Analysis Seminar

Speaker: Michael Francis (Western)

"The b-Newlander-Nirenberg theorem"

Time: 14:30

Room: MC108

Melrose introduced the formalism of b-geometry as a tool for studying partial differential operators on a smooth manifold M that suffer a first order degeneracy along a given hypersurface Z. The b-tangent bundle is the vector bundle whose sections are smooth vector fields defined on all of M and tangent along Z. Many of the classical geometries admit "b-analogues" in which the b-tangent bundle fills the role of the usual tangent bundle (so one has symplectic b-geometry, Riemannian b-geometry, etc). Complex b-geometry was introduced by Mendoza. In this talk, we will discuss the "b-Newlander-Nirenberg theorem": every complex b-manifold is locally isomorphic to some standard model. This allows one to define complex b-manifolds in the way one might hope, in terms of appropriately defined "b-holomorphic charts". This is joint work with Tatyana Barron.

Geometry and Topology

Speaker: Matthias Franz (Western)

"An $A_\infty$-version of the Eilenberg-Moore theorem"

Time: 15:30

Room: MC 107

I will discuss a new product structure on the two-sided bar constructions of singular cochains. This bar construction is used in the Eilenberg-Moore theorem to compute the cohomology of pull-backs of fibrations. The product structure is based on so-called homotopy Gerstenhaber operations on singular cochains, transforming the two-sided bar construction into an $A_\infty$-algebra. This type of algebra is associative up to a strong form of homotopy. The new product includes those previously defined by Baues, Gerstenhaber-Voronov, Kadeishvili-Saneblidze, and Carlson-Franz as special cases. Consequently, the multiplicative cohomology isomorphism from the Eilenberg-Moore theorem is elevated to a quasi-isomorphism of $A_\infty$-algebras.

Also, please make sure the event does not appear twice on the calendar. Thanks!

Colloquium

Speaker: Lyle Muller (Math department, Western)

"Spatiotemporal dynamics in neural systems: from data to mathematical models and computation"

Time: 15:30

Room: MC 107

Neurons in cortex are connected in intricate patterns, with local- and long-range connections and time delays for transmitting signals. In recent work, we have found that spontaneous and stimulus-driven waves travel over these networks, changing excitability of the neurons and shaping perceptual sensitivity. Understanding how these networks generate these sophisticated dynamics, however, remains an open problem. This is due, in part, to the fact that connecting the specific structure of networks to the nonlinear dynamics that will result is a difficult problem in general. Further, experiments suggest one mechanism for these waves could be the distance-dependent time delays due to transmitting spikes along the axons connecting neurons across these networks. Analyzing the underlying network mechanism for these waves thus represents an additional challenge, as we need to consider systems with many time delays.

In this talk, I will present recent results from my research team connecting the structure of individual networks to the resulting dynamics in systems of nonlinear Kuramoto oscillators. We introduce a complex-valued approach to the Kuramoto model that allows connecting the eigenspectrum of the graph adjacency matrix to the nonlinear dynamics that result in individual simulations of this system. This approach allows predicting the specific spatiotemporal pattern that will result from the connectivity pattern in an individual network. An extension of this approach allows predicting the specific spatiotemporal patterns generated by distance-dependent time delays from spike transmission in these systems. Finally, I will present our latest efforts to understand computation with spatiotemporal dynamics in neural systems using these nonlinear network models.

Graduate Seminar

Speaker: Prakash Singh (Western)

"The Hofer diameter problem for rational symplectic manifolds"

Time: 15:30

Room: MC 107

In general, Lie groups do not admit bi-invariant metrics, and infinite dimensional Lie groups should not admit such metrics either. But surprisingly, Ham admits one such metric (in fact, unique in a sense), called the 'Hofer metric', discovered by Hofer in the 90s. People have been studying the large-scale geometry properties of this metric for a long time, but such studies were restricted to either 2-dimensions, monotone symplectic manifolds, or to aspherical manifolds. In particular, it is widely conjectured that the hofer diameter is infinite for every closed symplectic manifold, and this conjecture has been settled for the above-mentioned manifolds. I will talk about the diameter problem associated with this metric for some rational ruled manifolds like CP2, S2 x S2, and their blow-ups, using methods from quantum homology and spectral invariants on them. I will prove the conjecture for CP2 and S2 x S2, and I will prove it under a mild assumption (but unproven) for S2 x S2 blown up once.