James Glimm

Stony Brook

Title: From Constructive Quantum Field Theory to Type Ia Supernova

Abstract:  We sketch some of the major developments for constructive quantum field theory. We note the fundamental obstacle at 4 space time dimensions, and continue with discussion of renormalization group ideas. Here we find analogies to fluid turbulence useful. Turbulence, as with Quantum Chromodynamics, has the simplifying property of asymptotic freedom. We assume a Kolmogorov 1941 style bound and prove as a result existence of solutions to the Euler equation. This can be thought of as  a handle on the inessential variables in the RNG terminology.

The Euler equation solutions are not unique and in need of an admissibility condition. Here a major scientific controversy occurs. We comment on the solution of the deflagration to detonation transition in SN Ia, and its relation to this controversy. 

Vaughan Jones

Vanderbilt University

Title: Factors and subfactors in one dimensional quantum physics

Abstract: Factors and subfactors arise as algebras of observables in quantum spin chains and conformal field theory. The subfactors in the two cases appear to be the same up to a type III factor. Whether the CFT subfactors can be obtained as limits of the quantum spin chain ones is a  brutal open question. As is the question of which subfactors arise in CFT. The techniques of constructive quantum field theory offer the only hope of progress in this area.

Giovanni Felder

ETH Zurich

Title: Representation homology and supersymmetric gauge theory

Abstract: 

Edward Witten

IAS

Title: Nonsupersymmetric D-Branes

Abstract: Nonsupersymmetric D-branes were discovered by Sen 20 years ago. I will explain a perspective on them that is based on a discrete anomaly.

Zhengwei Liu

Harvard

Title: Geometric Quantum Error-Correcting Codes

Abstract: I give a new geometric construction of quantum error-correcting codes using the quon language. This provides new pictorial insights for Kitaev’s toric code. It also gives a geometric realization of all stabilizer codes, which represents both the stabilizer group and stabilizer states in a geometric way simultaneously. In particular, I construct the 5,1,3 code using the complete graph K5 which encodes a S5 Symmetry. 

Cumrun Vafa

Harvard

Title: String Theory and the Unity of Quantum Toplogical Invariants

Abstract: I review the connections between quantum systems and topological invariants in low dimensions and explain how a stringy perspective not only gives a unified description of them, but also leads to new invariants for 4-manifolds as topological modular forms.

David Evans

Cardiff University

Title: Conformal Field Theory: the search for the exotic and reconstruction.

Abstract: Modular tensor categories arise in Conformal Field Theory through conformal nets of von Neumann algebras, vertex operator algebras and twisted equivariant K-theory. I will discuss the issue of constructing models with specific modular data, including the double of the Haagerup subfactor. This is joint work with Terry Gannon.

Anke Pohl 

University of Bremen

Title: Automorphic functions twisted with non-expanding cusp monodromies, and dynamics

Abstract: Automorphic functions play an important role in several subareas of mathematical physics, e.g., in rational conformal field theories. The correspondence principle between quantum und classical mechanics suggests that automorphic functions are closely related to 
geometric and dynamical entities of the underlying locally symmetric spaces. Despite intensive research, the extent of this relation is not yet fully understood.

A seminal and very influencial result in this direction has been provided by Selberg, showing that for any hyperbolic surface X of finite area, the Selberg zeta function (a generating function for the geodesic length spectrum of X) encodes the spectral parameters of the untwisted automorphic forms for X among its zeros. Subsequently, this type of relation was generalized to automorphic forms twisted by unitary representations, to resonances and to spaces of infinite area by various 
researchers.

Recently, a deeper relation could be established by means of transfer operator techniques. For the so-called Maass cusp forms, these methods allow us to provide a purely dynamical characterization of these automorphic functions themselves, not only of their spectral parameters.

The structure of this approach indicates that an extension to automorphic forms with well-behaved, also non-unitary twists should be expected. While such a generalization seems to be a long-term goal, we could already show first steps in this direction on the level of the Selberg zeta functions.

After surveying the transfer operator techniques, we discuss the current state of art regarding dynamical approaches towards automorphic forms with non-unitary twists.

Jennifer Chayes 

Microsoft

Title: Graphons: A Nonparametric Method to Model, Estimate, and Design Algorithms for Large-Scale Networks

Abstract: Many systems are naturally represented as networks, from off-line and on-line social networks, to bipartite networks, like Netflix and Amazon, between consumers and products, to proteomic networks in biological contexts. Graphons, developed as limits of graphs, are a natural, nonparametric method to describe and estimate (machine learn) large networks like Facebook and LinkedIn. Here I describe the development of the theory of graphons, for both dense and sparse networks, over the last decade. I also give theorems showing that we can consistently estimate graphons from massive networks in a wide variety of models. Finally, I show how to use graphons to estimate missing links in a sparse network, which has applications from estimating social and information networks in development economics, to rigorously and efficiently doing collaborative filtering with applications to movie recommendations in Netflix and product suggestions in Amazon, to inferring missing links in biological networks.

Shamil Shakirov 

Harvard & Uppsala University

Title: Higher genus analogues of Jack polynomials

Abstract: We construct systems of commuting differential operators associated with quivers with g loops, whose common eigenfunctions may be called genus g Jack polynomials.

Christoph Schweigert 

University of Hamburg

Title: Bulk Fields in Conformal Field Theory

Abstract: Two-dimensional conformal field theories are examples of quantum field theories that allow for mathematically precise statements about the field content and the existence of consistent sets of correlators.

In this talk, we present recent results on fields and on correlators for conformal field theories beyond rational conformal field theories, for chiral data that are described by non-semisimple categories.
In particular, we explain a new compact formula for bulk fields that can be easily extended to disorder and defect fields.

Klaus Hepp

ETH

Title: Phase Transitions in the Brain

Abstract: First we analyse microsleep, which happens during drowsy driving and to students in our lectures. Then we relate the neuronal data on microsleep in the oculomotor system to non-equilibrium phase transitions, guided by our work with E Lieb on the reservoir-driven finite-mode laser. Finally
we discuss current concepts of consciousness in the brain.
 

Xiao-Gang Wen

MIT

Title: Emergce of higher symmetry in lattice (condensed matter) systems

Abstract: TBD

Mikhail Lukin

Harvard

Title: Exploring new frontiers in quantum science

Abstract: TBD

Christian Jaekel

University of Sao Paulo

Title: On reflection positivity, modular localisation and Connes cocycles

Abstract: The unitary irreducible representations of the Lorentz group carry an intrinsic notion of  localisation on de Sitter space, known as modular localisation. An extension of Araki’s perturbation theory of modular automorphisms can be used to define interacting representations of the Lorentz group, as well as the corresponding Haag-Kastler nets. The analyticity properties of the correlation functions allow us to extend these theories to “nets" of (non-abelian) von Neumann algebras on the sphere. Reflection positivity can be used to recover the interacting quantum (field) theories 
on the de Sitter space from the sphere. Explicit examples are scalar bosons with polynomial or exponential interactions in 1+1 space-time dimensions, but our aim is to classify all interacting quantum theories compatible with the space-time symmetries. The Minkowski space limit is the limit of space-time curvature to zero, which is well-behaved on the level of local von Neumann algebras.

Bas Janssens

Delft University of Technology

Title: Reflection Positivite Doubles

Abstract: Reflection Positive Doubles constitute a general framework for reflection positivity, covering a wide variety of systems in statistical physics and quantum field theory. In this context, we give sufficient (and, under extra assumptions, necessary) conditions for reflection positivity. The main novelty here is that the systems under consideration can be not only bosonic or fermionic, but also parafermionic. (Joint work with Arthur Jaffe)

Davis Lazowski

Harvard

Title: Lie Algebras and Quiver Representations 

Abstract: This talk will survey methods of recovering information about a Lie algebra from the quiver representation theory of its finite or affine ADE Dynkin diagram.

William Norledge

Penn State University

Title: The Combinatorial Lie Coalgebra of Permutohedral Cones

Abstract: We study permutohedral cones as real-valued functions on the chambers of the adjoint braid arrangement. The discrete differentiation of functions across hyperplanes equips the linear span of permutohedral cones with the structure of a combinatorial Lie coalgebra. Using this Lie structure, we show that permutohedral cones satisfy the Steinmann relations of axiomatic quantum field theory, and we give an algorithm for expressing any function which satisfies the Steinmann relations in terms of permutohedral cones. We show that the universal enveloping coalgebra is the dual Hopf algebra of the braid arrangement, which is a combinatorial analog of the shuffle algebra.

Yunxiang Ren

Harvard

Title: Locally compact planar algebras

Abstract: For a finite group G, it is well-known that the monodical category of G-graded vector spaces and that of the representations of G are Morita equivalent. In this talk, we introduce locally compact planar algebras with reflection positivity. Within this framework, we can generalize the two monodical categories for locally compact groups. Moreover, we formalize and prove their Morita equivalence for compact groups. Our definition and proof also work for locally compact groups R^n. It requires a deep understanding of the analytic properties, particularly the uncertainty principle, to prove this Morita equivalence for locally compact groups and locally compact quantum groups in general. Discrete or compact planar algebras are closely related to infinite-index subfactors. 

Michele Tienni

Harvard

Title: Algebraic and Topological Structures of Subfactors

Abstract: It is well-known that a (finite depth) subfactor is associated to a fusion category and a planar algebra. In this talk we will explore how these two algebraic structure give rise to TQFTs in dimension 2+1 and 1+1 respectively.