The 2016 annual MPS meeting took place October 20–21. It featured exciting talks about research at the frontiers of math, physics and theoretical computer science, as well as lively discussions among the heterogeneous crowd of attending scientists.
The keynote speaker, Mina Aganagic, talked about math and string theory duality. In string theory, it is often the case that there are two very different mathematical descriptions of the same physical situation. The two descriptions are dual to each other, and the dictionary that translates from one description to the other is highly non-trivial. This leads to very surprising conjectural equivalences between mathematical objects that seem to be very different. In many cases, mathematicians have been able to prove these equivalences. The equivalences are very powerful, since they can relate very difficult computations, such as counting curves of given degrees in an algebraic manifold to much simpler classical computations in the dual picture. The dualities can also be used for defining new quantities such as knot invariants. Aganagic surveyed several notable dualities and ended with the announcement of a recent proof, due to Frenkel, Okounkov and herself, of a quantum geometric Langlands correspondence, which generalizes the classical geometric Langlands correspondence, itself a major result. The proof relies on an electromagnetic duality theory in six dimensions and the introduction of a quantum version of K-theory.
Lisa Manning talked about jamming in biological tissue. Such tissues start out in the early embryonic stage as solid-like structures and later become fluid like. Such transitions are crucial at the developmental stage. Manning described how movement occurs in completely jammed media, with cells “pushing” their way through the interfaces between other cells, changing the cell arrangement. She then described the phase diagram for such motion and related it to geometric features of the cell. Such features can be measured in static pictures, so we can learn about cell dynamics from static data, which is more readily available. The model fits observations very well.
Dan Boneh showed how cryptographers are using sophisticated mathematical constructions to achieve amazing technical goals that are at the heart of electronic commerce. In particular, he showed how the Tate pairing, a bilinear product which originates in abstract algebraic number theory, is used in credit card chips for fast authentication. Other constructions of cryptographic primitives coming from lattices are used to construct homomorphic encryption schemes, which allow computations on encrypted data, such schemes could be of great value in cloud computing environments if we could make them work a bit faster. Boneh challenged the mathematicians in the crowd to come up with new constructions based on higher math, for example, trilinear forms with some nice mathematical and computational properties, and explained that they could lead to some spectacular new applications.
Paul Seidel described ongoing work related to mirror symmetry. This symmetry is one of the deep dualities originating in string theory, which was described in the talk by Aganagic. It relates two seemingly different areas of mathematics: the rigid world of algebraic geometry and the more flexible world of symplectic geometry. Understanding mirror symmetry is the subject of one of the current MPS collaborations. For suitable algebraic manifolds, the string theory of the manifold leads to a formal two-parameter partition function that “counts” holomorphic curves on the manifold. Seidel described calculations of the partition function in a particularly challenging non-perturbative regime, where one of the parameters is not infinitesimally small.
Andrea Alu described some new devices with magical optical or acoustic properties. A basic feature of optical and acoustical systems is time reversal symmetry. If light or sound can propagate from A to B, then it can be reversed to propagate from B to A. Alu explained how he designs devices that break time reversal symmetry by introducing a rotational asymmetry in the system. As a result, the devices can route sound or light in an asymmetrical fashion to different destinations. Another application is in cloaking, in which an object becomes invisible. Alu complemented these demonstrations with theoretical results which show that, while cloaking with respect to a particular wavelength is possible, it is impossible to achieve for a broad band of the spectrum under rather mild assumptions on the system, explaining why many attempts are doomed to failure.
Rick Schwartz discussed the classical and very difficult problem of finding arrangements of points on the sphere that minimize some given standard energy function, which are based on distances between the points of the arrangement. In particular, he discussed the case of five-point configurations, where he was able to prove the rather natural conjecture that the energy-minimizing configuration consists of the north and south poles and an equilateral triangle on the equator. The computer-assisted proof required breaking up the configurations space into many small pieces and carrying out delicate numerical estimates for the minimum possible energy in each piece.
Daniel Eisenstein talked about dark energy and cosmic sound. He described how acoustic signals from plasma that were set free a short time after the big bang and traveling at 57 percent of the speed of light can be used as a particularly accurate ruler for accurately measuring distances between objects in the universe.
Julia Hartman talked about basic invariants of fields. Fields are very basic objects of study in algebra; they consist of systems with operations of addition and multiplication that satisfy all the usual properties we are familiar with. Of central interest in mathematics is the study of solutions to polynomial equations in a given field. The case of linear equations is covered completely by linear algebra. The next case of a single quadratic equation in several variables is already very challenging. A basic invariant of the field, the u-invariant is the largest number of variables for which one can find a quadratic equation with no non-zero solution in the field. Hartman described works from the last few years that have allowed the computation of u-invariants in many cases of interest.
Agenda & Slides
Thursday, October 20
- Keynote Address:
Mina Aganagic, UC Berkeley
String Duality and Mathematics
- Lisa Manning, Syracuse University
Jamming in Biological Tissues
Slides with videos (150MB PDF)
Slides without videos (35MB PDF)
- Julia Hartmann, University of Pennsylvania
Geometry and Algebra: From Local to Global
- Dan Boneh, Stanford University
Recent Developments in Cryptography
- Daniel Eisenstein, Harvard University
Dark Energy and Cosmic Sound
Friday, October 21
- Paul Seidel, Massachusetts Institute of Technology
Symplectic Topology Away from the Large Volume Limit
- Andrea Alù, The University of Texas at Austin
Breaking Reciprocity and Time-reversal Symmetry with Metamaterials
- Richard Schwartz, Brown University
Five Points on a Sphere
Download the documents below:
Elise Volpe, Protravel International
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Registration, Hotel and General Meeting Assistance
Senior Executive Assistant, Simons Foundation
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Program Associate, Simons Foundation
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