Theory and applications of total positivity
July 24 to July 28, 2023
at the
American Institute of Mathematics,
Pasadena, California
organized by
Shaun Fallat,
Dominique Guillot,
and Apoorva Khare
Original Announcement
This workshop will be devoted to the strong connections and major trends of the classical notion of total positivity across the subjects of analysis, matrix theory, combinatorics, and related applied fields. While the concept of a function / matrix being totally positive has had a very rich history, many important longstanding connections and interactions across a number of disciplines continue to be actively explored today.
The main topics for the workshop are:
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Study transformations that preserve Pólya frequency (PF) sequences and TNp sequences, including those sequences with finitely many terms.
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Determine if there is a determinant-free characterization of TNp functions.
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Density of totally positive kernels in totally nonnegative ones.
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What matrix properties (e.g., rank, principal rank, spectral information, Jordan Forms, determinantal inequalities, etc.) are distinguished by considering the bidiagonal factorization versus the Cauchon Algorithm?
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Explore instances of "upgradation" from numerical positivity to monomial positivity phenomena by considering various families of combinatorial polynomials.
Material from the workshop
A list of participants.
The workshop schedule.
A report on the workshop activities.
A list of open problems.
Workshop videos
Papers arising from the workshop:
Matrix positivity preservers in fixed dimension. II: positive definiteness and strict monotonicity of Schur function ratios
by Alexander Belton, Dominique Guillot, Apoorva Khare, Mihai Putinar
Plücker inequalities for weakly separated coordinates in totally nonnegative Grassmannian
by Daniel Soskin, Prateek Kumar Vishwakarma
Sufficient conditions for total positivity, compounds, and Dodgson condensation
by Shaun Fallat, Himanshu Gupta, Charles R. Johnson
An analog of multiplier sequences for the set of totally positive sequences
by Olga Katkova, Anna Vishnyakova
Generalized Diagonals in Positive Semi-Definite Matrices
by Robert Angarone, Daniel Soskin
Equality cases of the Alexandrov--Fenchel inequality are not in the polynomial hierarchy
by Swee Hong Chan, Igor Pak
