Disclaimer: Handwritten notes posted here are what I used for the actual talk, but they were never intended to be seen by the public: most contain unreferenced results, and likely contain errors.
These talks can potentially expose you to ideas known to the State of California to cause confusion.
Higher Entropy
Symposium on Categorical Semantics of Entropy ("scsy"); CUNY graduate center; May 13, 2022
N=2 SU(3) Super-Yang-Mills possesses a rather unexpected property: the number of BPS states of mass less than M grows exponentially with M; moreover, the states contributing to this large growth in the density of states represent bound states that become arbitrarily large in size.
We will discuss the techniques used to derive this result (which may generalize far beyond N=2 SU(3) SYM)—focusing on the machinery of spectral networks—and some possible consequences.
Other Miscellaneous Talks
A Probability Talk That Spaces Out
Arizona State University; Differential Geometry and Control Systems Seminar; Jan. 18, 2018.
That guy who retired at 25 to Sedona to practice reading auras of upside-down cerebral crystal nudists practicing cannabis yoga might have accidentally said something relevant to physics: space can emerge from the connectedness of things.
I will describe how a topological space emerges from a question about independence of random variables; the topology/geometry encodes information about correlations.
This story is the classical version of a quantum mechanical one, and is inspired by research into the relationship of entangled states and non-trivial geometries of space-time.
Just don't encourage that guy.
(Dr.) Strangeduality or: how I learned to stop dozing off and learned to love (the) Boolean algebras
Arizona State University; Differential Geometry and Control Systems Seminar; Nov. 4, 2016.
Boolean algebras: your grandma has been hyping them up since the 1930's.
But you're still not convinced that you can afford the quantity coffee that would keep you awake for her logic lectures, in-between her snoring sessions on the couch.
That is, until you hear about Stone duality: every Boolean algebra lives a double life as an oddball topological space.
It doesn't get much attention, but Stone duality is part of the larger theme of the duality between algebras and spaces.
During my talk, I'll channel your grandma to describe Stone duality and, time-permitting, it's relation to measure theory as well as how it relates to other duality theorems.
Morse(t) I listen to this Talk?
Arizona State University; Differential Geometry and Control Systems Seminar; Nov. 11, 2015.
1982: Freddie Mercury is still the lead vocalist for Queen and Ed Witten provides a physical interpretation of the Morse-Smale complex in "Supersymmetry and Morse Theory".
I'll give a colloquium-style overview of this historic paper (from a modern, retrospective viewpoint); outlining how de Rham cohomology of a compact smooth manifold can be extracted from the study of the (supersymmetric) quantum mechanics of a point particle living on that manifold, and how Morse homology arises as a "classical limit" of this theory.
Such ideas formed the basis for the development of Floer homology, among other things.
Familiarity with physics terminology will be useful, but not necessary; similarly an appreciation for synthesizers and reverence of Reagan are not required.
Functional Equations and DT-invariants from Spectral Networks: Revenge of the m-herds
Emphasis Year Workshop on Rep. Theory, Integrable Systems, and Quantum Fields (Northwestern); May 14, 2014.
Nowadays the idea of entanglement being related to non-trivial geometries or topologies is about as surprising as a solitary carrot.
Typically global/topologically non-trivial data can be extracted from a (co)homology theory.
I will describe some ideas and partial results relating to why (multipartite) entanglement may be encoded in a homology theory and the dream of extracting entanglement entropy (more accurately: mutual information) as the Euler characteristic of a chain complex.
This talk will be just linear algebra, not an attempt to put you to sleep with a jargon lullaby.
Big Machinery in Quantum Mechanics: An Intro to Cohomology as a Tool for Quantifying Multipartite Entanglement.
In the futuristic year 2017 Piscataway, NJ is a depressing metropolis filled with urban decay. An ex-Cohomological cop (Harrison Ford) is pulled out of retirement to track down obstructions to the calculation of expectation values in terms of local quantities, caused by the presence of entangled states that came to the planet Earth in search of non-local correlations. With the help of C*-algebra representation theory of Gelfand, Naimark, and Segal, he closes in on a chain complex outputting obstructions to factorizability in the form of tuples of multi-body non-locally correlated operators---but his hatred for numerical valued quantities is called into question when he realizes the Euler characteristic is related to (q-deformations of) multivariate mutual information.
Quantum Chern Simons
West Coast Algebraic Topology Summer School (UBC); July 10, 2014.
This was a talk for a seminar on Derived Algebraic Geometry co-organized by Richard Derryberry and Aaron Royer. Details and references can be found here.
Bethe Subalgebras, counting, and integrability
Geometry and String Theory Seminar (UT Austin); March 20, 2013.
We will continue Pavel's discussion from last time, fitting the XXX spin chain into a larger set of models with a Yangian-module structure. Integrability will follow by introducing a commutative subalgebra of the Yangian and outsourcing the Hamiltonian counting problem to a local elementary school.
This one weird trick has algebraic functions generating Donaldson-Thomas invariants from home! Sit in this talk to see why!
Kansas State Mathematics M-seminar; November 4, 2014
It is a result of Kontsevich and Soibelman that generating functions for Donaldson-Thomas (DT) invariants associated to the m-Kronecker quiver are algebraic functions over the rationals. Using a technique from supersymmetric physics-- "spectral networks" -- one can see algebraicity of such functions directly and construct the associated algebraic equations in an algorithmic manner. We will discuss some interesting corollaries of these algebraic equations-- namely asymptotics on DT-invariants and Euler-characteristics of (Kronecker) quiver moduli-- and (if time allows) give a brief overview of the spectral network technique.
Fourier-Mukai: A perspective from the village idiot
Student Seminar on Geometric Langlands (UT Austin); February 17, 2015
I will attempt to convey a superficial understanding of the Fourier-Mukai transform (in particular for abelian varieties).
This is the third talk in a series on GL_1 Geometric Langlands.