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Einladung_mit_Abstract_20080506
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Fulltext:
Sommersemester 2008
Einladung zum
MATHEMATISCHEN KOLLOQUIUM
Am Dienstag, dem 13. Mai 2008
spricht
Prof. Dr. Gleb Koshevoy
Russian Academy of Scienes, Moskau
Āüber
The Horn problem and its reļ¬nements
The Horn problem (although I.M. Gelāfand stated this problem much earlier) deals with the following
question: What can we say about the spectrum of the sum of two Hermitian (or real symmetric)
matrices if we known the spectra of A and B? The spectrum of a Hermitian matrix, that is the set
of its eigenvalues, consists of real numbers, which are usually arranged in decreasing order. Let A
have spectrum Ī± = (Ī±(1) ā„ Ī±(2) ā„ . . . ā„ Ī±(n)) (where n is the size of the matrices), and let B and
C = A + B have spectra Ī² and Ī³ of a similar form. In 1962, Horn conjectured that the set H(Ī±, Ī²) of
possible values of Ī³ was a convex polyhedron speciļ¬ed by linear inequalities of the form
kāK
Ī³(k) ā¤
iāI
Ī±(i) +
jāJ
Ī²(j),
where I, J and K are some subsets (of the same size) in [n]. Horn suggested a recursive procedure
for constructing such triples (I, J, K). The Horn problem was solved in 1996 by Klyachko modulo
Saturation Conjecture and the latter was solved by Knutson and Tao in 1998.
I will suggest a somewhat diļ¬erent statement of the solution to the Horn problem and a signiļ¬cantly
more elementary proof. Speciļ¬cally, we prove that the existence of the required triple of matrices
(A, B, C) for given (Ī±, Ī², Ī³) is equivalent to the existence of a so-called discretely concave function
on the triangular grid ā(n) with boundary value-increments Ī±, Ī², and Ī³. Our proof involves only
one non-elementary issue, namely, a reference to the convexity of the image of a moment mapping.
The Horn original linear inequalities can be obtained from this form of the Horn problem by fairly
elementary (but non-trivial) means of linear programming. Several reļ¬nements of the Horn problem
will be proposed (some are still open problems). Reported results are obtained jointly with V.Danilov.
Der Vortrag ļ¬ndet statt um 17 Uhr c.t. im Raum 7260, 7. Ebene des
Mehrzweckhochhauses (MZH) der UniversitĀät Bremen, Bibliothekstr.
Zuvor gibt es Kaļ¬ee/Tee und GebĀäck im Raum 7140.
Alle Interessierten sind herzlich eingeladen.
Marc KeĆebĀöhmer als Kolloquiumsbeauftragter.
Invitation_with_Abstract_20080506 Einladung_mit_Abstract_20080506
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