Noncommutative Geometry

Speaker: (Western)

"Zeta regularized determinants"

Time: 11:00

Room: MC 108

I will show how to compute the zeta regualrized determinants of differential operators, notably Jacobi operators in differential geometry and the Sturm-Liouville operators on the line. Note: this talk was cancelled last week. So this is the first part.

Homotopy Theory

Speaker: Mitchell Riley (Western)

"(2)-Category Theory in HoTT"

Time: 13:30

Room: MC 107

In this talk I will describe the development of category theory within HoTT and what it means for a category to be univalent.

Geometry and Combinatorics

Speaker: Matthias Franz (Western)

"Equivariant cohomology of smooth toric varieties, II"

Time: 16:00

Room: MC 108

We continue by proving Brion's theorem identidying the equivariant cohomology of a smooth toric variety with the ring of piecewise polynomials on the associated fan. Then we compare piecewise polynomials to the Stanley-Reisner ring. We conclude with some remarks about non-equivariant cohomology.

Noncommutative Geometry

Speaker: (Western)

"Zeta regularized determinants II"

Time: 11:00

Room: MC 107

Comprehensive Exam Presentation

Speaker: Chris Henry (Western)

"Identities of Lie algebras with actions"

Time: 14:00

Room: MC 107

Polynomial identity algebras (or PI-algebras) can be seen as generalizations of both commutative and finite dimensional algebras which maintain many ``nice'' properties. In this talk, we give a short introduction to Lie PI-algebras, focusing on the study of multilinear identities. Using these techniques, we demonstrate the existence of an identity for Lie algebras having certain actions on them; such actions arise naturally, for example, when the Lie algebra is graded by a finite group.

Homotopy Theory

Speaker: Jason Brennan (Western)

"POSTPONED: Impartial Game Theory in Coq"

Time: 13:30

Room: MC 107

I will provide a brief introduction to impartial game theory consisting of examples of games as well as elementary operations on them. This will be followed by a presentation of their syntactic counterparts in Coq, culminating in formal proofs of several basic theorems.

Geometry and Combinatorics

Speaker: Matthias Franz (Western)

"Non-equivariant cohomology of smooth toric varieties"

Time: 15:00

Room: MC 108

Having studied the equivariant cohomology of smooth toric varieties, we will see to what extent it determines the non-equivariant cohomology. In the equivariantly formal case, one gets the Jurkiewicz-Danilov theorem (with integer coefficients). In general, a complete answer is only known for rational coefficients.

Noncommutative Geometry

Speaker: Babak Beheshti (Western)

"Stationary phase approximation, Duistermaat-Heckman formula, and localization"

Time: 11:00

Room: MC 108

Colloquium

Speaker: Jon Borwein

"Continued Logarithms and Associated Continued Fractions"

Time: 13:30

Room: MC 108

Noncommutative Geometry

Speaker: Babak Beheshti (Western)

"Duistermaat-Heckman localization and stationary phase approximation II"

Time: 11:00

Room: MC 108

Colloquium

Speaker: Jonathan Rosenberg (Maryland)

"From dualities in string theory to K-theory isomorphisms"

Time: 11:00

Room: MC 107

An amazing discovery of physicists is that there are many seemingly quite different quantum field theories that lead to the same observable predictions. Such theories are said to be related by dualities. A duality leads to interesting mathematical consequences; for example, certain K-theory groups on the two spacetime manifolds have to be isomorphic. We will explain how some of these K-theory isomorphisms predicted by physics correspond to certain cases of the Baum-Connes Conjecture, or to equivalences of derived categories of twisted coherent sheaves.

PhD Thesis Defence

Speaker: Mitsuru Wilson (Western)

"A Gauss-Bonnet-Chern theorem for the noncommutative 4-sphere (PhD Public Lecture)"

Time: 14:00

Room: MC 107

We introduce pseudo-Riemannian calculus of modules over noncommutative algebras in order to investigate as to what extent the differential geometry of classical Riemannian manifolds can be extended to a noncommutative setting. In this framework, it is possible to prove an analogue of the Levi-Civita theorem. It states that there exists at most one connection, which satisfies a torsion-free condition and a metric compatibility condition on a given smooth manifold. More significantly, the corresponding curvature operator has the same symmetry properties as in the classical curvature tensors. In my talk, I will discuss pseudo-Riemannian calculi over the noncommutative 4-sphere for a conformal class of the round metric and their corresponding scalar curvatures. Lastly, in this setting it is possible to prove a Gauss-Bonnet-Chern type theorem.

Colloquium

Speaker: Vasileios Nestorides (University of Athen)

"Primitives on general planar domains"

Time: 15:30

Room: MC 108

We will start by showing that generically for all holomorphic functions f on a planar simply connected domain every order's antiderivative is unbounded. This leads to consider multivalued antiderivatives of any order on every non simply connected domain. If on such a domain V a holomorphic function f has a one-valued antiderivative of any order in V, then f has a holomorphic extension on the simply connected envelop of V. On a non simply connected domain generically every holomorphic function deos not have a one-valued primitive.

Can the space H(V) of all holomorphic functions be replaced in the above results by other spaces of holomorphic functions?

Noncommutative Geometry

Speaker: Ludwik Dabrowski and Andrzej Sitarz (Trieste and Warsaw)

"Twisted reality structure for spectral triples"

Time: 11:00

Room: MC 107

The reality condition for spectral triples is a noncommutative generalization of the charge conjugation for Dirac spinors, but not always it is satisfied on interesting examples. Motivated by conformal deformations of spectral triples and a spectral triple construction on quantum cones, we propose a new twisted reality condition.

Noncommutative Geometry

Speaker: Kenny De Commer (Brussels)

"A field of quantum upper triangular matrices"

Time: 14:00

Room: MC 107

t is well-known that the Poisson-Lie group dual of a compact semi-simple Lie group G with its standard Poisson-Lie structure can be identified with the solvable part AN of the Iwasawa decomposition of its complexification. In the setting of quantum groups, this entails that the dual of the standard q-deformation of G can be seen as a q-deformation of AN. While this statement has been made rigorous by Drinfeld within the setting of formal series Hopf algebras, a corresponding general statement in the operator algebraic setting seems lacking at the moment. In this talk, we will discuss in detail the simplest case of G equal to SU(2). We show how the function algebras on the duals of the quantum SU(2) groups fit naturally into a continuous field of C*-algebras with classical limit the function algebra on the group AN, in this case the group of special upper triangular 2 by 2-matrices with positive diagonal. The global C*-algebra of sections can be described as a crossed product of a classical space by a partial automorphism (in the sense of Exel). We next show compatibility with the coproduct structure. This is joint work with M. Floré.

Colloquium

Speaker: Piotr Hajac and Thomas Maszczyk (IMPAN, Warsaw)

"FROM CLASSICAL TO QUANTUM QUATERNIONIC PROJECTIVE SPACES"

Time: 15:30

Room: MC 107

Our main theorem is that the pullback of an associated noncommutative vector bundle induced by an equivariant map of quantum principal bundles is a noncommutative vector bundle associated via the same finite-dimensional representation of the structural quantum group. On the level of K_0-groups of vector bundles, we realize the induced map by the pullback of explicit matrix idempotents. Finally, we construct quantum quaternionic projective spaces together with noncommutative tautological quaternionic line bundles and their duals. As a key application of the main theorem, we show that these bundles are stably non-trivial as noncommutative complex vector bundles.

Colloquium

Speaker: Jason Lotay (University College, London, UK)

"Adventures in 7 dimensions"

Time: 11:00

Room: MC 107

Ricci-flat spaces are analogues of space-times satisfying Einstein's vacuum field equations of General Relativity. Unlike their flat cousins, these spaces exhibit fascinating geometric properties and are poorly understood. In odd dimensions the only non-trivial Ricci-flat spaces we can find are 7-dimensional and they appear in theoretical physics, making geometry in 7 dimensions particularly compelling to study. I will describe in an elementary way some of the recent endeavours to understand this geometry and several key open questions.