Schedule, Notes/Handouts & Videos

Chern-Simons invariants of knots have many applications in mathematics and in physics. Khovanov showed  in ‘99 that the simplest such invariant, the Jones polynomial, arizes as the Euler characteristic of a homology theory. The knot categorification problem is to find a general construction of knot homology groups, and to explain their meaning: what are they homologies of?

Mirror symmetry is another important strand in the interaction between mathematics and physics. Homological mirror symmetry, formulated by Kontsevich in ’94, naturally produces hosts of homological invariants. Sometimes, it can be made manifest, and then its striking mathematical power comes to fore. Typically though, it leads to invariants which have no particular interest outside of the problem at hand.

I showed recently that there is a vast new family of mirror pairs of manifolds, for which homological mirror symmetry can be made manifest. They do lead to interesting invariants: in particular, they solve the knot categorification problem.

Fracton order describes the peculiar phenomena that point excitations in certain strongly interacting systems either cannot move at all or are only allowed to move in a lower dimensional sub-manifold. It has recently been discovered in various lattice models, tensor gauge theories, etc. In this talk, I will discuss how the 2+1D Chern-Simons (CS) gauge theory can be used to provide new insights and explore new possibilities in 3+1D fracton order. 2+1D U(1) gauge theories with a CS term provide a simple and complete characterization of 2+1D Abelian topological orders. To study 3+1D fracton order, we extend the theory by taking the number of component gauge fields to be infinity. In the simplest case of infinite-component CS gauge theory, different components do not couple to each other and the theory describes a decoupled stack of 2+1D fractional Quantum Hall systems with quasi-particles moving only in 2D planes -- hence a fractonic system. More interestingly, we find that when the component gauge fields do couple through the CS term, more varieties of fractonic orders are possible. For example, they may describe foliated fractonic systems which extends the framework found in exactly solvable models. At the same time, we find examples which lie beyond the foliation framework, characterized by 2D excitations of infinite order and braiding statistics that are not strictly local.

In the first part of the talk we will review  Andreas Floer's construction, from the mid 1980's,  of 3-manifold invariants: applying the ideas of Morse Theory to the Chern-Simons functional. We will recall how these Floer homology groups  fit into a partial topological field theory and some of the many developments in this area over the past four decades.In the latter part of the talk we will discuss  variants of these ideas in higher dimensions and some of the analytical questions involving singularity formation which arise. 

In 1992 at Howard University, the first descriptionS of maximally extended supersymmetric Chern-Simons Theory were formulated. This initiated a research program combining SUSY with topology that in 2021 resolved a forty year-old puzzle about eleven dimensional supergravity. The discovery arc of this journey is presented in this talk.

The golden age of extreme gravitational astrophysics has just begun with the gravitational wave observations of merging compact binaries and time-resolved X-ray observations of rotating neutron stars. These observations carry signatures of the behavior of gravity in a regime in which this interaction is simultaneously strong, non-linear and dynamical. The search for deviations from Einstein's theory of general relativity in this uncharted extreme gravity regime had not been possible until now. In this talk, I will describe a few astrophysical signatures of dynamical Chern-Simons gravity, a classical extension of Einstein's theory that is well-motivated from string theory, loop quantum gravity and effective theories of inflation. I will show how this theory introduces a dynamical pseudo scalar that couples to the curvature of spacetime in a parity-violating way, thus avoiding solar system and binary pulsar constraints, while introducing signatures in the extreme gravity regime.

I will review work over the past few years on Chern-Simons theories in dimensions four, five and six, with applications to integrability and holography.

There have in recent years been a number results in number theory and arithmetic geometry suggesting the possibility of applying the ideas of physics, (topological) quantum field theory in particular, to difficult arithmetic questions and theories. This talk will give a brief indication of these ideas and suggest avenues for research.

We will talk about link homology theories that categorify the Jones polynomial and related quantum invariants and their relation to a variety of structures in algebra and geometry.

I will explore a toy model in which spontaneous symmetry breaking – acting on the level of operators (not states) – can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. The spirit is to look for the simplest context from which the rich physics of our universe might have emerged.  The talk is based on joint work with Modj Shokrian Zini,  see arXiv:2011.05917 and arXiv:2108.12709

In this talk we will explore the improvisational and musical nature of mathematics and physics.

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Stephon Alexander is a theoretical physicist, computational physicist, and author whose work is at the interface between cosmology, particle physics, quantum gravity and music cognition.  His expertise lies in constructing new theories and models of the early universe that have predictions for the universe at present, such as dark energy and dark matter.  He also combines mathematics and tools from theoretical physics into machine learning, the geometry and cognition of musical perception, biophysics, signal processing and computational algorithms. 

Alexander is a Professor of Physics at Brown University, and the President of the National Society of Black Physicists.  Alexander is also the Executive Director of the Harlem Gallery of Science.  He had previous appointments at Stanford University, Imperial College, Penn State, Dartmouth College and Haverford College.  Alexander is a specialist in the field of string theory and cosmology, where the physics of superstrings are applied to address longstanding questions in cosmology. In 2001, he co-invented the model of inflation based on higher dimensional hypersurfaces in string theory called D-Branes. In such models the early universe emerged from the destruction of a higher dimensional D-brane which ignites a period of rapid expansion of space often referred to as cosmic inflation.

In his critically acclaimed book, The Jazz of Physics, Alexander revisits the ancient interconnection between music and the evolution of astrophysics and the laws of motion. He explores new ways music, in particular jazz music, mirrors modern physics, such as quantum mechanics, general relativity, and the physics of the early universe. He also discusses ways that innovations in physics have been and can be inspired from “improvisational logic” exemplified in Jazz performance and practice. Alexander is also a professional touring jazz musician.