Geometry and Topology

Speaker: John Harper (Western)

"On a Whitehead theorem for topological Quillen homology of algebras over operads"

Time: 15:30 - 16:30

Room: MC 107

In Haynes Miller's proof of the Sullivan conjecture on maps from classifying spaces, Quillen's derived functor notion of homology (in the case of commutative algebras) is a critical ingredient. This suggests that homology for the larger class of algebraic structures parametrized by an operad O will also provide interesting and useful invariants. Working in the context of symmetric spectra, we prove a Whitehead theorem for topological Quillen homology of algebras and modules over operads. This is part of a larger goal to attack the problem: how much of an O-algebra can be recovered from its topological Quillen homology? We also prove analogous results for algebras and modules over operads in unbounded chain complexes. This talk is an introduction to these results (joint with K. Hess) with an emphasis on several of the motivating ideas.

Analysis Seminar

Speaker: Ilya Kossovskiy (Western)

"Real Submanifolds in a Complex Space III"

Time: 15:30 - 16:30

Room: MC 107

The Theory of Real Submanifolds in a Complex Space (which is sometimes called, in some more general settings, "CR-geometry") goes back to H.Poincare and was deeply developed in further works of E.Cartan, N.Tanaka, S.Chern and J.Moser. In the present series of lectures we consider the classical aspects of this theory, as well as some recent results, focusing mainly on the holomorphic equivalence problem, groups of holomorphic symmetries and the holomorphic extension problem for real submanifolds in a complex space.

Operads Seminar

Speaker: Marcy Robertson (Western)

"Operadic Algebra I"

Time: 14:30 - 15:30

Room: MC 107

We will define operads, algebras over operads, modules over operads, and give various examples.

Algebra Seminar

Speaker: Parker Lowrey (Western)

"Autoequivalences and stability conditions"

Time: 15:30 - 16:30

Room: MC 107

I will discuss how stability conditions and well adapted autoequivalences can be used to understand geometric information in derived categories. Following this discussion, I will provide an example of the usefulness of these techniques. In particular, I will show how to classify all compactifications of stable bundles on a class of genus 0 singular reducible curves.

Analysis Seminar

Speaker: Vincent Grandjean (Fields Institute)

"Gradient trajectories on isolated surface singularities do not oscillate at their limit point"

Time: 15:30 - 16:30

Room: MC 107

Consider $\mathbb{R}^n$ equipped with a real analytic Riemannian metric ${\bf g}$. Let $f : \mathbb{R}^n\to\mathbb{R}$ be a real analytic function singular at $O$ the origin. We would like to understand the dynamics of $\nabla f$ in a neighbourhood of the critical point $O$, where $\nabla f$ stands for the gradient vector field of the function $f$ associated with the metric ${\bf g}$. We are particularly interested in the oscillating/non-oscillating behaviour in a neighbourhood of $O$ of any gradient trajectory accumulating on $O$.

We prove that if a trajectory lies in a real analytic surface with an isolated singularity at $O$, then it cannot oscillate at $O$.

In the first talk, I will recall elementary and well known facts and ideas about the gradient problem. In the second one, I will sketch the proof of our theorem.

This is joint work with Fernando Sanz (Valladolid).

Colloquium

Speaker: Igor Dolgachev (Michigan)

"Configuration spaces of real and complex spheres"

Time: 15:30 - 16:30

Room: MC 107

Abstract: I will discuss configuration spaces of complex and real spheres modulo the inversive space transformations. A classical construction of S. Lie and F. Klein allows one to treat these spaces as configuration spaces of points in the projective space of one dimension higher modulo the orthogonal transformations. Some special cases can be described very explicitly using subvarieties in toric varieties.

Algebra Seminar

Speaker: John Harper (Western)

"On a homotopy completion tower for algebras over operads"

Time: 15:30 - 15:30

Room: MC 107

We introduce a completion tower for algebras over operads in unbounded chain complexes (resp. symmetric spectra) and prove that under appropriate connectivity conditions this tower interpolates between Quillen homology and the identity functor. This talk will focus on the chain complex version of these results, beginning with a short introduction to Quillen's notion of derived abelianization, and followed by a sketch of the proof with an emphasis on several of the homotopical arguments. We will illustrate the tower results in the special case of commutative differential graded algebras and André-Quillen homology. These results are joint work with K. Hess.

Geometry and Topology

Speaker: Rick Jardine (Western)

"Dynamical systems and diagrams"

Time: 15:30 - 16:30

Room: MC 107

A dynamical system is a map of spaces $X \times S \to X$, and a map of dynamical systems $X \to Y$ over $S$ is an $S$-equivariant map. There is both an injective and a projective model structure for this category.

These model structures are special cases of injective and projective model structures for space-valued diagrams $X$ defined on a fixed category $A$ enriched in simplicial sets. Simultaneously varying the parameter category $A$ (or parameter space $S$) along with the diagrams $X$ up to weak equivalence is more interesting, and requires new model structures for $A$-diagrams having weak equivalences defined by homotopy colimits, as well as a generalization of Thomason's model structure for small categories to a model structure for simplicial categories.

Analysis Seminar

Speaker: Vincent Grandjean (Fields Institute)

"Gradient trajectories on isolated surface singularities do not oscillate at their limit point"

Time: 15:30 - 16:30

Room: MC 107

Consider $\mathbb{R}^n$ equipped with a real analytic Riemannian metric ${\bf g}$. Let $f : \mathbb{R}^n\to\mathbb{R}$ be a real analytic function singular at $O$ the origin. We would like to understand the dynamics of $\nabla f$ in a neighbourhood of the critical point $O$, where $\nabla f$ stands for the gradient vector field of the function $f$ associated with the metric ${\bf g}$. We are particularly interested in the oscillating/non-oscillating behaviour in a neighbourhood of $O$ of any gradient trajectory accumulating on $O$.

We prove that if a trajectory lies in a real analytic surface with an isolated singularity at $O$, then it cannot oscillate at $O$.

In the first talk, I will recall elementary and well known facts and ideas about the gradient problem. In the second one, I will sketch the proof of our theorem.

This is joint work with Fernando Sanz (Valladolid).

Pizza Seminar

Speaker:

"Achieving the Unachievable"

Time: 16:30 - 17:30

Room: MC 105B

One of the most fascinating enigmas of modern art is the empty circle left at the centre of "Print Gallery", an engraving by Dutch artist M. C. Escher. In 1956, Escher challenged the laws of perspective with Print Gallery and found himself trapped behind an impossible barrier. This uncompleted masterpiece quickly became the most puzzling enigma of Modern Art, for both artists and scientists. Half a century later, mathematician Hendrik Lenstra took everyone by surprise by drawing a fantastic bridge between the intuition of the artist and his own, shattering the Infinity Barrier.

Operads Seminar

Speaker: Fatemeh Bagherzadeh (Western)

"TBA"

Time: 14:30 - 15:30

Room: MC 107

Colloquium

Speaker: Zinovy Reichstein (UBC)

"Simplifying polynomials by Tschirnhaus transformations, old and new"

Time: 15:30 - 16:30

Room: MC 107

In this talk I will revisit the classical topic of polynomial equations in one variable and Tschirnhaus transformations. I will discuss 19th century theorems of Hermite, Joubert and Klein, recent results in this area, and several open problems.

Analysis Seminar

Speaker: Javad Masreghi (Laval)

"Hilbert transform of Lipschitz functions and its generalization"

Time: 13:30 - 14:30

Room: MC 108

The classical theorem of Privalov says that if $u$ is $Lip_\alpha$ with $0 < \alpha <1$, then its Hilbert transform $\tilde{u}$ is also $Lip_\alpha$. However, this result fails for $Lip_1$ functions. In this case, the modulus of continuity of $\tilde{u}$ behaves like $t \log 1/t$ as $t \to 0^+$. We introduce the “generalized Lipschitz class” $Lip_{\alpha(t)}$, which certainly coincides with the classical case when $\alpha(t) \equiv \alpha$, and then show that the above results, as well as some other classical results of Hardy—Littlewood, hold for $Lip_{\alpha(t)}$ functions.

Algebra Seminar

Speaker: Zinovy Reichstein (UBC)

"Essential dimension"

Time: 14:30 - 15:30

Room: MC 107

Informally speaking, the essential dimension of an algebraic object is the minimal number of independent parameters one needs to define it. In the past 15 years this notion has been investigated in several contexts by a range of techniques, and has been found to have interesting and surprising connections to many problems in algebra and algebraic geometry. I will survey some of this research in my talk.

Geometry and Topology

Speaker: Mikael Vejdemo-Johansson (Stanford)

"Persistent cohomology, circle-valued coordinates and periodicity"

Time: 15:30 - 16:30

Room: MC 107

From the topological fact that the circle is the representing space for the functor $X \to H1(X,\mathbb Z)$ follows that by computing degree 1 cohomology and picking cocycle representatives corresponds to computing equivalence classes of continuous maps $X\to S1$. In a research project with Vin de Silva and Dmitriy Morozov, we use this in a data analysis context to produce intrinsic coordinate functions with values on the circle.

We shall discuss the derivation of circle-valued coordinates for point clouds using persistent cohomology to distinguish useful coordinate functions from functions appearing from noise in the data set, and discuss applications to the analysis of periodic signals and periodic dynamical systems.

Analysis Seminar

Speaker: Martin Pinsonnault (Western)

"Symplectic packings of rational ruled surfaces"

Time: 15:30 - 16:30

Room: MC 107

After a short introduction on the symplectic packing problem, we will explain how recent results of B‐H. Li, T.‐J. Li, A. K. Liu, and Gao on symplectic cones lead to a concrete understanding of symplectic packings for rational ruled surfaces. If time permits, we will also explain how this relates to recent work of M. Hutchings on embedded contact homology.

This is joint work with O. Buse.

Colloquium

Speaker: Andrew Nicas (McMaster)

"The horofunction boundary of the Heisenberg group"

Time: 15:30 - 16:30

Room: MC 107

The horofunction compactification of a proper metric space (X,d), also known as the Busemann compactification, is obtained by using the distance function d to embed X into the space of continuous real valued functions on X and taking the closure. The horofunction boundary of X is the complement of the image of X in its horofunction compactification.

We explicitly find the horofunction boundary of the (2n+1)-dimensional Heisenberg group with the Carnot-Caratheodory metric and show that it is homeomorphic to a 2n-dimensional disk. We also show that the Busemann points correspond to the (2n-1)-sphere boundary of this disk and that the compactified Heisenberg group is homeomorphic to a (2n+1)-dimensional sphere. As an application, we find all isometries of the Carnot-Caratheodory metric. This is joint work with Tom Klein.

Algebra Seminar

Speaker: Marcy Robertson (Western)

"Introduction to operads"

Time: 14:30 - 15:30

Room: MC 107

We will give an introductory talk to the theory of operads, giving basic definitions and examples. We will focus on examples in topology, category theory, and computer science.