Analysis Seminar

Speaker: Hadi Seyedinejad (Western)

"Irreducibility in real algebraic geometry (Part I)"

Time: 15:30

Room: MC 108

The notion of irreducibility in the conventional Zariski topology is too coarse for real algebraic sets. For example, the hyperbola xy=1 is an irreducible algebraic set which is not even connected in the Euclidean topology. Cartan umbrella is another irreducible algebraic set, which is connected, but decomposes into the union of a 'sheet' and a 'stick.' Inspired by Nash and his notion of 'sheets,' one might require then to distinguish, as called by Kurdyka, the 'rigid pieces' of a real algebraic set. We will review different approaches to defining irreducibility and irreducible components in real algebraic geometry, in which more 'good' functions than just polynomials should be considered. We may work with the ring of Nash functions, continuous rational functions, or, most notably, arc-analytic functions. Nash functions are able to detect two components for the hyperbola xy=1. Continuous rational functions are able to detect a sheet and a stick component for Cartan umbrella. But we find examples in which only the phenomenon called 'arc-symmetricity' can among the others realize a decomposition.

Speaker's web page: http://www.math.uwo.ca/index.php/profile/59/

Noncommutative Geometry

Speaker: Bogdan Nica (McGill University)

"Hyperbolic groups and Noncommutative Geometry"

Time: 12:30

Room: MC 107

C*-algebras associated to groups provide some of the most interesting and most important examples in Noncommutative Geometry. In this respect, hyperbolic groups have, time and again, proved to be particularly exciting. I will discuss three vignettes illustrating this idea.

Homotopy Theory

Speaker: James Richardson (Western)

"Presentable infinity categories"

Time: 13:00

Room: MC 107

I will introduce presentable quasicategories and discuss some of their properties. I will then discuss their relationship with combinatorial model categories.

Colloquium

Speaker: Pinaki Mondal (School of Mathematics, Physics and Technology at The College of the Bahamas)

"Milnor number, intersection multiplicity and number of zeroes of systems of polynomials"

Time: 15:30

Room: MC 107

We talk about two of the original problems that shaped the theory of Newton polyhedra: the problem of computing the Milnor number of the singularity at the origin of a generic polynomial, and computing the number of zeroes of generic polynomials. The former was addressed by Kushnirenko, who gave a beautiful formula in terms of Newton diagrams in a special case.Bernstein (following work of Kushnirenko) solved completely the latter problem for the case of (C^*)^n. In the case of C^n the problem was partially solved following the work of Khovanskii, Huber-Sturmfels, and many others. We give complete solution to both these problems. A common theme in our solution to both problems is the computation of intersection multiplicity at the origin of the hypersurfaces determined by n generic polynomials.

Noncommutative Geometry

Speaker: Masoud Khalkhali (Western)

"Random Matrix Theory (IV)"

Time: 12:30

Room: MC 106

Geometry and Topology

Speaker: Cihan Okay (Western)

"Spherical posets from commuting elements"

Time: 15:30

Room: MC 107

I will show that the universal cover of the commutative classifying space has the homotopy type of a wedge of spheres when $G$ is an extraspecial $p$-group. Commutative classifying space is a certain subspace of the usual classifying space whose set of $n$-simplices is given by the set of commuting $n$-tuples in $G$. Its universal cover can be described as a partially ordered set obtained from the collection of abelian subgroups of $G$. Such objects are closely related to Tits buildings associated to algebraic groups. I will also mention about possible applications of such objects in quantum computation.

Analysis Seminar

Speaker: Hadi Seyedinejad (Western)

"Irreducibility in real algebraic geometry (Part II)"

Time: 15:30

Room: MC 108

The notion of irreducibility in the conventional Zariski topology is too coarse for real algebraic sets. For example, the hyperbola xy=1 is an irreducible algebraic set which is not even connected in the Euclidean topology. Cartan umbrella is another irreducible algebraic set, which is connected, but decomposes into the union of a 'sheet' and a 'stick.' Inspired by Nash and his notion of 'sheets,' one might require then to distinguish, as called by Kurdyka, the 'rigid pieces' of a real algebraic set. We will review different approaches to defining irreducibility and irreducible components in real algebraic geometry, in which more 'good' functions than just polynomials should be considered. We may work with the ring of Nash functions, continuous rational functions, or, most notably, arc-analytic functions. Nash functions are able to detect two components for the hyperbola xy=1. Continuous rational functions are able to detect a sheet and a stick component for Cartan umbrella. But we find examples in which only the phenomenon called 'arc-symmetricity' can among the others realize a decomposition.

Speaker's web page: http://www.math.uwo.ca/index.php/profile/59/

Noncommutative Geometry

Speaker: Masoud Khalkhali (Western)

"Random Matrix Theory (V)"

Time: 12:30

Room: MC 106

Homotopy Theory

Speaker: Pal Zsamboki (Western)

"Equivalent notions of infinity-topoi"

Time: 13:00

Room: MC 107

Let $X$ be a quasicategory. Then it is an $\infty$-topos, if it is an accessible left exact localization of the presheaf category of a small quasicategory. We will introduce two sets of intrinsic conditions which are equivalent to being an $\infty$-topos:

1) the $\infty$-categorical Giraud axioms, and

2) colimits in $X$ are universal, and it has small object classifiers for large enough regular cardinals,

and we discuss the equivalences.

Noncommutative Geometry

Speaker: Piotr M. Hajac (IMPAN)

"NONCOMMUTATIVE BORSUK-ULAM-TYPE CONJECTURES REVISITED"

Time: 11:00

Room: MC 108

Let H be the C*-algebra of a non-trivial compact quantum group acting freely on a unital C*-algebra A. Baum, Dabrowski and Hajac conjectured that there does not exist an equivariant *-homomorphism from A to the equivariant noncommutative join C*-algebra A*H. When A is the C*-algebra of functions on a sphere, and H is the C*-algebra of functions on Z/2Z acting antipodally on the sphere, then the conjecture becomes the celebrated Borsuk-Ulam theorem. Recently, Chirvasitu and Passer proved the conjecture when H is commutative. The main goal of this talk is to show how to extend the Chirvasitu-Passer result to a far more general setting assuming only that H admits a character different from the counit. Also, assuming that our compact quantum group is a q-deformation of a compact connected semisimple Lie group, we prove that there exists a finite-dimensional representation of the compact quantum group such that, for any C*-algebra A admitting a character, the finitely generated projective module associated with A*H via this representation is not stably free. (Based on joint work with L. Dabrowski and S. Neshveyev.)

Noncommutative Geometry

Speaker: Christian Voigt (University of Glasgow)

"The Plancherel formula for complex quantum groups"

Time: 13:30

Room: MC 108

We explain the structure of the reduced group C∗-algebras of complex semisimple quantum groups, and discuss a connection to the Baum-Connes assembly map for classical complex groups.

Noncommutative Geometry

Speaker: Sajad Sadeghi (Western)

"A Scalar Curvature Formula for the Noncommutative 3-Torus"

Time: 15:00

Room: MC 108

This talk is based on my joint paper with Masoud Khalkhali and Ali Moatadelro (arXiv:1610.04740). First I will recall Gilkey’s theorem on asymptotic expansion of heat kernels for the special case of Laplacians. I will also introduce the noncommutatvie 3-torus (NCT3) and then I will conformally perturb the standard volume form on it. Then the corresponding perturbed Laplacian will be discussed, and using Connes’ pseudodifferential calculus, I will define the scalar curvature of NCT3. Finally, introducing a rearrangement lemma I will compute an explicit formula for the scalar curvature of the curved noncommutative 3-torus.

Noncommutative Geometry

Speaker: Masoud Khalkhali (Western)

"Random Matrix Theory (VI)"

Time: 15:00

Room: MC

Geometry and Topology

Speaker: Jeffrey Carlson (Univ. of Toronto)

"Equivariant formality in rational cohomology and $K$-theory"

Time: 15:30

Room: MC 107

An action of a group $G$ on a space $X$ is said to be equivariantly formal if the induced map from Borel equivariant cohomology of the action to singular cohomology of $X$ is surjective. This situation is much to desired in various geometric settings, where it can allow the integral of an invariant function to be reduced to one over a lower-dimensional or even finite fixed-point set.

It turns out (work of Fok) one can approach the question in terms of equivariant $K$-theory: the action of a compact Lie group $G$ on a finite $G-CW$ complex $X$ is equivariantly formal with rational coefficients if and only if some each power of every vector bundle over $X$ admits a stable equivariant structure. This correspondence allows us an alternate proof, with a slightly stronger conclusion, of a result of Adem–Gómez on the equivariant $K$-theory of actions with maximal-rank isotropy.

In the case of the left translation ("isotropy") action of a connected group $H$ on a homogeneous space $G/H$, the correspondence also allows us to more simply reobtain results of Goertsches–Noshari on generalized symmetric spaces. In the realm of rational homotopy theory, we are able to show that equivariant formality of the isotropy action implies $G/H$ is a formal space, allowing us to improve a sufficient condition of Shiga–Takahashi to an equivalence.

This work is joint with Chi-Kwong Fok.

Noncommutative Geometry

Speaker: Sajad Sadeghi (Western)

"Rearrangement Lemma"

Time: 12:30

Room: MC 106

Rearrangement Lemma is a very deep result that first was stated and proved by Connes. He proved it to integrate a vector valued function while studying the geometry of noncommutative 2-tori with a conformally perturbed metric. In our recent work joint with Masoud Khalkhali and Ali Moatadelro, on the noncommutative 3-torus, we have proved a slightly different version of that. In this talk I will give a proof of that version of Rearrangement Lemma.