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Summer Graduate School

A Window into Zeta and Modular Physics June 16, 2008 - June 27, 2008
Organizers Floyd Williams (University of Massachusetts) and Klaus Kirsten (Baylor University)
Lecturer(s) No Lecturers Assigned Yet.
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Description Speaker list :Klaus Kirsten (co-organizer, Baylor University), Geoffrey Mason (UCSC), Audrey Terras (UCSD), Michael Tuite (National University of Ireland), and Floyd Williams (organizer,University of Massachusetts). In recent years,a noteworthy and very fruitful interlacing of number theory and physics has emerged. As indicated in the September 2007 issue of the AMS Notices, for example, a new journal "Communications in Number Theory and Physics" has just been launched to follow significant interactions and dynamics between these two fields. Several books are now available, in addition to an array of conference and workshop activity, that accent this fortunate merger of "pure" mathematics and physical theory-with applications that range from field theory (conformal and topological), extended objects (strings and branes) cosmology and black hole physics, to Bose-Einstein condensation and the theory of relativistic gases. The workshop is designed to provide students a bridge, or a window, into this vast, interesting, rapidly-developing, interactive arena. Some special attention would be given to zeta and modular aspects of the interactions. The students would be provided with some lecture notes and hand-outs, and with very ample opportunities to engage in discussion/question sessions with the lecturers, apart from the two hours per day lectures.In some cases, web material will be available prior to the workshop-all in a concerted effort to afford maximal learning situations. Prof.Terras, for example, already has some introductory notes on line to help students get a head start towards her lectures on quantum chaos: http://www.math.ucsd.edu/~aterras/newchaos.pdf; also compare http://www.math.ucsd.edu/~aterras/newbook.pdf The students would be required to have a good working knowledge of complex variables,including familiarity with infinite products and the gamma function.It is suggested they should read some introductory material on the Riemannn zeta function ( say, from a text on complex variables), and it would be helpful for them to do some minimal reading on the Dedekind eta functions. The material in Tom Apostol's Springer Graduate Text on "Modular Functions and Dirichlet Series in Number Theory" (chapter 3) , for example, is more than needed for a basic understanding of the Dedekind eta function. Alternately, one can obtain the same information on zeta and eta (without proofs) by simply going to Google. The organizer has emphasized, repeatedly, to the speakers that our aim is to keep the lectures elementary, expository, and self-contained as possible. Vertex Operators (PDF 165KB) Modular Forms in Vertex Operator Algebras (PDF 228KB) Audrey Terras - Lecture 1 (PDF 1.27MB) Audrey Terras - Lecture 2 (PDF 860KB) Audrey Terras - Lecture 3 (PDF 1.91MB) First week : Monday: Riemann zeta function(analytic continuation,special values,and infinity factorial as a simple example of zeta regularization)/Floyd/ introduction to Hurewitz,Epstein,Barnes zeta/Klaus Tuesday: A simple example of a Casimir enengy calculation(another example of zeta regularization)/Klaus/introduction to Selberg,Ihara,Ruelle zeta/Audrey Wednesday: conformal field theory (CFT) for beginners/Geoff and Michael/ modular forms of positive weight and their Fourier coefficients/Floyd Thursday: zeta zeros/poles and distribution of primes/Audrey/CFT/Geoff and Michael Friday: Dedekind eta and its reciprocal with some string theory remarks/Floyd/computation of functional determinants via contour integrals/Klaus Second week Monday: polylog expansion of a partition function with some intuitive remarks on Bose-Einstein condensation/Klaus/application of Epstein-Barnes zeta to gravity in extra dimensions/Floyd Tuesday: modular forms of negative weight (more on the reciprocal of the Dedekind eta function,and a statement of the Rademacher-Zuckerman formula for Fourier coefficients)/Floyd/CFT/Geoff and Michael Wednesday: quantum chaos (some applications of Audrey's earlier lectures)/Audrey/CFT/Geoff and Michael Thursday: generalized Cardy formula for black hole entropy(applications of forms of non-positive weight)/Floyd/unscheduled time for review,miscellaneous material,extra time for speakers,etc. Friday: unscheduled time Photo of workshop participants (.98MB)

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Schedule
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Jun 16, 2008
Monday
09:30 AM - 10:30 AM
  Riemann's zeta function
Floyd Williams
11:00 AM - 12:00 PM
  Introduction to Hurewitz,Epstein,Barnes zeta
Klaus Kirsten (Baylor University)
Jun 17, 2008
Tuesday
09:30 AM - 10:30 AM
  Conformal field theory (CFT) for beginners
Geoff Mason (University of California, Santa Cruz)
11:00 AM - 12:00 PM
  A simple example of a Casimir enengy calculation(another example of zeta regularization)
Klaus Kirsten (Baylor University)
Jun 18, 2008
Wednesday
09:30 AM - 10:30 AM
  Modular forms of positive weight and their Fourier coefficients
Floyd Williams
11:00 AM - 12:00 PM
  Conformal field theory (CFT)
Geoff Mason (University of California, Santa Cruz)
Jun 19, 2008
Thursday
09:30 AM - 10:30 AM
  Introduction to Selberg,Ihara,Ruelle zeta
Audrey Terras
11:00 AM - 12:00 PM
  zeta zeros/poles and distribution of primes
Audrey Terras
Jun 20, 2008
Friday
09:30 AM - 10:30 AM
  Dedekind eta and its reciprocal with some string theory remarks
Floyd Williams
11:00 AM - 12:00 PM
  Computation of functional determinants via contour integrals
Klaus Kirsten (Baylor University)
Jun 23, 2008
Monday
09:30 AM - 10:30 AM
  Polylog expansion of a partition function with some intuitive remarks on Bose-Einstein condensation
Klaus Kirsten (Baylor University)
11:00 AM - 12:00 PM
  Quantum chaos
Audrey Terras
Jun 24, 2008
Tuesday
09:30 AM - 10:30 AM
  Modular forms of negative weight (more on the reciprocal of the Dedekind eta function,and a statement of the Rademacher-Zuckerman formula for Fourier coefficients)
Floyd Williams
11:00 AM - 12:00 PM
  CFT
Geoff Mason (University of California, Santa Cruz)
Jun 25, 2008
Wednesday
09:30 AM - 10:30 AM
  A Window into Zeta an Modular Physics talk I
Michael Tuite
11:00 AM - 12:00 PM
  A Window into Zeta an Modular Physics talk II
Michael Tuite
Jun 26, 2008
Thursday
09:30 AM - 10:30 AM
  Generalized Cardy formula for black hole entropy(applications of forms of non-positive weight)
Floyd Williams
11:00 AM - 12:00 PM
  On Relating d-dimensional FRLW Cosmology to Bose-Einstein Condensates
Jennie D'Ambroise
Jun 27, 2008
Friday
09:30 AM - 10:15 AM
  Zeta and L-functions
Paul Nelson
10:30 AM - 11:15 AM
  Arithmetic of String Compactification and Emergent Spacetime.
Savan Kharel
11:15 AM - 12:00 PM
  How a soliton Illuminates a Blackhole
Shabnam Beheshti (Rutgers University)