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.
Recorded Talks
Higher Entropy
Link to Video on YouTube
Handwritten Notes: PDF
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Is the frowzy state of your desk no longer as thrilling as it once was?
Are numerical measures of information no longer able to satisfy your needs?
There is a cure!
In this talk we'll learn about: the secret topological lives of multipartite measures and quantum states;
how a homological probe of this geometry reveals correlated random variables;
the sly decategorified involvement of Shannon, Tsallis, Réyni, and von Neumann in this larger geometric conspiracy;
and the story of how Gelfand, Neumark, and Segal's construction of von Neumann algebra representations can help us uncover this informatic ruse.
So come to this talk, spice up your entropic life, and bring new meaning to your relationship with disarray.
The Joy of Watching your BPS States Grow Up.
Link to Video on YouTube
Typed Notes: PDF
Handwritten Notes: PDF or tar.gz (PNG images)
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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 External/Invited Talks
Higher Information: The untold topological secrets of information
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In the futuristic year 2023, an ex cohomological cop (Harrison Ford) is pulled out of retirement to track down obstructions to the factorizability of multipartite measures: entangled states and their classical counterparts...joint probability measures.
A vast conspiracy is uncovered: associated to every multipartite measure is an emergent space whose topology encodes non-local correlations between various subsystems.
Probing this space with tools with homology, homotopy, and category theory reveals evidence that entropy and mutual information are Euler characteristics: puppets in a larger informatic ruse.
Join as we slowly uncover pieces of the big picture, unmasking filthy rich measures of shared information higher up on the food chain (complex).
A Probability Talk That Spaces Out
Handwritten Notes: PDF
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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
Handwritten Notes: PDF or tar.gz (PNG images)
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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?
Handwritten Notes: PDF or tar.gz (PNG images)
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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.
Handwritten Notes: PDF or tar.gz (PNG images)
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(Apparently I didn't make an abstract for this talk; feel free to invent your own.)
Quantum Chern Simons
Handwritten Notes: PDF or tar.gz (PNG images)
This one weird trick has algebraic functions generating Donaldson-Thomas invariants from home! Sit in this talk to see why!
Handwritten Notes: PDF or tar.gz (PNG images)
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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.
Internal (Local) Talks
Homological Toolkit for the Quantum Mechanic Part III
Rutgers NHETC Subgroup Meeting; February 14, 2019.
Handwritten Notes: PDF
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This talk abides by the Hollywood rule: it's not dead until you see the body.
We will recap the main ideas of the previous talks (so don't worry if you haven't seen them or cannot remember much).
We'll introduce presheaves of vector spaces from the data of a multipartite state and show how the associated Cech cochain complexes are the complexes predicted from previous talks.
The cohomology of such complexes outputs tuples of operators that exhibit non-local correlations caused by entanglement.
If there is time we will make brief mention about the classical analogues of this construction, where the cohomology is the cohomology of an actual space.
Why waste time seeking tinder hookups on Valentine's day when you can \hookrightarrow yourself into the NHETC common room?
Homological Toolkit for the Quantum Mechanic Part II
Rutgers NHETC Subgroup Meeting; October 11, 2018.
Handwritten Notes: PDF
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This is the second talk in a three part talk series.
See the abstract for Part I below.
Homological Toolkit for the Quantum Mechanic Part I
Rutgers NHETC Subgroup Meeting; September 27, 2018.
Handwritten Notes: To Appear (Available on Request)
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This is the first part of an N-part series about entanglement and homological algebra; here N is large and equal to 2 or 3.
After a short overview, we will encounter the mutual information of a multipartite density state as measure of "entanglement" (more precisely: lack of factorizability).
It leads a secret life as the limit of a triholomorphic index that is invariant under local unitary (or invertible) transformations and plays nicely with tensor products and "classical sums".
What else is it hiding?
Is its trihomolorphic persona the Euler characteristic of something?
Find out next week in another episode of States of our Lives.
Big Machinery in Quantum Mechanics: An Intro to Cohomology as a Tool for Quantifying Multipartite Entanglement.
Rutgers NHETC Subgroup Meeting; October 13, 2017.
Handwritten Notes: PDF or tar.gz (PNG images)
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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.
Entanglement is a Global Conspiracy Encoded in a Graded Vector Space
Rutgers NHETC Group Meeting; March 17, 2016
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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.
Fourier-Mukai: A perspective from the village idiot
Student Seminar on Geometric Langlands (UT Austin); February 17, 2015
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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.
What is a Formal Derived Stack?
Handwritten Notes: PDF or tar.gz (PNG images)
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.
Get Stoked about Stokes Groupoids
Handwritten Notes: PDF
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The Riemann-Hilbert correspondence is a statement about a correspondence between the study of flat connections on bundles over a manifold X, and representations of its fundamental group(oid).
In the case that X is a complex curve, however, many folks take pleasure in the study of the fancier world of flat connections with singularities bounded by some divisor D < X.
Gualtieri, Li, and Pym (1305.7288) state that in this world, the appropriate Riemann-Hilbert correspondence tells us that the study of such flat connections is the study of representations of a groupoid \Pi(X,D).
Despite all of the "oiding," the results are quite simple and reveal a beautiful (and "functorial") way to take formal power series solutions (of flat sections) to honest solutions by looking at pull-backs to the groupoid \Pi(X,D).
Clifford Algebras and Dirac Operators
Handwritten Notes: PDF
Attendee Notes (Richard Derryberry): PDF
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This was a talk given as part of a student-run seminar on the Atiyah-Singer Index Theorem, organized by
Richard Derryberry.
A link to the seminar series, which includes various notes taken by Richard is
here.
In case of bit-rot, an internet archive link is
here.
Understanding Theory X via Discrete Light Cone Quantization
Handwritten Notes: PDF
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The year is 1997.
N M5 branes have collided and the fate of the world(volume theory) rests on understanding Theory X: the 6D (2,0) theory.
In this talk we'll sketch the ideas behind discrete light cone quantization (DLCQ), Matrix models for M-theory (the BFSS conjecture) , and the subsequent application of DLCQ to Theory X (of type A_{N-1}).
You won't even need to hang up your rotary phone to access the internet.
Bethe Subalgebras, counting, and integrability
Geometry and String Theory Seminar (UT Austin); March 20, 2013.
Handwritten Notes: PDF or tar.gz (PNG images)
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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.