Conference on Number Theory, Geometry, Moonshine & Strings II

  • Organized by
  • Quantrell Award winner Jeff Harvey, a physics professor, in the Gordon Center Monday, May 20, 2013, at the University of Chicago.    (Photo by Robert Kozloff)Jeffrey Harvey, Ph.D.Professor, University of Chicago
Date & Time


The second conference was held March 14-16, 2018, and consisted of a series of seminars and a public lecture, “The Riemann Hypothesis,” by Ken Ono. The conference was organized by Jeff Harvey, John Duncan, Shamit Kachru and Ken Ono. There were 25 supported participants and a smaller number of people who came with their own support including some postdocs and graduate students. Several mathematicians from local universities attended parts of the conference including Michael Harris (Columbia), Jim Lepowsky (Rutgers), Steve Miller (Rutgers), and Karen Taylor (NYU).

The conference brought together number theorists, geometers and physicists and many people commented on the unusual opportunities this presented. A number of new collaborations were started at the conference and others benefited from new points of view. Some examples: a new collaboration between Ken Ono and Sergei Gukov involving relations between ongoing work of Gukov and collaborators on generating functions for mock modular forms and three-manifold invariants and earlier work of Ono on quantum modular forms; a new collaboration between Shamit Kachru and Ken Ono connecting number theoretic aspects of quadratic forms to counting of black holes on K3 _ T 2; discussions between Kachru, Jim Bryan, John Duncan, Sheldon Katz, Albrecht Klemm and Georg Oberdieck on the definition of and predictions for twined Gromov-Witten invariants; fruitful discussions between Chris Beem and Jeff Harvey on the modularity of Schur indices in N = 2 superconformal field theories in four dimensions; a new collaboration between Duncan, Harvey, Michael Griffin and Brandon Rayhaun on moonshine for “Pariah” sporadic groups connected to weight 3=2 modular forms. These examples are just those I know of or have been made aware of. I am sure there are many other such examples.

In an email following the conference, Shamit Kachru said “This meeting helps show them [graduate students] that within the sphere of mathematical string theory, there are other directions, related to or quite distinct from what they have seen before, being pursued by other people who are fun to talk to. Interactions at such meetings have helped lead to projects and new collaborators for Sarah Harrison, Natalie Paquette, and Nathan Benjamin in the past, and continue to help people (e.g. Brandon Rayhaun). It would be hard for me to imagine raising mathematical physicists without such meetings.”

Sergei Gukov said in email to me that “many of the talks were directly or indirectly related to my current projects. For example, Martin’s talk about mock modular forms was directly related to above mentioned work in progress with Miranda (Cheng) et.al. (And, from Martin’s talk, I learned about a different perspective on mock modular forms relative to that I used so far.) Even your own talk on characters was extremely useful to my line of work on Vertex Operator Algebras labeled by 4-manifolds (that arise in compactification of 6d (0,2) theory on 4-manifold). I can go on with this list, but overall, it is hard to point to another two to three days in this academic year that would be equally helpful and, at the same time, relatively relaxed and enjoyable!”

Minhyong Kim said, “The second conference on number theory, geometry, moonshine and strings was a great success from my point of view. The first conference gave a clear sense of the kind of topics to read about in the past months, so I was able to follow the physics lectures much better this time around. I found most of the lectures greatly illuminating and tantalizing, especially those of Jeff Harvey, Shamit Kachru, Greg Moore and Sameer Murthy. All these people were very responsive to questions as well. In both number theory and high- energy physics, the ubiquity of modularity is a phenomenon at the exciting boundary of understanding and mystery, and there is much to be gained by continued interaction between the two communities.”

Although there were a number of notable presentations, one that impressed me the most regarding the potential for new relations between string theory and number theory was the talk by Albrecht Klemm on relations between masses of D-brane states in Calabi-Yau geometry, arithmetic geometry and values of L-functions. The work he presented is forging new connections between some very deep aspects of string theory and number theory.

The third conference is scheduled for February 27-March 1, 2019. It is too soon to set the agenda for this meeting, but I hope that some of the collaborations and new avenues of research that were promoted by the first two meetings will reach fruition and that part of the meeting will be devoted to hearing new results that were discovered as a result of the first two meetings. The first two meetings have strengthened my belief (and that of others) that these new interactions between mathematicians and physicists involving the topics of number theory, geometry, moonshine and string theory central to this series of conferences will lead to important, new results and connections between these areas. The support provided by the Simons Foundation has been crucial for the development of these new research directions.
 

  • Agenda & Slidesplus--large

    WEDNESDAY, MARCH 14

    9:30 AMShamit Kachru | Black Holes, BPS Jumping, and Special Loci
    11:00 AMSameer Murthy | Black Hole Bound States and Modularity
    (Download Slides PDF)
    1:30 PMSheldon Katz | Refined BPS Invariants and Modular Forms
    (Download Slides PDF)
    3:00 PMMartin Raum | Vector-Valued and Mock Modular Forms
    5:00 PMPUBLIC LECTURE | Ken Ono | The Riemann Hypothesis
    (Download Slides PDF)

    THURSDAY, MARCH 15

    9:30 AMGreg Moore | More about Four-Manifolds and Supersymmetric Gauge Theory
    11:00 AMJohn Duncan | Moonshine and Geometry
    (Download Slides PDF)
    1:30 PMAlbrecht Klemm | Periods and Quasiperiods of Modular Forms and D-Brane Masses of the Quintic
    (Download Slides PDF)
    3:00 PMSergei Gukov | Mock Modular Forms and BPS State Counting
    4:30 PMGeorg Oberdieck | Counting Curves in K3 Geometries and Jacobi Forms

    FRIDAY, MARCH 16

    9:30 AMKen Ono | Polya's Program for the Riemann Hypothesis and Related Problems
    (Download Slides PDF)
    11:00 AMJeff Harvey | Hecke Relations Between RCFT Characters
    (Download Slides PDF)
    1:00 PMMinhyong Kim | Arithmetic Gauge Fields in Diophantine Geometry
    (Download Slides PDF)
  • Participantsplus--large
    Victor ArichetaEmory University
    Chris BeemOxford University
    Lea BeneishEmory University
    Nathan BenjaminStanford University
    Jim BryanUniversity of British Columbia
    Gautam ChintaCUNY
    John DuncanEmory University
    Rebecca FrankelNew York University
    Dorian GoldfeldColumbia University
    Joao GomesUniversity of Amsterdam
    Michael GriffinBrigham Young University
    Sergei GukovCalifornia Institute of Technology
    Michael HarrisColumbia University
    Sarah HarrisonHarvard University
    Jeff HarveyUniversity of Chicago
    Herve JacquetColumbia University
    Shamit KachruStanford University
    Sheldon KatzUniversity of Illinois, Urbana Champaign
    Maryam KhaqanEmory University
    Minhyong KimOxford University
    Albrecht KlemmHausdorff Center for Mathematics
    Jim LepowskyRutgers University
    Stephen MillerRutgers University
    Greg MooreRutgers University
    David MorrisonUC Santa Barbara
    Sameer MurthyKing's College, London
    Georg OberdieckMIT
    Ken OnoEmory University
    Natalie PaquetteCalifornia Institute of Technology
    Martin RaumUniversity of Gothenburg
    Brandon RayhaunStanford University
    Meng-Chwan TanNational University of Singapore
    Karen TaylorCUNY
    Arnav TripathyHarvard University
    Ian WagnerEmory University
    Shou-Wu ZhangPrinceton University
    Max ZimetStanford University
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