BEGIN:VCALENDAR
VERSION:2.0
PRODID:-//Talks.cam//talks.cam.ac.uk//
X-WR-CALNAME:Talks.cam
BEGIN:VEVENT
SUMMARY:Topology of the Ritz energy landscape - Gregory Berkolaiko (Texas 
 A&M University)
DTSTART:20260422T100000Z
DTEND:20260422T110000Z
UID:TALK246037@talks.cam.ac.uk
DESCRIPTION:Motivated by questions from quantum chemistry and spectral opt
 imization\, this talk explores the topology of the Ritz energy landscape. 
 Ritz values are the eigenvalues of a larger matrix (or operator) B compres
 sed to a smaller trial subspace S.&nbsp\; The subspace S is viewed as a pa
 rameter\, varying over the Grassmannian Gr(n\,s)\, and the function of int
 erest is the k-th Ritz value for some arbitrary but fixed k.We show that f
 or any k\, the k-th Ritz value is a perfect Morse function --- once the de
 finition of "perfection" is suitably adjusted.&nbsp\; A Morse function is 
 called perfect if it describes the topology of its domain in the most effi
 cient way possible\, meaning the number of its critical points of each typ
 e exactly matches the corresponding Betti number of the space.&nbsp\; Whil
 e the k-th Ritz value is Lipschitz rather than smooth and while its critic
 al points are not isolated --- and not even Morse-Bott --- its critical po
 int count is nevertheless well-defined and reflects the topology of the Gr
 assmannian in a minimal\, perfect way.The proof proceeds by introducing a 
 suitable perturbation which ensures that points of non-smoothness are not 
 critical (using a theorem of Zelenko and the presenter) and that the criti
 cal points that remain are isolated.&nbsp\; We use Thom's Isotopy to show 
 that the notion of perfection we introduce ("homological perfection") is c
 losed under such perturbations.Based on a joint work with Mark Goresky (IA
 S).
LOCATION:Seminar Room 2\, Newton Institute
END:VEVENT
END:VCALENDAR