January 25
1:00-1:20 pm Registration
1:20-1:30 pm Introduction and Welcome
1:30-2:30 pm Talk 1 Dennis Sullivan (Graduate Center and Stony Brook University )
Title: Combinatorial Hydrodynamics in 3D
Abstract:If the viscosity of an incompressible fluid is vanishingly small, the time evolution is determined by the mantra “the curl of velocity at each time is transported along by the fluid motion.” Assuming a cubical cell decomposition of periodic three space we translate the mantra into combinatorial topology as follows:
• the divergence free velocity translates to a one cochain X which is a cycle when construed as a one chain using the cellular basis.
• the curl of the velocity translates to the coboundary Y of the combi- natorial velocity X.
• the transport by the motion is effected by constructing for an appropriate time interval dT the corresponding cochain mapping via Cartan’s magic formula: u is mapped to u + {dS + Sd}[u]dT where S is the transpose of the operator v is mapped to -X ∧ v. Here ∧ is the combinatorial analogue of the wedge product defined naturally in cubical complexes.
The implied algorithm is being tested with Scott Wilson, who used results of Dodziuk in his thesis. Dodziuk’s theme also resonates with the ideas here.
2:30-3:00 pm Break
3:00-4:00 pm Talk 2 Varghese Mathai (The University of Adelaide)
Title: Spectral gap-labelling conjecture for magnetic Schrödinger operators
Abstract: Given a constant magnetic field on Euclidean space Rp determined by a skew-symmetric (p×p) matrix Θ, and a Zp-invariant probability measure μ on the disorder set Σ, we conjecture that the corresponding Integrated Density of States of any self-adjoint operator affiliated to the twisted crossed product algebra C(Σ) ⋊ σ Zp takes on values on spectral gaps in an explicit Z-module involving Pfaffians of Θ and its sub-matrices that we describe, where σ is the multiplier on Zp associated to Θ.
4:00-4:30 pm Break
4:30-5:30 pm Talk 3 Matthias Keller (University of Potsdam)
Title: On Cheeger’s inequality for graphs
Abstract: In 1984 Dodziuk proved a discrete analogue to Cheeger’s famous estimate relating the spectrum of the Laplacian to an isoperimetric constant. However, in contrast to its continuum counterpart, very basic geometric notions such as the volume or the area of the boundary are not 1 canonically determined in the discrete case. This led to a variety of different estimates following up on Dodziuk’s work. We give an overview of this development and present a systematic approach for general weighted graphs.
Time TBD: Banquet at Ravagh Midtown
January 26
10:00-11:00 am Talk 1 Alexander Grigor’yan (Bielefeld University)
Title: Heat kernels on ultra-metric spaces
Abstract: We discuss a family of jump processes on ultra-metric spaces and obtain estimates of their heat kernels.
11:00-11:30 am Break
11:30-12:30 pm Talk 2 Christina Sormani (Lehman College and CUNYGC)
Title: Limits of Manifolds with Positive Scalar Curvature
Abstract: Recall that the scalar curvature at a point p in a Riemannian manifold can be determined using the infinitesimal behavior of the volume of balls about that point. Gromov has suggested that sequences of Riemannian manifolds with positive scalar curvature converge in the intrinsic flat sense to limit spaces with some generalized notion of positive scalar curvature. In joint work with Basilio and Dodziuk, we make this conjecture more precise after providing counter examples to the simplest interpretation of this suggestion. Our counter example is constructed using a method we call sewing that requires precise estimates on the diameter and volume of Gromov-Lawson Schoen-Yau tunnels. The sequences of manifolds we construct converge in the intrinsic flat, Gromov-Hausdorff and metric measure sense to limit spaces which fail the infinitesimal volume condition. A series of related conjectures were presented at the Fields Institute this summer and resulting work with students and postdocs there may be presented as well as subsequent work with Basilio. These and all other papers on intrinsic flat convergence appear here; 2 https://sites.google.com/site/intrinsicflatconvergence
12:30-2:30 pm Lunch break
2:30-3:30 pm Talk 3 Hugo Parlier (University of Luxembourg)
Title: Quantifying isospectral finiteness
Abstract: Associated to a closed hyperbolic surface is its length spectrum, the set of the lengths of all of its closed geodesics. Two surfaces are said to be isospectral if they share the same length spectrum. The talk will be about the following questions and how they relate:
- How many questions do you need to ask a length spectrum to determine it?
- How many different surfaces can be isospectral to a surface of a given genus?
The approach will include finding adapted coordinate sets for moduli spaces and exploring McShane type identities.
3:30-4:00 pm Break
4:00-5:00 pm Talk 4 Radoslaw Wojciechowski (York College and the Graduate Center)
Title: Stochastic completeness of graphs – ten years after
Abstract: In my thesis defended under the guidance of Józef Dodziuk approximately ten years ago, we studied three interrelated problems for the graph Laplacian: essential self-adjointness, discreteness of the spectrum and stochastic completeness or conservativeness. I will focus on the third of these and present a historical overview of developments since the time of the thesis. This includes finding an analogue to a volume growth criterion for stochastic completeness of manifolds proven by Grigor’yan in 1986. In the graph case, this was first shown by Folz, reproven using analytical tools by Huang and, more recently, a new approach was given by Huang, Keller and Schmidt. Finally, we will present a criterion for stochastic completeness using a recently developed notion of curvature for graphs proven in joint work with Florentin Münch.