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SUMMARY:Tomasz Kania (Institute of Mathematics\, Czech Academy of Sciences
\, Czech Republic)
DTSTART;VALUE=DATE-TIME:20200527T130000Z
DTEND;VALUE=DATE-TIME:20200527T140000Z
DTSTAMP;VALUE=DATE-TIME:20200812T040136Z
UID:PortMATHS/1
DESCRIPTION:Title: Estimating Kalton's constant: from twisted sums of Bana
ch spaces to fine-tuning algorithms\nby Tomasz Kania (Institute of Mathema
tics\, Czech Academy of Sciences\, Czech Republic) as part of Portsea Math
s Research Webinar\n\n\nAbstract\nWe shall try to draw a quite unexpected
connection between Kalton and Roberts' work in the theory of twisted sums
of Banach spaces (and quasi-linear maps) that originated with the construc
tion of twisted Hilbert spaces and fine-tuning certain optimisation algori
thms. Kalton and Roberts [Trans. Amer. Math. Soc. 1983] proved a stability
result for 1-additive maps asserting that there exists a universal consta
nt K not smaller than 44.5 such that for any set algebra F\, for every sca
lar-valued 1-additive map f defined thereon\, there is a 0-additive map (a
finitely additive signed measure) whose distance to f is at most K. Pawli
k [Colloq. Math. 1987] noticed that in general K cannot be smaller than 1.
5. We shall present a class of positive 1-additive maps\, which witnesses
that K cannot be smaller than 3. If time permits\, we shall mention certai
n results due to Feige\, Feldman\, and Talgam-Cohen [SIAM J. Comput. 2020]
illustrating the sensitivity of certain machine-learning algorithms to es
timates for K. \n\nThis is joint work with M. Gnacik (UoP) and M. Guzik (U
BS) [Proc. Amer. Math. Soc. 2020+].\n
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