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Stream: event: Topos Colloquium

Topic: Joachim Kock: "Noncrossing hyperchords and free probability"

Tim Hosgood (Apr 07 2021 at 17:55):

Abstract:

Free probability is a noncommutative probability theory introduced by Voiculescu in the 1980s, motivated by operator algebras and free groups, and useful in random matrix theory. Where classical independence relates to the tensor product of algebras, free independence relates to the free product of algebras. Speicher discovered the combinatorial substrate of the theory: noncrossing partitions. He derived the free cumulant-moment relations from Möbius inversion in the incidence algebra of the lattice of noncrossing partitions, and used it, via two reduction procedures, to model free multiplicative convolution. A crucial ingredient, which has no analogue in the classical setting, is the notion of Kreweras complement of a noncrossing partition. In this talk, after a long introduction to these topics, I will explain some more categorical viewpoints. A first step is an operad of noncrossing partitions. A second step is a decomposition space (2-Segal space) Y of noncrossing hyperchords, whose simplicial structure encodes higher versions of Kreweras complementation. The incidence bialgebra of Y is a direct combinatorial model for free multiplicative convolution. It is related to the previous models by the standard simplicial notion of decalage: the first decalage of Y gives the (two-sided bar construction of the) operad, and the second decalage gives the lattice. These two decalages encode precisely Speicher's two reductions.

This is joint work with Kurusch Ebrahimi-Fard, Loïc Foissy, and Frédéric Patras.

Tim Hosgood (Apr 07 2021 at 17:55):

https://researchseminars.org/talk/ToposInstituteColloquium/7/

Tim Hosgood (Apr 07 2021 at 17:56):

this will be on Thursday the 8th of April at 17:00 UTC

Tim Hosgood (Apr 07 2021 at 17:56):

YouTube link: https://youtu.be/_AueymcE2Eg
Zoom link: https://topos-institute.zoom.us/j/5344862882?pwd=Znh3UlUrek41T3RLQXJVRVNkM3Ewdz09

Tim Hosgood (Apr 08 2021 at 16:47):

starting in 15 minutes!

Tim Hosgood (Apr 11 2021 at 18:06):

a question that i've been thinking about since this talk is the following: the relations between moments and cumulants are entirely polynomial, so what do the algebraic varieties defined by these relations look like?

Tim Hosgood (Apr 11 2021 at 18:08):

that is, the $n$ th moment $m_n$ is a degree-$n$ polynomial in the first $n$ cumulants $\kappa_1,\ldots,\kappa_n$, so what does $\operatorname{Spec}(\mathbb{C}[y,x_1,\ldots,x_n]/(y-f_n(x_1,\ldots,x_n))$ look like, where $f_n$ is the polynomial such that $m_n=f_n(\kappa_1,\ldots,\kappa_n)$

Tim Hosgood (Apr 11 2021 at 18:09):

e.g. the first few varieties would be defined by

1. $y-x_1$
2. $y-x_2-x_1^2$
3. $y-x_3-3x_2x_1-x_1^3$

Tim Hosgood (Apr 11 2021 at 18:10):

in fact, these might even make more sense as varieties in weighted projective space, with $\deg x_i=i$, since then they are homogeneous

Joachim Kock (Apr 12 2021 at 13:58):

Tim Hosgood said:

the relations between moments and cumulants are entirely polynomial, so what do the algebraic varieties defined by these relations look like?

I don't know of any work in that direction. If there is anything it is more likely to be in the commutative case, where the polynomials you list are the (complete) Bell polynomials. It may be relevant in that direction to know that the Bell polynomials can be given as determinants in various ways. I agree that weighted projective spaces seem like a good setting for this question.

fosco (Apr 13 2021 at 09:40):

These polynomials are similar (but not equal) to the cumulants: image.png

why? (Naive question from someone who does not know a thing :smile: )

Joachim Kock (Apr 13 2021 at 09:46):

Hi Fosco, the polynomials you list count the ways to split a set into cycles. The Bell polynomials (those appearing in the classical moment-cumulant relations) count partitions, i.e. just splitting into (nonempty) subsets. (Well, and then there is a normalising factorial factor, which you also sometimes include in the Bell polynomials, depending on how they appear.)

fosco (Apr 13 2021 at 12:27):

oh, I see, yes