Essays on mathematical physics and theoretical biology

The dilemma of modern biology, and in particular molecular biology, is that staggering amounts of data have been and still are being accumulated and stored, but the question of what to actually do with this vast data set is still in its infancy. A great deal of emphasis in biological science is placed upon bioinformatics , essentially storing information and data in library catalogs that can be retreived. The main examples here are the mapping of the human genome and the vast stores of information on protein structure that are held in various protein data banks. The dna sequences of hundreds of organisms have been decoded and stored in this way, and massive sequencing efforts seek to identify mutations in a variety of genes in cancers. The vast volume of data that is produced requires automated/computer systems to read it and to compare sequencing results; consequently there have been considerable advances in computational biology and bioinformatics/biostatistics with major efforts in gene finding, sequence alignment, genome assembly, protein structure etc.

This is of course very important but in itself may never reach the deeper understanding of what actually makes life function, and how these processes can go awry and produce pathological states, disease and cancer. The list of ‘working parts’ is essentially complete but a deep and fundamental mathematical-physical understanding of how the parts actually function together to generate the underlying processes of life is still essentially lacking–the whole is still much greater than the sum of the parts. The explosion of information produced by the genomics revolution will be difficult to understand without the continued application of powerful mathematical methods. Knowing how to properly describe and fully utilize this data using such methods could also open up new and powerful applications in medicine, genetic engineering, drug design and cancer therapy.

Read the rest of this entry »

Every so often a textbook comes along that takes a very broad and very difficult subject and explains it with a marvellous clarity of exposition. Many topics and material that in the past had left you somewhat confused or even totally stymied are suddenly made crystal clear, or at the least are greatly clarified.  Such is the case with the beautiful new string theory textbook by John Schwarz and Melanie and Katrin Becker called String Theory and M-Theory–A Modern Introduction.

String theory has become much maligned in recents years with its detractors claiming that a string-theoretic construction of ‘a theory of everything’ has failed, that it is ‘not physics’ and that it cannot hope to ever connect with reality. However, most of them fail to appreciate that nature is already full of examples of fluctuating line-like or string-like objects: these include defects in crystals and liquid crystals and condensed medias, vortex lines in superfluids and turbulence, flux tubes in superconductors and qcd, polymers in solutions, biopolymers like polypetide chains/proteins and dna, and even viruses. (Path integrals are usually particularly useful tools to describe such fluctuating line-like structures like polymers.) Biology in particular, over many length scales, seems to be built from fluctuating string-like structures and membranes or ‘branes’. It is therefore still reasonable to suspect (in my opinion) that this is a hint that at the most fundamental level the building blocks of the universe might actually be string-like and brane-like structures, even if we cannot yet fully understand, develop or manipulate such a theory.

Read the rest of this entry »

Stanislaw Ulam can be compared to some of the best mathematical minds of the 20th century but is probably somewhat less well known or remembered than many of his contemporaries. In his early career he made important contributions to ergodic theory, set theory, number theory, measure theory and algebraic topology. A few of these include the Ulam spiral, which leads to a mysterious pattern for the prime numbers; the Borsuk-Ulam theorem, an important result in topology; the Mazur-Ulam theorem and many more. Often he provided the seeds of ideas which others then developed and took the main credit for. Later, during the war, he switched over to applied mathematics and together with his friend John von Neumann was one of the first to advocate the important role computers could play in studying and solving difficult mathematical and scientific problems. He realized that one could perform powerful ‘mathematical experiments’ on a computer; for example, he was among the first to initiate a computer study of a nonlinear dynamical system in the Fermi-Pasta-Ulam experiment.

His most important work in this regard however, is the Monte Carlo method, perhaps the most powerful and versatile tool ever developed in applied mathematics, which has a plethora of applications over a very wide range of fields. These are a class of stochastic algorithms that utilize random sampling to solve otherwise intractable problems. Modern applications can now utilize the power and speed of modern computers and supercomputers. They include: computational physics and chemistry, nuclear engineering, reactor design and radiation shielding, molecular dynamics, biological physics, cancer radiation dosimetry, astrophysics, particle physics(QCD), statistical physics, simulated annealing, computer science, statistics, operations research, finance and financial engineering, image processing, traffic flow problems, aerodynamics, and engineering problems in general. The list is hardly exhaustive. But if the Monte Carlo method was his most famous contribution to science then his most infamous is one he let another man take most of the credit for. Personal tragedy and fate were to place him in a situation, at a crucial juncture in history, where his powerful abilities provided the key ideas that helped solve a dark problem; one which for the most part had totally stymied his peers–and a problem many of whom had actually hoped was a scientific and technological impossibility. But the solution changed history and made possible the construction of an awesome and terrifying weapon of virtually unlimited power.

Stan Ulam was born in 1909 in Lwow into a well-to-do family that was positioned as high in the social ladder as a Jewish family could be at that time in that part of Europe. (Lwow was part of Austria-Hungary in 1909, was part of Poland from 1918 and part of the Ukraine from 1945.) He was a privileged child from one of the richest families in Lwow and the Ulam name was synonomous with banking wealth in central Europe, much like the Rothschilds were in western Europe. He was tutored and educated at the best schools and showed an early interest in mathematics and science, easily becoming a top student with little effort. At age 17 he attended the Lwow Polytechnic Institute but soon discovered that the real mathematical action was to be found at one of the large cafes in town known as the ‘Scottish Cafe’. Here, each day Lwow mathematicians would gather to talk shop, play chess and drink coffee or brandy, and they would pose and often solve some of the most outstanding mathematical problems and conjectures of their time. Read the rest of this entry »

Steven Weinberg once remarked that there is a tradition in theoretical physics that affected him, but ‘which by no means affected everyone’, namely that the strong (nuclear) interactions are ‘too complicated for the human mind’. Fortunately, he went on instead to clarify our understanding of the weak nuclear interactions and their connection with electromagnetism. (Whether he actually did say this I am not sure. Physics is full of folklore but I have read this statement at least twice over the years.) Those who did take up the challenge to break their heads on explaining the strong interactions included Murray Gell-Mann, who brought in powerful group-theoretic ideas; David Gross, David Politzer and Frank Wilczek, who demonstrated asymptotic freedom; and Kenneth Wilson, who provided important technical tools and demonstrated how the resulting theory of quantum chromodynamics (QCD) could be put on large computers and solved numerically. All were rightly awarded the famous Swedish prize. (As was Weinberg too of course, although Wilson was actually awarded the prize for his work on critical phenomena and phase transitions.)

It became clear in the late 60s and early 70s that quarks and gluons constituted the ‘machinary’ inside protons and neutrons, the nuclear building blocks of the familiar forms of matter in the universe; from ordinary hydrogen and helium to carbon, iron, gold and everything else in the periodic table. It also become clear that the strength of the gluon-exchanging color forces holding quarks together in subnuclear matter was immense; giving rise to both the stability of ordinary matter and–almost as a ‘feeble’ byproduct–the ferocious infernos in the cores of the sun and stars, powering them for billions of years.

Despite the successes of QCD, many crucial aspects of the strong or color interactions and the Yang-Mills gauge theories underpinning them, still remain elusive; indeed, one of the Clay Millennium prizes is for a rigorous formulation of quantum Yang-Mills theory on , which currently remains beyond reach. This does not imply that little progress has been made: Yang-Mills theories, with their deep geometric significance, have played a crucial role in mathematics, as have their supersymmetric extentions. Work in string theory has also revealed a deep and tantalizing connection between string theory on one space and a QFT without gravity defined on the boundary of that space, as embodied in the celebrated AdS/CFT conjecture. For example, supersymmetric Yang Mills theory living on the 4-dimensional boundary of a five-dimensional anti-deSitter space and Type IIB string theory on , where is the 5-dimensional sphere. This is a very promising development within string theory and the conjecture has certainly emphasised once again the central importance of gauge theories. (A more detailed discussion on the importance of the duality for studying QCD can found here.) One of the main problems has always been (and still is) the need for development of tractable quantitative analytical nonperturbative methods. However, for all practical purposes, the main approach to effectively dealing with strongly coupled Yang-Mills theories and QCD has been (and still is) via powerful lattice computer simulations using numerical methods.

A central conjecture and open problem in quantum field theory is the following, and from a mathematical physics perspective the one that intrigues and interests me the most

Let be a simple compact Lie group and let be a representation of . In (3+1) dimensions and in (2+1) dimensions , the pure gauge theory with connection or 1-form , curvature , gauge group and action exhibits confinement.

The specific case of the conjecture that is relevent for strong interactions is and , and is tantamount to confinement of quarks within subnuclear matter. A rigorous analytical proof of confinement is a classical and long-standing open problem within gauge theory and QCD, and essentially states that quarks cannot be isolated: color-charged quarks are confined together in pairs (mesons) or triplets (baryons) with an overall neutral net color. There is a linearly rising potential between quarks of the form , for a constant , and consequently no free quarks exist in nature.

Read the rest of this entry »

Blogging has to take a low priority I am afraid due to time constraints, but topics I would like to try and write about this year are tentatively listed below, and in no particular order. Some will be ‘lay person reviews’, some will be semi-technical, while others (denoted with *) will be technical articles at the text book or journal level perfused with some LaTex. I certainly don’t claim to have expertise in all of these areas (although quite a few I think I can discuss with some authority). I do however have a wide range of interests reflecting my background. Primarily I am writing about research topics and areas I have either worked in or have studied or read about that I find especially interesting; or simply summarizing something I am trying to understand. Other posts will include book reviews, primers on a few topics, and reviews of preprints or journal articles. Comments, input etc. is welcome as are corrections of the dumb errors I will invariably make. (My only request is that any discourse is kept civil.)

Despite difficulties, string theory remains a compelling area of mathematical research.

Statistical topology of polymer tangles: applications of Wilson loops and Chern-Simons theory (*)

Stochastic Analysis I: Markov processes, diffusions and Brownian motions. (*)

Stochastic Analysis II: some illustrative applications in finance and quantum mechanics. (*)

Topological  field theory and molecular biology: dna knots and superhelicity.

Brief primer: Lax pairs, integrability and the Bethe ansatz. (*)

Brief primer: characteristic functions and the linked cluster decomposition. (*)

Freedom and slavery in the subnuclear realm: quarks, gluons and confinement. (*)

Inferno: the thermonuclear processes that fire the sun and the stars.

Dissipative stochastic processes and the statistical mechanics of folding proteins. (*)

The power of statistical sampling: history and applications of the Monte Carlo method.

John Von Neumann and Subramanyan Chandrasekhar: great intellects of the 20th century.

Singular terminal indecomposable past sets: a class of conformal factors restoring geodesic completeness. (*)

Light scattering and radiative transfer in liquid suspensions of polymers and bioparticles: borrowing mathematical methods from nuclear and astrophysics. (*)

Strings, branes and black holes: a tantalizing connection. (*)

The Kerr solution of the vacuum Einstein equations–the most remarkable exact solution in mathematical physics? (*)

Possibility of pycnonuclear reactions in superdense matter, liquid metal hydrogen and collapsed stars. (*)

“I know how to make it work…and it will change the course of history…”: Mathematician Stanislaw Ulam and the hydrogen superbomb.

Wave propagation on stochastic geometries. (*)

The surface physics and chemistry of artificial polymer-blood interations.

Notes on differential geometry, gravitation and black hole mechanics (*)

Classical Book Review: The Mathematical Theory of Black Holes. S. Chandrasekhar.

Book review: String Theory and M-theory–a Modern Introduction. Becker, Becker and Schwarz.

Book review:Inventing Money. The Story of LTCM and the legends behind it. N Dunbar.

Stochastic games and Markov games: are casinos beatable?

Submissions to J. Math. Physics for 2007.