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Dror Bar-Natan:
Talks:

# Computation without Representation

**Abstract.** A major part of "quantum topology" (you don't have to
know what's that) is the definition and computation of various knot
invariants by carrying out computations in quantum groups (you don't have
to know what are these). Traditionally these computations are carried out
"in a representation", but this is very slow: one has to use tensor powers
of these representations, and the dimensions of powers grow exponentially
fast.

In my talk I will describe a direct-participation method for carrying out these
computations without having to choose a representation and explain why in
many ways the results are better and faster. The two key points we use are
a technique for composing infinite-order "perturbed Gaussian" differential
operators, and the little-known fact that every semi-simple Lie algebra can
be approximated by solvable Lie algebras, where computations are easier.

This is joint work with Roland van der Veen and continues work by Rozansky and Overbay.

**Handout:**
CWOR.html,
CWOR.pdf,
CWOR.png.

**Talk Video**
(better version at StonyBrook-1805).

**Links:**
Full
NCSU
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talks

**Sources:** pensieve.

**Aborted Idea:**

## Locally Euclidean Knotted Objects, Meta-Hopf Algebras, and
Circuit Algebras

**Abstract.** Seeing that I have nothing to say about operads, I'll
talk about other "generalized algebraic structures" that I like.
Specifically, I will explain what are locally Euclidean knotted objects and
how they form a "meta-Hopf algebra" (along the way explaining what is that
latter notion). I will then describe the "circuit algebra" of linearized
circuits and explain how to use it to construct a "Yang-Baxter meta-Hopf
algebra" which generalizes the Alexander polynomial. I will have no time to
explain, yet I'll sketch, how "solvable approximation of semi-simple Lie
algebras" lead to more sophisticated Yang-Baxter meta-Hopf algebras which
lead to very powerful poly-time computable knot polynomials.

**Handout:**
LEKO.html,
LEKO.pdf,
LEKO.png.

**Sources:** pensieve.