Geometry and Topology

Speaker: Matthias Franz (Western)

"Equivariant cohomology and syzygies"

Time: 15:30

Room: MC 107

The GKM method is a powerful way to compute the equivariant (and ordinary) cohomology of many spaces with torus actions. So far it has been applied to so-called equivariantly formal $T$-spaces, which include compact Hamiltonian $T$-manifolds.

In this talk I will explain that the GKM method is valid for a much larger class of $T$-spaces. The explanation is based on a new interpretation of a sequence originally due to Atiyah and Bredon, and involves the notion of syzygies as used in commutative algebra. I will also exhibit a surprising relation between the equivariant Poincaré pairing and the GKM description.

This is joint work with Chris Allday and Volker Puppe.

Analysis Seminar

Speaker: Seyed Mohammad Hadi Seyedinejad (Western)

"Testing local regularity of complex analytic mappings by fibred powers, II"

Time: 14:40

Room: MC 107

This two-session talk will be concerned with holomorphic mappings between complex analytic sets (or more generally, analytic spaces). Local regularity of such a mapping can be measured by uniformity (or lack thereof) of the family of its fibres. In the first part of the talk, we will discuss the general idea of testing local regularity (like openness or flatness) by passing to fibred powers of a given map. The second session will be devoted to a recent joint work with Janusz Adamus: We establish an analytic version of flatness descent to prove a criterion for flatness of a holomorphic mapping with singular target. Previously, the best analogous result had been known only for the case of smooth targets.

Graduate Seminar

Speaker: Zack Wolske (Western)

"Techniques in Algebraic Number Theory"

Time: 16:30

Room: MC 107

We introduce some standard techniques in algebraic number theory to investigate solutions of polynomials. If we consider the integers mod p, and our polynomial has no solution there (local), then it has no integer solution (global). But if there are solutions for every p, can we find a global solution? More generally, we can ask for rational solutions, and consider completions of the rationals as localizations. This is called the Hasse local-global principle. We will introduce and use Henselian lifting, the class number, and the Minkowski bound to give an example of a polynomial which does not satisfy the Hasse principle.

Noncommutative Geometry

Speaker: Jason Haradyn (Western)

"Einstein Manifolds and Distinct 7-Manifolds Admitting Positively Curved Riemannian Structures"

Time: 10:30

Room: MC 108

An Einstein manifold is a smooth manifold whose Ricci tensor is proportional to the metric. Many homogeneous spaces can be realized as Einstein manifolds, and have been widely studied for general existence and nonexistence of Einstein metrics. In this talk we will give examples of homogeneous and Einstein manifolds and discuss some of the general underlying theory related to these spaces. We will also briefly discuss how this can be extended to the noncommutative case. Finally, we will show that if we are given a closed, connected, one-dimensional subgroup H of SU(3) that has no nonzero fixed points, then SU(3)/H admits an SU(3)-invariant Riemannian structure of strictly positive curvature. This result was first proven in 1975 by Aloff and Wallach, and it was here that the famous Aloff-Wallach spaces were introduced.

Noncommutative Geometry

Speaker: Mohammad Hassanzadeh (UNB)

"A new class of ASYD modules for Hopf cyclic cohomology"

Time: 13:30

Room: MC 108

We show that the category of coefficients for Hopf cyclic cohomology has two proper subcategories where one of them is the category of stable anti Yetter-Drinfeld modules. Generalizations of suitable coefficients for Hopf cyclic cohomology are introduced. The notion of stable anti Yetter-Drinfeld modules is extended based on underlying symmetries. We show that the new introduced categories for coefficients of Hopf cyclic cohomology and the category of stable anti-Yetter-Drinfeld modules are all different. (This is joint work with Bahram. Rangipour and Dan. Kucerovsky )

Algebra Seminar

Speaker: Marcy Robertson (Western)

"Introduction to derived Hall algebras"

Time: 14:30

Room: MC 107

Roughly speaking, the Hall algebra $H(A)$ of a (small) Abelian category $A$ is the algebra of finitely supported functions on the moduli space of objects of $A$ (i.e. the set of isoclasses of objects of $A$ with the discrete topology). Interest in Hall algebras exploded in the early 1990's when Ringel discovered that the Hall algebra associated to the category of $F_q$-representations of a Dynkin quiver $Q$ provides a realization of the positive part of the (quantized) enveloping algebra of the (simple) complex Lie algebra associated to the same Dynkin diagram. To\"{e}n and Bergner have used the theory of model categories to obtain Hall algebras on triangulated categories. In this talk we will survey these constructions and, time permitting, explain some open problems in this area which are being studied via homotopy theory.