Transformation Groups Seminar

Speaker: Grigory Solomadin (HSE Moscow)

"Homotopy decomposition for quotients of moment-angle manifolds and its applications to cohomology"

Time: 09:30

Room: https://westernuniversity.zoom.us/s/93798234275, Passcode: 520011

In this talk we present computational tools for singular and equivariant cohomology of orbit spaces for moment-angle complexes with respect to any closed subgroup in the naturally acting torus (quotients). The tools are the Bousfield-Kan spectral sequence collapse, commutation of cohomology and colimit, and the diagram of Koszul resolutions. These ideas stem from the recent homotopy decomposition for any such quotient obtained by the author joint with I. Limonchenko, and are motivated by previous works of Notbohm and Ray; M. Franz; Lambrechts, Tourchine and Volic.

We will discuss proofs and applications of this toolkit, namely: the new formulas for the equivariant cohomology ring for a certain class (*) of non-free quotients; the equivariant and singular (Hohster-type) cohomology ring and group, respectively; formulas for the quotients by coordinate tori; Eilenberg-Moore spectral sequence construction and collapse for (*); a comparison spectral sequence between EMSS and BKSS second pages.

Colloquium

Speaker: Rajesh Pereira (University of Guelph)

"Approaches to the doubly stochastic spectral region problem"

Time: 15:30

Room: Zoom or MC107

We consider the open problem of characterizing the set of all possible complex numbers that can be the eigenvalue of an n by n doubly stochastic matrix. We look at the solution of the corresponding problem for stochastic matrices by Karpelevich and interpret this solution in terms of partial orders. We then look at possible partial order approaches to the doubly stochastic problem motivated by the connection between doubly stochastic matrices and the majorization order. Some related results in quantum information theory and possible connections to group representations are also discussed. ------------- Zoom Link:https://westernuniversity.zoom.us/j/92796051710 Password:colloq

Geometry and matrix analysis

Speaker: Rukmini Dey (International Centre for Theoretical Sciences, Bengaluru)

"Berezin-type quantization on compact even dimensional manifolds"

Time: 09:00

Room: zoom

We will first work out a local description of Berezin quantization on ${\mathbb C}P^d$. We show that a Berezin-type quantization can be achieved on a compact even dimensional manifold $M^ {2d}$ by removing a skeleton $M_ 0$ of lower dimension such that what remains is diffeomorphic to $R^{ 2d}$ which we identify with ${\mathbb C}^ d$ and embed in ${\mathbb C}P^ d$ . A local Poisson structure and Berezin-type quantization are induced from ${\mathbb C}P^ d$ . This construction depends on the diffeomorphism. However, suppose $X = M \setminus M_ 0$ has a complex structure and we have from $X \setminus X_0$ , (X 0 a set of measure zero or empty) a biholomorphism from it to ${\mathbb C}^d \setminus N_ 0$ , (where $N_ 0$ is of measure zero or empty). As before we embed ${\mathbb C}^d \setminus N_ 0$ in ${\mathbb C}^d and then into ${\mathbb C}P^ d$ and we have a Berezin-type quantization induced from ${\mathbb C}P^ d$ . If we use another biholomorphism, we have a map of the two Hilbert spaces under consideration such that the reproducing kernel of one maps to the reproducing kernel of the other and we have an equivalent quantization. We have a similar construction where we consider an arbitrary complex manifold and use local coordinates to induce the quantization from ${\mathbb C}P^ d$ . We study the possibility of defining a global Berezin quantization on compact complex manifolds. Finally we give a simple construction of pullback coherent states on compact smooth manifolds.