minimal Higgs?

Question 9 – Quantum Mechanics
How should we think about the foundations of quantum mechanics? What is meant by a "measurement" in quantum mechanics? Does "wavefunction collapse" actually happen as a physical process? If so, how and under what conditions? If not, what happens instead? Can we make sense of the theory as it stands today? If not, can we invent a new theory that does make sense?

Master's Level Mathematics Curriculum

2004-10-02T16:59:00.002-04:00

Algebra:
This recommendation assumes that a year-long undergraduate course at the level of A First Course in Abstract Algebra by Fraleigh has been completed.

Algebra by Lang
Algebra by Hungerford

Analysis:
This recommendation assumes that a year-long undergraduate course at the level of "Baby Rudin" has been completed.

Real and Functional Analysis and Complex Analysis by Lang
Real and Complex Analysis and Functional Analysis by Rudin

Topology:
This recommendation assumes that a year-long undergraduate course in point-set topology at the level of Topology by Munkres has been completed.

A Basic Course in Algebraic Topology by Massey, Algebraic Topology by Fulton, and Elements of Algebraic Topology by Munkres
Differential Topology by Guillemin & Pollack

Master's Level Physics Curriculum

2004-10-02T15:21:00.003-04:00

Mathematical Physics:
This recommendation assumes that a text at the level of Arfken or Butkov has been completed as an undergraduate.

Principles of Advanced Mathematical Physics Volumes I & II by Robert Richtmyer

Mechanics:

Mechanics by Landau & Lifshitz is a good place to start. Brilliant exposition, but too short; this can serve as a supplement to the standard Goldstein text.
Classical Mechanics (3rd Edition) by Herbert Goldstein et al
Theoretical Mechanics of Particles and Continua by Fetter & Walecka. The first half (on particles) is standard, the material is the same as (but the exposition inferior to) Goldstein or L&L. The second part (on continua) provides excellent coverage of a great deal of material that is not covered in ANY of the other standard graduate mechanics texts.
Problems and Solutions on Mechanics: Major American Universities Ph.D. Qualifying Questions and Solutions by Yung-Kuo Lim

Electrodynamics:

Classical Electrodynamics by John David Jackson
Electrodynamics of Continuous Media by Landau & Lifshitz
Problems and Solutions on Electromagnetism (Major American Universities Ph.D. Qualifying Questions and Solutions) and Problems and Solutions on Optics (Major American Universities Ph.D. Qualifying Questions and Solutions) by Yung-Kuo Lim

Quantum Mechanics:

Unfortunately, undergraduate quantum mechanics does not adequately prepare you for the material you'll have to tackle in graduate quantum mechanics. Invariably there are (sometimes amazingly large) gaps which must be filled in before moving on. Two outstanding texts for doing this are Quantum Mechanics (3rd Edition) by Leonard Schiff and Quantum Mechanics: Non-Relativistic Theory, Third Edition by Landau & Lifshitz
Once that's accomplished, you can move on to "relativistic quantum mechanics" and quantum field theory. The standard treatment is in Relativistic Quantum Mechanics and Relativistic Quantum Fields by Bjorken & Drell, which is a bit dated but still quite valuable. A much more accessible (and consequently less thorough) treatment can be found in Relativistic Quantum Mechanics: Wave Equations, Quantum Electrodynamics and Field Quantization by Walter Greiner.
Quantum Electrodynamics by Landau & Lifshitz contains material not covered in either of the treatments mentioned in the previous bullet point, in particular with respect to phenomenology and calculational techniques.
Quantum Field Theory by Itzykson & Zuber rounds out and brings up to the date the material in the previously mentioned books.
Problems and Solutions on Quantum Mechanics: Major American Universities Ph. D. Qualifying Questions and Solutions (Major American Universities Ph. D. Qualifying Questions and Solutions) and Problems and Solutions on Atomic, Nuclear and Particle Physics: Major American Universities Ph.D. Qualifying Questions and Solutions by Yung-Kuo Lim, just to prove you know what you're talking about.

Statistical Mechanics:

Statistical Mechanics by Donald McQuarrie
Statistical Mechanics: A Set of Lectures by Richard Feynman
Statistical Physics and Physical Kinetics by Landau & Lifshitz
Problems and Solutions on Thermodynamics and Statistical Mechanics (Major American Universities Ph.D. Qualifying Questions and Solutions) by Yung-Kuo Lim

General Relativity:
I'm not sure why so few graduate programs include general relativity. It's really hard to consider yourself a fully educated physicist without having a grounding in this subject.

Gravitation by Misner, Thorne & Wheeler for unparalleled breadth
General Relativity by Robert Wald for added depth in the more important parts and a more modern treatment than MTW
The Classical Theory of Fields by Landau & Lifshitz. This book has several very good chapters on electrodynamics, but it is for its discussion of relativity that you should devour this book.
Problems and Solutions on Relativity and Miscellaneous Topics (Major American Universities Ph. D. Qualifying Questions and Solutions) by Yung-Kuo Lim

An Undergraduate Mathematics Curriculum

2004-10-02T14:01:00.000-04:00

The indispensable lower division core consists of the following three courses.

Calculus by Howard Anton
Linear Algebra and Its Applications by David Lay
Elementary Differential Equations by Derrick & Grossman

A standard problem in all mathematics curricula is how to transition from lower-division problem-based courses such as the three above to the upper-division proof-based courses. The following trio of books accomplishes this quite well.

How to Prove It: A Structured Approach by Daniel Velleman
Introduction to Analysis by Edward Gaughan
Set Theory and Logic by Robert Stoll

The following constitute the standard trio of upper-division courses that all mathematics majors should cover to be considered mathematically mature (as well as mathematically literate).

A First Course in Abstract Algebra by John Fraleigh
Principles of Mathematical Analysis by Walter Rudin
Topology by James Munkres

Some standard elective courses in the area of applied mathematics are the following.

Freund's Mathematical Statistics by Miller & Miller
Numerical Analysis by Burden & Faires
Partial Differential Equations : An Introduction by Walter Strauss

Some standard elective courses in the area of pure mathematics are the following.

Introduction to Modern Set Theory by Judith Roitman
Introduction to Mathematical Logic by Mendelson & Mendelson
Introductory complex analysis and applications by William Derrick

One topic that is seldom covered in the undergraduate curriculum is geometry. This course is required for most mathematics education majors, but not mathematics majors. This is a shame since this course, even though it doesn't lead into any of the major fields of mathematics research, provides a most insightful foray into how an axiomatic mathematical system should function. I found it utterly fascinating and am very glad I took the opportunity to include it in my coursework.

Euclidean and non-Euclidean geometries: Development and history by Marvin Greenberg

An Undergraduate Physics Curriculum

2004-10-02T13:40:00.007-04:00

Introductory Physics:

Sears and Zemansky's University Physics by Hugh Young. This book is much better than the standard text by Halliday & Resnick. You’ll probably want to pick up the study guide and solution manual to help you navigate through the text.
Modern Physics by Llewellyn & Tipler
Feynman Lectures On Physics by Feynman. You will never learn the material just from this book, but in conjunction with a more accessible book such as Young, Feynman’s Lectures will yield an insight on every page.

Mathematical and Computational Physics:

Mathematical Physics by Eugene Butkov. This book is better than the standard text by Arfken.
A First Course in Computational Physics by David Yevick. In this day and age, computational physics is no longer a luxury.

Classical Mechanics:

Classical Dynamics of Particles and Systems by Marion & Thornton
Classical Mechanics and Classical Mechanics: Systems of Particles by Walter Greiner. Not good enough to stand alone but a great supplement to Marion & Thornton.

Classical Electromagnetism:

Electromagnetic Fields and Waves by Lorrain, Corson & Lorrain. This book contains two fascinating chapters covering the relativistic description of electricity and magnetism as a unified phenomenon that I have not seen in any other undergraduate text.
Classical Electrodynamics by Walter Greiner. Not good enough to stand alone but a great supplement to Lorrain, Corson & Lorrain.

Quantum Mechanics:

Quantum Mechanics (2nd Edition) by Bransden & Joachain. Good quantum mechanics textbooks are hard to find; this is one of the better ones out there. (Griffiths is a good alternative.)
Quantum Mechanics: An Introduction, Quantum Mechanics: Symmetries, and Quantum Mechanics: Special Chapters by Walter Greiner is an excellent series.

Thermodynamics & Statistical Mechanics:

Fundamentals of Statistical and Thermal Physics by Frederick Reif. Good thermodynamics textbooks are hard to find; this is one of the better ones out there.
Thermodynamics and Statistical Mechanics by Walter Greiner. Not good enough to stand alone but a great supplement to Reif.

Relativity:

A First Course in General Relativity by Bernard Schutz. Special relativity is usually sufficiently covered in Modern Physics classes (and the suggested book by Llewellyn & Tipler does it well), but general relativity seems to be never covered in the undergraduate curriculum. This book is a good introduction for the undergraduate. (Hartle is a good alternative.)

Started another blog

2004-10-01T18:06:00.000-04:00

Decided to create a second blog for my mathematics and physics observations. I have much more of a vision for this one. I'll probably post something new in it later today or tomorrow at the latest.