- Library Home /
- Search Collections /
- Open Collections /
- Browse Collections /
- BIRS Workshop Lecture Videos /
- Noncommutative polynomials describing convex sets
Open Collections
BIRS Workshop Lecture Videos
BIRS Workshop Lecture Videos
Noncommutative polynomials describing convex sets Klep, Igor
Description
The semialgebraic set $D_f$ determined by a noncommutative polynomial $f$ is the closure of the connected component of $\{(X,X^*): f(X,X^*)\succ0\}$ containing the origin. When $L$ is a linear pencil, the semialgebraic set $D_L$ is the feasible set of the linear matrix inequality $L(X,X^*)\succeq 0$ and is known as a free spectrahedron. Evidently these are convex and by a theorem of Helton \& McCullough, a free semialgebraic set is convex if and only it is a free spectrahedron. \\\\ In this talk we solve the basic problem of determining those $f$ for which $D_f$ is convex. The solution leads to an effective algorithm that not only determines if $D_f$ is convex, but if so, produces a minimal linear pencil $L$ such that $D_f=D_L$. Of independent interest is a subalgorithm based on a Nichtsingulà ¤rstellensatz: given linear pencils $L,L'$, it determines if $L'$ takes invertible values on the interior of $D_L.$ Finally, it is shown that if $D_f$ is convex for an irreducible noncommutative polynomial, then $f$ has degree at most two, and arises as the Schur complement of an $L$ such that $D_f=D_L$. This is based on joint work with Bill Helton, Scott McCullough and Jurij Volà ià .
Item Metadata
Title |
Noncommutative polynomials describing convex sets
|
Creator | |
Publisher |
Banff International Research Station for Mathematical Innovation and Discovery
|
Date Issued |
2019-05-27T15:30
|
Description |
The semialgebraic set $D_f$ determined by a noncommutative polynomial $f$ is the closure of the connected component of $\{(X,X^*): f(X,X^*)\succ0\}$ containing the origin. When $L$ is a linear pencil, the semialgebraic set $D_L$ is the feasible set of the linear matrix inequality $L(X,X^*)\succeq 0$ and is known as a free spectrahedron. Evidently these are convex and by a theorem of Helton \& McCullough, a free semialgebraic set is convex if and only it is a free spectrahedron. \\\\ In this talk we solve the basic problem of determining those $f$ for which $D_f$ is convex. The solution leads to an effective algorithm that not only determines if $D_f$ is convex, but if so, produces a minimal linear pencil $L$ such that $D_f=D_L$. Of independent interest is a subalgorithm based on a Nichtsingulà ¤rstellensatz: given linear pencils $L,L'$, it determines if $L'$ takes invertible values on the interior of $D_L.$ Finally, it is shown that if $D_f$ is convex for an irreducible noncommutative polynomial, then $f$ has degree at most two, and arises as the Schur complement of an $L$ such that $D_f=D_L$.
This is based on joint work with Bill Helton, Scott McCullough and Jurij Volà ià .
|
Extent |
32.0 minutes
|
Subject | |
Type | |
File Format |
video/mp4
|
Language |
eng
|
Notes |
Author affiliation: The University of Auckland
|
Series | |
Date Available |
2019-11-24
|
Provider |
Vancouver : University of British Columbia Library
|
Rights |
Attribution-NonCommercial-NoDerivatives 4.0 International
|
DOI |
10.14288/1.0385849
|
URI | |
Affiliation | |
Peer Review Status |
Unreviewed
|
Scholarly Level |
Faculty
|
Rights URI | |
Aggregated Source Repository |
DSpace
|
Item Media
Item Citations and Data
Rights
Attribution-NonCommercial-NoDerivatives 4.0 International