The availability of very large datasets and the striking progress in artificial intelligence are revolutionizing the way scientists approach their disciplines. The deployment of state-of-the-art techniques in machine learning and statistical inference to study large datasets is leading to unprecedented discoveries and narrowing the gap between physics, mathematics, biology, computer science, and statistics. Despite this practical success, little is known about the general principles governing neural networks learning and dynamics and the geometry of data manifolds. The Simons Program on “Neural networks and the Data Science Revolution” will bring together researchers from the theoretical physics, artificial intelligence, and computational neuroscience communities to discuss foundational and theoretical aspects of neural networks, and highlight challenges and opportunities in their application to specific open problems along an axis of topics encompassing string theory, machine learning, big data science, and brain science. The program will begin with a workshop on the interface between theoretical physics, geometry and data science, with a focus on synthetic datasets arising in the classification of manifolds and knots, quantum and statistical field theories, and vacuum configurations of string theory. The program will end with a workshop on the physics of neural circuits, discussing ways to bridge the gap between neural network models and the large experimental datasets nowadays available in neuroscience.

This program will also be hosting a workshop: Physics of neural circuits and network dynamics: January 27-31, 2020

]]>The goal of the program is to bring together mathematicians and physicists working on various aspects of renormalization in dynamical systems. The idea of Renormalization group emerged in Quantum Field Theory. Later, in the 1960s, it became a major tool in Statistical Mechanics in analysis of phase transitions and critical phenomenon. One can say that the ideas of renormalization group have revolutionized the field. This development culminated in Wilson’s expansion based on his ideas on intrinsic relation between physical parameters in different scales.

In the 1970s the renormalization ideology was transferred to Dynamics in the context of Universality discoveries by Feigenbaum, Coullet and Tresser, and has since become one of the most powerful tools of understanding small scale structure of a large variety of systems. It has become particularly well (and rigorously) developed in the Conformal context, in particular, in the geometric problems related to the celebrated MLC Conjecture on the local connectivity of the Mandelbrot set.

Today, the renormalization ideas have penetrated deeply into many areas of Mathematics and Physics, but an explicit relation between various areas often remains elusive. One of our goals is to look for a unifying approach that would cover various manifestations of the renormalization.

In the first part of the program we will focus on conformal aspects that would include small scale structure of the dynamical and parameter loci of conformal dynamical systems, small scale properties of Brownian motions and harmonic measures, SLE, random conformal welding, and other relevant systems. We shall also discuss other aspects of dynamical renormalization, including related area of renormalization for quasi-periodic Schr ̈odinger operators.

In the second part, we will explore renormalization in various physical situations: QFT, fluid dynamics, and statistical mechanics, including KPZ phenomenon, with the emphasis on the underlying stochastic mechanisms responsible for the statistical

scaling invariance.

There are two workshops associated with this program: Analysis, Dynamics, Geometry and Probability: March 2-6, 2020 and Many Faces of Renormalization: June 1-5, 2020

]]>Organized by: Luis Alvarez-Gaume, Simeon Hellerman, Domenico Orlando, and Susanne Reffert

QFT is the basic paradigm for the description of condensed matter physics and high-energy particle physics. Despite decades of research, strongly coupled regimes of QFT are largely inaccessible to analytic methods, even though important progress has come from the conformal bootstrap and non-Lagrangian methods.

This program will be dedicated to exploring the asymptotic directions of the space of amplitudes in strongly coupled QFT, particularly the case of large quantum number under an internal and/or rotational symmetry. We will make use of many different methods to explore such limits, including the conformal bootstrap, Monte Carlo simulations, supersymmetric localization and recursion relations, and the most recently-applied method, the use of effective field theories that describe the large-charge limit of strongly interacting systems, with

the total charge as a loop-suppressing parameter. Working in sectors of large global charge allows us to perform a perturbative expansion for strongly coupled theories without any small parameters, with the

inverse of the total charge itself, as the perturbation parameter suppressing quantum fluctuations and unknown terms in the effective action.

Another aim of the program is to better understand the connections between the different approaches to large quantum number physics that are emerging. In particular understanding the connection between the study of systems at large spin via the light-cone bootstrap and the large-charge approach would be of major importance.

The study of the large-quantum-number regime is a subject still in its infancy, with a number of rapidly developing directions and applications. Apart from the issues explicitly mentioned here, the LQN expansion appears to be a special case of a more general set of phenomena in which strongly coupled systems self-classicalize in various limits of “observable space”. Such examples include resurgence theory, Regge theory, supersymmetric localization, and the macroscopic limit of quantum thermodynamics. A key goal of the program is to discover the most natural context containing the large quantum-number expansion in its current form.

We hope that this program aimed to develop these connections and applications will meet with wide interest!

]]>Organized by: Jennifer Cano, Dominic Else, Andrey Gromov, Siddharth Parameswaran, and Yizhi You

Topological phases of matter are a long-standing subject of interest in the condensed matter community, and increasingly relevant to issues in high-energy physics. A topological phase is traditionally defined to be one which is “non-trivial” (cannot be deformed to a trivial insulator without a phase transition), but where the non-triviality cannot be ascribed merely to spontaneous symmetry breaking. Another interpretation is that the word “topological” in “topological phase” is supposed to suggest that the low-energy physics of this phase (that is, the infrared fixed point controlling the phase in the renormalization group sense) is “topological”, meaning that it can be defined on any background space-time and is sensitive only to the background topology. This program will be devoted to exploring the connections, and tension, between these two distinct notions of “topological”. The program will focus on encouraging cross-fertilization between three rapidly developing, interconnected research areas:

The past decade witnessed an explosion of activity in research on symmetry-protected topological phases. Traditionally, the symmetry group in question is either time-reversal or an internal, “on-site’’ symmetry. However, in the past several years, topological phases protected by spatial (crystalline) symmetries have emerged. Unlike topological phases described by topological quantum field theories, these phases a priori can only be defined on the particular background space-time on which the crystalline symmetries act. Consequently, the traditional approach of topological quantum field theory has to be supplemented with additional information, such as the properties of lattice defects,

which are a kind of geometric response of the system. A generic bulk-edge correspondence for these states is absent because, unlike on-site symmetries, crystal symmetries are typically broken on boundaries. In some instances, the usual correspondence is replaced by a “higher-order’’ correspondence between the d-dimensional bulk state and symmetry-preserving (d-2)-dimensional edges. Developing a systematic theory of such phases, and discovering physical examples, is an important open question.

Several years ago, a conceptually new type of gapped phases was discovered. These phases are known as fracton phases, due to the presence of topologically non-trivial excitations that can only move on lower-dimensional submanifolds, or cannot move at all. Like spatial symmetry-protected topological phases, fracton phases challenge our notion of what “topological order” means, because the low-energy theory depends on some non-topological features of space. In fact, these phases also appear to have a very complicated relationship with the geometry of the space where they reside. Intuitively, the exotic features of the excitations in these models can be viewed as stemming from a non-trivial interplay between translation symmetry and topological order. Alternatively, these phases can be viewed as higher-rank gauge theories obtained by gauging the subsystem symmetries — the symmetries which act along lower-dimensional subspaces. Fractons have attracted a broad interdisciplinary interest due to their potential relationship to lattice gauge theory, quantum computation and memory, elasticity, glassy dynamics and emergent gravity in condensed matter.

Fractional quantum Hall states exhibit a non-trivial geometric response. Since FQH phases are liquids, with continuous rotational and translational symmetries, these geometric properties are intuitively related to those of topological phases with spatial symmetries. Developing a rigorous unified approach to the geometric properties of FQH phases on equal footing with the topological phases with crystalline symmetrie will be another focus of the program. Separately, the breaking of discrete and/or continuous crystalline symmetries underpins a remarkable class of unconventional nematic quantum Hall liquids — with unusual properties analogous to those of liquid crystals familiar from classical soft condensed matter physics. Concurrently, there has been recent progress on the FQH physics that goes beyond the topological order paradigm. This builds on pioneering work by Haldane, who has argued that certain collective modes supported by a FQH liquid can be described by a fluctuating geometry. This area of study has recently witnessed further progress simultaneously on three fronts: in terms of trial states, Matrix Models, and effective theory. Quantitative properties of these modes are related to the geometric responses.

Geometry plays a central role in these topics, but currently there is no coherent picture that unifies them. Nevertheless, there are some tantalizing hints of possible close connections. We expect that the program will lead to identification of the common themes and cross-fertilization of these fields.

The program is accompanied by a workshop New directions in topological phases: from fractons to spatial symmetries, to be held between April 13-17, 2020.

]]>Organized by: Boris Altshuler, Anatoly Dymarsky, Lea Santos, and Jacobus Verbaarschot

Dynamics of quantum many-body systems is attracting extensive attention across different fields of physics: theoretical and experimental condensed matter, AMO physics, high energy theory, quantum information, and others. Remarkable progress has been achieved in the past years understanding thermalization, or lack thereof, of isolated quantum systems, paving new ways to quantitatively probe quantum chaos and connecting many-body systems to quantum gravity. However, this progress is still fragmentary and there are very important basic questions which are not well understood. The program will address, among others, the following questions.

1) What are the characteristic timescales of relaxation and thermalization for many-body quantum systems? The interplay between different timescales and their manifestation in various observables is not clear. One highly contested question is the behavior of the so-called Thouless time, which marks the scale of applicability of Random Matrix Theory to describe correlations of energy spectrum. For the disordered single-particle systems the latter was identified as the timescale of diffusion. For various many-body systems (local 1D systems, Floquet, SYK model) the Thouless time was shown recently to behave very differently.

2) What is the relation between out-of-time-ordered correlators (OTOC) and the timescales marking the onset of universal behavior? This question is in its infancy with very few papers making explicit connection between dynamics of thermalization and quantitative manifestations of quantum chaos.

3) What constitutes ergodic behavior for quantum systems? Ergodicity for quantum systems is often equated with Eigenstate Thermalization, which is only concerned with the value of observables at equilibrium. Can something be said about the approach toward and fluctuations around the asymptotic value?

4) What is the scope of applicability of Random Matrix Theory to many-body systems? Random matrices are known to describe various aspects of quantum chaotic systems, e.g. statistics of energy levels. More recently, there have been several applications of random matrices to describe the late time behavior of various observables, including survival probability, correlation functions, Euclidean partition function, etc. An important question would be to understand which dynamical quantities and to what they extent may exhibit universal behavior and be described by the Random Matrix Theory.

The program is accompanied by a workshop Applications of Random Matrix Theory to many-body physics, to be held between September 16-20, 2019, which will be focused on universal behavior of many-body systems.

]]>Organized by Lara Anderson and Laura Schaposnik

Hitchin systems have found remarkable applications in many different areas of mathematics and physics. In particular, Hausel and Thaddeus related Higgs bundles to mirror symmetry, and in the work of Kapustin and Witten, Higgs bundles were used to give a physical derivation of the geometric Langlands correspondence. Moreover, in the last year many advances were made in the study of branes inside the moduli space of Higgs bundles, opening paths to promising new research directions.

The overall goal of the program is to bring together mathematicians and physicists working on areas close to Higgs bundles: the four interrelated themes of the program are: (A) Higgs bundles and geometric structures; (B) Higgs bundles and representation theory; (C) Higgs bundles and mirror symmetry; (D) Higgs bundles and singular geometry. Both (A) and (B) are closely interrelated, and concern in particular the study of the geometry of the moduli space, and the understanding of limits and geometric structures on this space. In particular, since many geometric structures are parametrised by the Hitchin fibration, and limits are taken in this context too, we hope that the workshop will lead to the understanding of these questions in the context of (B) and (C). Indeed, physicists have long been interested in the appearance of the Coulomb Branch (an Abelianized limit of an associated physical gauge theory), the Hitchin base, and Seiberg-Witten curves, the spectral covers of Higgs bundles. Finally, there should be algebraic connections between the above limits and Donaldson-Thomas type of invariants and wall-crossings, either in the Hitchin base or in the total space, and hence algebraic geometers can develop useful tools from this perspective. Among possible directions related to (C) is the study of the moduli space of Higgs bundles and its branes through the Hitchin fibration, appearing in mathematical physics and Langlands duality. It is from this perspective that mathematical physicists and theoretical physicists will play a key role in bringing attention to the most urgent problems in the area, and also for developing new methods centred around spectral data of Higgs bundles. Moreover, since one can use equivariant characters of certain coherent sheaves on the Hitchin moduli space to further understand the physics proposal of dualities between branes, interactions with the algebraic geometers will be of significant importance during the Workshop: Holomorphic Differentials in Mathematics and Physics February 4-8, 2018. Finally, (D), which links Calabi-Yau and Hitchin Integrable systems, is another rapidly developing area that we hope to highlight. Recent work has brought to light a remarkable isomorphism between (non-compact) Calabi-Yau integrable systems and those of Hitchin. Extensions of this correspondence to compact Calabi-Yau geometry have recently been found in the context of string compactifications but many open questions remain. In addition, links between Hitchin and Calabi-Yau moduli spaces have the potential to shed light on a number of important physical questions regarding string vacuum spaces. We expect the program to have a strong broader impact, emphasizing the presence of women and other minorities working in the area, and promoting the area amongst younger researchers. Please note that there will also be two additional workshops held during this program:

Workshop: Challenges at the Interface of Hitchin Systems and String Theory: March 18 – 22, 2019

Graduate Summer School on the Mathematics and Physics of Hitchin Systems: May 27 – 31, 2019

This program has two weekly meetings:

** A Journal Club every Friday from 12-1 (brown bag lunch) in room 313. Co-ordinated by Paul Oehlmann.

** A weekly thematic seminar every Thursday from 1-2 in 313. Full schedule can be found below (for exceptional dates please see the calendar).

Date | Speaker | Title |

Jan 24 | Introductory Talk | |

Jan 31 | Paul Oehlmann | 6D Discrete Charged Superconformal Matter from F-theory on Quotient Threefolds |

Feb 8 | Florent Schaffhauser | Higher Teichmüller spaces for orbifolds |

Feb 14 | Andy Sanders | Opers, flat connections, and variations of Hodge structure |

Feb 21 | Fabio Apruzzi | Non-flat elliptically fibered Calabi-Yau threefolds in F/M-theory and phases of five-dimensional superconformal field theories |

Feb 28 | Andre Lukas | Line Bundle Cohomology and Machine Learning |

March 8 | Katrin Wendland | On Invariants shared by Geometry and Conformal Field Theory |

March 14 | Florian Beck | Hyperholomorphic Line Bundles and Energy Functionals |

March 28 | Motohico Mulase | Gaiotto’s conformal limit of Higgs bundles and its geometry |

March 29 | Marina Logares | A TQFT on representation varieties |

April 4 | Ian McIntosh | Minimal surfaces in hyperbolic spaces and their Higgs bundles |

April 5 | Vishnu Jejjala | Experiments with Machine Learning in Geometry & Physics |

April 11 | Marcos Jardim | Branes on moduli spaces of sheaves |

April 18 | Philip Boalch | Higgs bundles on the affine line and wild surface groups |

April 19 | Paul Ziegler | p-adic integration for the Hitchin system |

April 25 | Andres Collinucci | |

May 1 | Nigel Hitchin | |

May 9 | Marco Gualtieri | |

May 23 | Fernando Marchesano | |

May 24 | Misha Verbitskiy | |

June 6 | John Loftin | |

June 7 | Ljudmila Kamenova | |

The purpose of this interdisciplinary program is to investigate connections between string theory, automorphic forms, mock modular forms and beyond. Automorphic representations form a cornerstone of the Langlands program, while at the same time playing a crucial role in understanding the structure of scattering amplitudes in string theory. Mock modular forms appear ubiquitously in studying quantum black holes and new moonshine phenomena. Investigations of string amplitudes have also given rise to new objects called “modular graph functions” whose mathematical structure is only beginning to emerge.

Higher-curvature corrections in string theory satify unconventional (‘Poisson-type’) differential equations that call for a generalization of the notion of automorphic form that goes beyond the current mathematical framework. The program will be dedicated to the exchange of idea and perspectives in these diverese fields, with the aim of breaking new ground in both physics and mathematics. This is a follow-up to the program “Automorphic forms, mock modular forms and string theory”, which ran in the fall of 2016. During the first week of the program there will be a kick-off workshop with the same title. For more information see: Automorphic Structures in String Theory.

Program Application Deadline: November 18, 2018 (or when event is at maximum capacity). Applicants will be contacted soon after this date

Shuttle Bus Schedule ]]>Since Heisenberg’s and Jordan’s “matrix formulation” of quantum mechanics about one hundred years ago, theoretical quantum physics has been a major incentive for mathematical research into operator algebras. The stimulus from physics has helped the development of this field, founded by von Neumann with the motivation to clarify the mathematical structure of quantum mechanics. At the same time, theoretical quantum physics has benefited tremendously from new results in operator algebras, making the link between the two fields one of the most fruitful interdisciplinary research programs of the 20th and 21st century.

Nowadays, both the quantum physics and operator algebras groups have grown into large and diverse communities with many subareas and new links to other fields. For example, in the last decades, we have witnessed connections between operator algebras and knot theory, free probability, K-theory, and further topics. In quantum physics, the most active fields that have a close link to operator algebras are quantum field theory and quantum statistical mechanics. More recently, also connections between quantum information theory and operator algebras began to emerge.

Despite its enormous phenomenological success, quantum field theory still contains many challenging open questions, for example regarding its proper place within mathematics in general, as well as the mathematical status of various field theoretic models, including in particular the types of models quantitatively describing the interactions of elementary particles. It is widely expected that new mathematical insights are needed in order to make further progress on these fronts.

The aim of this program is to bring together established experts and young researchers in these fields, focusing on operator-algebraic approaches to quantum field theory, subfactor theory, and quantum information theory.

There will also be a workshop associated with this program: Operator Algebras and Applications from June 17-21, 2019.

]]>

Organized by: Nathan Haouzi, Vladimir E. Korepin, Sergei L. Lukyanov, Nikita A. Nekrasov, Samson Shatashvili, and Alexander B. Zamolodchikov

Weekly program seminars are held on Mondays at 11:30am and Fridays at 11:00am. For the full upcoming schedule please visit our calendar: http://scgp.stonybrook.edu/calendar/full-calendar

Integrability is a traditional area of mathematical physics. For 1+1 dimensional field theory the inverse scattering method is an appropriate method. It is based on the zero-curvature representation. In quantum theory it leads to the Yang-Baxter algebras and quantum groups. These are useful for description of the connections between the supersymmetric gauge theories and quantum integrable systems. Bethe/gauge-correspondence [based on BPS/CFT-correspondence] relates Bethe ansatz solvable spin chains to the twisted chiral rings of the gauge theories with two dimensional Poincare supersymmetry. The planar maximal super-Yang-Mills theory in four dimensions is related to quantum and classical integrable systems [at the level of the anomalous dimensions of local operators and scattering amplitudes]. An important phenomena is the ODE/CFT correspondence. Many deep properties of representations of Yang-Baxter algebras in integrable Conformal Field Theories can be encoded in the monodromies of certain linear Ordinary Differential Equations. This can be extended to massive Integrable Quantum Field Theories: the ODE/IQFT correspondence. A related problem is the application of the ODE/IQFT method to non-linear sigma models, including the supersymmetric ones. The sigma model associated with the AdS side of the correspondence for the N = 4 theory was argued to be integrable. It is natural to start with simpler models, like principal chiral models, O(n) models, and such. Integrable structures of such symmetric models correspond to the Yangian reductions of the Yang-Baxter algebras. Previous experience with the ODE/IQFT approach shows that this reduction leads to a subtle limiting case on the ODE side of the correspondence. An intriguing generalization lies in study of a two-parameter deformation of the general principal chiral. In the SU(2) case, coincides with the Fateev sigma model. We believe it is the best testing ground for the ODE/IQFT approach in the sigma model context. The ultimate goal, is to extend the approach to sigma models of direct interest AdS/CFT duality and in condensed matter physics.

Spin chains are in the center of high energy physics, statistical mechanic, condensed matter, quantum optics and quantum information. Bethe Ansatz and Yang-Baxter equations helped to construct multiple examples. Some spin chains are solvable in a weaker cense: only the ground state can be described analytically. For example Fredkin model has high level of quantum fluctuations.

Another important development in statistical mechanics is the failure of van Hove theorem. The most notable case is six vertex model.

The goal of the program is to connect these traditional problems and methods to the recent developments, notably complex saddle points, resurgence, and the Bethe/gauge correspondence.

]]>Organized by: Vasily Pestun and Maxime Zabzine

The program will be focusing on the development of localization techniques in quantum field theories and its applications. In particular we want to concentrate on the developments in the field since 2007.

The main idea of different localization formulas is that the specific finite dimensional integral can be evaluated exactly by summing up over fixed point contributions. The proof of the Berline-Vergne-Atiyah-Bott formula can be recasted in terms of supergeometry and finite dimensional version of supersymmetry. This allows to discuss the possible generalizations of the Berline-Vergne-Atiyah-Bott formula to the infinite dimensional setup.

Since the discovery of localization formula in math there have been different attempts to implement it in infinite dimensional setting, in particular in the context of path integral. Localization of the path integral to various interesting finite-dimensional geometrical moduli spaces was pioneered by Witten (1982) and later was applied to two-dimensional topological sigma model, four dimensional topological gauge theory, two-dimensional Yang-Mills theory. Further development on supersymmetric localization is related to the calculation of Nekrasov’s partition function, or equivariant Donaldson-Witten theory based on earlier works on the equivariant integration of the hyper-K\”ahler quotients by Losev-Moore-Nekrasov-Shatashvi

As for the non-topological supersymmetric theories, the localization technique, which captures the essential physical phenomena such as a $\beta$-function for running coupling constant, was developed in 2007 by the first organizer to compute the full partition function and the expectation value of supersymmetric Wilson loops on four dimensional sphere for N=2 supersymmetric field theories. After this paper it has been explosion of the works dealing with the localization of different supersymmetric theories in different dimensions. This explosion of exact results in quantum field theory led to many non-trivial developments in quantum field theory: the check of non-perturbative dualities, the further checks of AdS/CFT correspondence, the new look at the supersymmetric theories on the curved manifold and better analytical and algebraic understanding of the exact results (e.g., the partition functions). This also led to the new mathematical results, e.g. the new ways of calculating the genus 0f Gromov-Witten invariants from Kahler potentials computed by the partition functions of gauge theories on a two-sphere. The program will concentrate on these developments during last 10 years.

Program Application Deadline: October 30, 2017 (or when event is at maximum capacity). Applicants will be contacted soon after this date.

Date | Time | Title | Speaker | Location |

Mon. Jan 18 | 10:00 am | 2d (2,2) ADE dualities and branches | Bruno Le Floch | SCGP 313 |

Date | Time | Title | Speaker | Location |

Tues. Jan 23 | 10:00 am | Coulomb branch operators from supersymmetric localization | Silviu Pufu | SCGP 313 |

Date | Time | Title | Speaker | Location |

Mon Jan 29 | 10:00 am | 4D exotic Donaldson-Witten theory from 5D super-Yang-Mills | Jian Qiu | SCGP 313 |

Tues. Jan 30 | 10:00 am | T[U(N)] duality webs: mirror symmetry, spectral duality and gauge/CFT correspondences | Sara Pasquetti | SCGP 313 |

Wed. Jan 31 | 11:00 am | 5-Brane Webs of SCFTs for N=1 G2 Theories | Kimyeong Lee | SCGP 313 |

Thurs. Feb 1 | 11:00 am | Localization on circle bundles | Brian Willett | SCGP 313 |

Fri Feb 2 | 11:00 am | Exotic equivariant cohomology of loop space, localisation and T-duality | Mathai Varghese | SCGP 313 |

Date | Time | Title | Speaker | Location |

Mon. Feb 5 | 11:00 am | 3d N=2 gauge theories on Seifert manifolds (or: how to localize on the Poincaré homology sphere) | Cyril Closset | SCGP 313 |

Tues. Feb 6 | 11:00 am | Fractional quiver gauge theory | Taro Kimura | SCGP 102 |

Wed. Feb 7 | 11:00 am | Localization on compact spaces, q-W algebras and modularity | Fabrizio Nieri | SCGP 313 |

Date | Time | Title | Speaker | Location |

Mon. Feb 12 | 11:00 am | 6d strings and exceptional instantons | Seok Kim | SCGP 313 |

Tues. Feb 13 | 11:00 am | Supersymmetric vortex defects in two dimensions | Takuya Okuda | SCGP 313 |

Wed. Feb 14 | 11:00 am | Between duality and symmetry in supersymmetric QFT | Shlomo Razamat | SCGP 313 |

Thurs. Feb 15 | 11:00 am | Joseph Minahan | SCGP 313 |