Transformation Groups Seminar

Speaker: Steven Amelotte (Western)

"Cohomology operations for moment-angle complexes and minimal free resolutions of Stanley-Reisner rings"

Time: 09:30

Room: MC 107

After reviewing some results from last week's talk concerning moment-angle complexes $\mathcal{Z}_K$ and their cohomology rings, I will describe some further structure on $H^*(\mathcal{Z}_K)$ given by cohomology operations induced by the standard torus action. Under the identification of $H^*(\mathcal{Z}_K)$ with the Koszul homology of the Stanley-Reisner ring of $K$, these operations assemble to give an explicit differential on the minimal free resolution of the Stanley-Reisner ring. Using this topological interpretation of the minimal free resolution, we give simple algebraic and combinatorial characterizations of equivariant formality for torus actions on moment-angle complexes. This is joint work with Benjamin Briggs.

Analysis Seminar

Speaker: Blake J. Boudreaux (Western)

"Rational Convexity of Totally Real Sets"

Time: 14:30

Room: MC 107

A compact set $X\subset\mathbb C^n$ is said to be rationally convex if for every point $z\not\in X$ there is a polynomial $P$, depending on $z$, so that $P(z)=0$ but $P^{-1}(0)\cap X=\varnothing$. In view of the Oka-Weil theorem, any function holomorphic on a rationally convex compact $X$ can be approximated uniformly on $X$ by rational functions with poles off $X$. A totally real manifold $M$ is one whose tangent space has no complex structure, i.e., $J(T_pM)\cap T_pM=\{0\}$ for all $p\in M$. $$ $$ By a classical result of Duval-Sibony, a totally real manifold $M$ in $\mathbb{C}^n$ is rationally convex if and only if there exists a Kähler form $dd^c\varphi$ for which $M$ is isotropic. Under a mild technical assumption, we generalize this necessary and sufficient condition to the setting of totally real sets (zero loci of strictly plurisubharmonic functions).

Basic Notions Seminar

Speaker: Taylor Brysiewicz (Western)

"Basic Notions: Numerical Algebraic Geometry"

Time: 15:30

Room: MC 107

Numerical algebraic geometry is a computational framework for studying solution sets to polynomial equations (called varieties) using numerical algorithms. In contrast to symbolic methods (e.g. Grobner bases) which manipulate polynomials, numerical algorithms manipulate points on varieties. In this sense, numerical algebraic geometry may be thought of as the geometric side of computational algebraic geometry. In this talk, I will explain the basics ideas underlying the theory of numerical algebraic geometry.

Transformation Groups Seminar

Speaker: Tao Gong (Western)

"Integral cohomologies of classifying spaces of Kac-Moody groups"

Time: 09:30

Room: MC 107

In this seminar, I will introduce the Kac-Moody group associated with a Cartan matrix, then how to construct the homotopy colimit of classifying spaces of parabolic subgroups directly. Theoretically, we can know the classifying space of a Kac-Moody group from the previous homotopy colimit. I will apply that method to detailed computation of integral cohomologies of those classifying spaces with cases of exceptional Lie groups and projective unitary groups, in which spectral sequences are widely used.

Colloquium

Speaker: Yakov Shlapentokh-Rothman (Univesity of Toronto)

"Self-Similarity and Naked Singularities for the Einstein Vacuum Equations"

Time: 15:30

Room: MC107

We will start with an introduction to the weak cosmic censorship conjecture and the problem of constructing naked singularities for the Einstein vacuum equations. Then we will explain our discovery of a new type of self-similarity, and explain how this allows us to construct the desired naked singularity solutions.

Geometry and Combinatorics

Speaker: Matt Larson (Stanford University)

"Stellahedral geometry of matroids"

Time: 15:30

Room: Zoom

The stellahedral toric variety is a toric variety whose cohomology ring and K-ring are both closely related to matroids. We construct an integral isomorphism from the K-ring to the cohomology ring of the stellahedral toric variety, and use this to prove that three equivalence relations on matroids, valuative equivalence, numerical equivalence, and homological equivalence, coincide. Based on joint work with Chris Eur and June Huh.

Transformation Groups Seminar

Speaker: Anton Ayzenberg (HSE Moscow)

"Face posets of equivariantly formal torus actions and applications"

Time: 09:30

Room: online -- ask Matthias for details

Consider an effective smooth action of a compact torus on a connected closed smooth manifold X having isolated fixed points. We introduce the finite graded poset S(X) called the face poset of the action. If X is a toric variety or a quasitoric manifold, then S(X) is the face poset of the moment polytope X/T. However, S(X) is defined for actions of any complexity, in which case the local structure of S(X) is determined by the linear matroids of tangent weights.

If an action on X is equivariantly formal, we prove that the geometrical realization |S(X)| has some degree of acyclicity, depending on tangent weights. This statement gives a homological obstruction for particular actions to be equivariantly formal. As a motivating example, we study canonical conjugation actions on the manifolds of isospectral Hermitian matrices, having zeroes at prescribed positions. We prove a complete classification, which of these manifolds are equivariantly formal.

This talk is based on several works written jointly with V.Buchstaber, V.Cherepanov, M.Masuda, G.Solomadin, and K.Sorokin.

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.

Transformation Groups Seminar

Speaker: Xin Fu (Ajou University)

"The integral cohomology ring of four-dimensional toric orbifolds"

Time: 09:30

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

Toric orbifolds introduced by Davis and Januszkiewicz are topological analogs of projective toric varieties. When a toric orbifold is smooth, its integral cohomology ring is isomorphic to a quotient ring of the Stanley-Reisner ring. Such a formula holds for the singular case over rational coefficients, but integrally it becomes more complicated. For instance, the cohomology of a weighted projective space is additively isomorphic to the cohomology of a complex projective space, but the ring structure differs. In this talk, we focus on toric orbifolds $X$ in four dimensions. If $X$ has a smooth fixed point, we construct a basis for its integral cohomology and present their cup products in a matrix whose entries are explicitly determined by the characteristic function. This is joint work with Tseleung So and Jongbaek Song.