# Physics

## Combinatorial structures in perturbative quantum field theory

I will give an overview of a few places where combinatorial structures have an interesting role to play in quantum field theory and which I have been involved in to varying degrees, from the Connes-Kreimer Hopf algebra and other renormalization Hopf algebras, to the combinatorics of Dyson-Schwinger equations and the graph theory of Feynman integrals.

For other events in this series see the quanTA events website.

## The topology and geometry of the space of gapped lattice systems

Recently there has been a lot of progress in classifying phases of gapped quantum many-body systems. From the mathematical viewpoint, a phase of a quantum system is a connected component of the “space” of gapped quantum systems, and it is natural to study the topology of this space. I will explain how to probe it using generalizations of the Berry curvature. I will focus on the case of lattice systems where all constructions can be made rigorous. Coarse geometry plays an important role in these constructions.

## The Infinite HaPPY Code

I will construct an infinite-dimensional analog of the HaPPY code as a growing series of stabilizer codes defined respective to their Hilbert spaces. These Hilbert spaces are related by isometries that will be defined during this talk. I will analyze its system in various aspects and discuss its implications in AdS/CFT. Our result hints that the relevance of quantum error correction in quantum gravity may not be limited to the CFT context.

For other events in this series see the quanTA events website.

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## Conformal field theories and quantum phase transitions: an entanglement perspective

Quantum phase transitions occur when a quantum system undergoes a sharp change in its ground state, e.g. between a ferro- and para-magnet. I will present a remarkable set of transitions, called quantum critical, that are described by conformal field theories (CFTs). I will focus on 2 and 3 spatial dimensions, where the conformal symmetry is powerful yet less constraining than in 1 dimension. We will probe these scale-invariant theories via the structure of their quantum entanglement. The methods will include large-N expansions, the AdS/CFT duality from string theory, and large-scale numerical simulations. Finally, we’ll see that certain quantum Hall states, which are topological in nature, possess very similar entanglement properties. This hints at broader principles that relate very different quantum states.

For other events in this series see the quanTA events website.

## Integers in many-body quantum physics

Although integers are ubiquitous in quantum physics, they have different mathematical origins. In this colloquium, I will give a glimpse of how integers arise as either topological invariants or as analytic indices, as is the case in the so-called quantum Hall effect. I will explain the difficulties arising in extending well-known arguments when one relaxes the approximation that the particles effectively do not interact with each other in matter. Recent advances have made such realistic generalizations possible.

## From Euler to Born and Infeld, Fluids and Electromagnetism

As the Euler theory of hydrodynamics (1757), the Born-Infeld theory of electromagnetism (1934) enjoys a simple and beautiful geometric structure. Quite surprisingly, the BI model which is of relativistic nature, shares many features with classical hydro- and magnetohydro-dynamics. In particular, I will discuss its very close connection with Moffatt’s topological approach to Euler equations, through the concept of magnetic relaxation.

The Marsden Memorial Lecture Series is dedicated to the memory of Jerrold E Marsden (1942-2010), a world-renowned Canadian applied mathematician. Marsden was the Carl F Braun Professor of Control and Dynamical Systems at Caltech, and prior to that he was at the University of California (Berkeley) for many years. He did extensive research in the areas of geometric mechanics, dynamical systems and control theory. He was one of the original founders in the early 1970’s of reduction theory for mechanical systems with symmetry, which remains an active and much studied area of research today.

This lecture is part of the Centre Interfacultaire Bernoulli Workshop on Classic and Stochastic Geometric Mechanics, June 8-12, 2015, which in turn is a part of the CIB program on

Geometric Mechanics, Variational and Stochastic Methods, 1 January to 30 June 2015.

## From the Adinkras of Supersymmetry to the Music of Arnold Schoenberg

The concept of supersymmetry, though never observed in Nature, has been one of the primary drivers of investigations in theoretical physics for several decades. Through all of this time, there have remained questions that are unsolved. This presentation will describe how looking at such questions one can be led to the 'Dodecaphony Technique' of Austrian composer Schoenberg.

Jim Gates is a theoretical physicist known for work on supersymmetry, supergravity and superstring theory. He is currently a Professor of Physics at the University of Maryland, College Park, a University of Maryland Regents Professor and serves on President Barack Obama’s Council of Advisors on Science and Technology.

Gates was nominated by the US Department of Energy to present his work and career to middle and high school students in October 2010. He is on the board of trustees of Society for Science & the Public, he was a Martin Luther King Jr. Visiting Scholar at MIT (2010-11) and was a Residential Scholar at MIT’s Simmons Hall. On February 1, 2013, Gates received the National Medal of Science.

## General Relativity, Differential Geometry and Differential Equations; Stories From a Successful Menage-a-trois

**N.B. Due to problems with the microphone, the audio quality of this video is significantly lower than expected.**

It is well known that Einstein's general theory of relativity provides a geometrical description of gravity in terms of space-time curvature. Einstein's theory poses some fascinating and difficult mathematical challenges that have stimulated a great deal of research in geometry and partial differential equations. Important questions include the well-posedness of the evolution problem, the definition of mass and angular momentum, the formation of black holes, the cosmic censorship hypothesis, the linear and non-linear stability of black holes and boundary value problems at conformal infinity arising in the analysis of the AdS/CFT correspondence. I will give a non-technical survey of some significant advances and open problems pertaining to a number of these questions.

## An Octahedral Gem Hidden in Newton's Three Body Problem (2012 Marsden Memorial Lecture)

Richard Montgomery, University of California, Santa Cruz will deliver a talk entitled, "An Octahedral Gem Hidden in Newton's Three Body Problem." The lecture will take place on July 25, 2012 at the Fields Institute, as part of the conference on "Geometry, Symmetry, Dynamics, and Control: The Legacy of Jerry Marsden."

Richard Montgomery received undergraduate degrees in both mathematics and physics from Sonoma State in Northern California. He completed his PhD under Jerry Marsden at Berkeley in 1986, after which he held a Moore Instructorship at MIT for two years, followed by two years of postdoctoral studies at University of California, Berkeley.

Montgomery's research fields are geometric mechanics, celestial mechanics, control theory and differential geometry and is perhaps best known for his rediscovery - with Alain Chenciner - of Cris Moore's figure eight solution to the three-body problem, which led to numerous new 'choreography' solutions. He also established the existence of the first-known abnormal minimizer in sub-Riemannian geometry, and is known for investigations using gauge-theoretic ideas of how a falling cat lands on its feet. He has written one book on sub-Riemannian geometry.

The PIMS Marsden Memorial Lecture Series is dedicated to the memory of Jerrold E Marsden (1942-2010), a world-renowned Canadian applied mathematician. Marsden was the Carl F Braun Professor of Control and Dynamical Systems at Caltech, and prior to that he was at the University of California, Berkeley, for many years. He did extensive research in the areas of geometric mechanics, dynamical systems and control theory. He was one of the original founders in the early 1970s of reduction theory for mechanical systems with symmetry, which remains an active and much studied area of research today.

The inaugural Marsden Memorial Lecture was given by Alan Weinstein (University of California, Berkeley) in July of 2011 at ICIAM in Vancouver.

## Gauge Theory and Khovanov Homology

After reviewing ordinary finite-dimensional Morse theory, I will explain how Morse generalized Morse theory to loop spaces, and how Floer generalized it to gauge theory on a three-manifold. Then I will describe an analog of Floer cohomology with the gauge group taken to be a complex Lie group (rather than a compact group as assumed by Floer), and how this is expected to be related to the Jones polynomial of knots and Khovanov homology.