Supported by Tulane's Mathematics Department and DTRA this mini series of lectures will be held in the Mathematics Department at Tulane University, New Orleans, from March 19 to 21, 2012. The principal speaker will be Professor Shmuel Weinberger from the University of Chicago, who will deliver four lectures centered around the topic of the title. In addition we we have twelve supporting talks by distinguished speakers in their respective fields listed below. We have also invited a speaker from DTRA (Defense Threat Reduction Agency) who will talk about DTRA's problems, and challenges for the mathematical community and topologists in particular. He will also describe funding opportunities with DTRA and their sponsored programs.

Speakers

  • Shmuel Weinberger (Chicago)
  • Yuliy Baryshnikov (UIUC)
  • Vin de Silva (Pomona)
  • John Etnyre (Georgia Tech)
  • Steve Ferry (Rutgers)
  • Robert Kusner (UMass)
  • Sanjeevi Krishnan (Penn)
  • Elizabeth Meckes (Case Western)
  • Erik Guentner (U Hawaii)
  • Jerry Kaminker (UC Davis)
  • Vivian Ferry (Berkeley)
  • Monica Nicolau (Stanford)
  • Christian Whitchurch (DTRA)
  • Yevgeniy Liokumovich (Toronto)

General abstract

Although topology is traditionally a qualitative field, for many reasons, both pure and applied, it has become important to reconsider basic ideas of connectedness, homotopy, and homeomorphism through a quantitative lens. These four lectures will be devoted to various sides of this idea, and will focus on the new problems pressed upon topology, some of what we know and what we don't know at all.

Lecture titles

  • The unreasonable effectiveness of algebraic topology

    Lecture one will discuss why one needs to understand sizes in algebraic topology. Mainly it will focus on the geometry of the Lipschitz constant as a functional on mapping spaces, and will both describe how large function spaces are, and discuss the implications of how algebraic topology analyzes them. In particular, we will discuss some conjectures of Gromov, and also connections between logic, entropy, and variational problems.

  • Persistent Homology of data, groups, function spaces and landscapes

    Lecture two will concentrate on one idea, persistent homology, that had been introduced and studied in the context of data analysis. We will also explain how this idea is related to geometric group theory, and also to the study of closed geodesics on Riemannian manifolds.

  • Bounded propagation on large networks with geometric applications

    Lecture three is about the bounded category. In both topology and global analysis there are strong implications to C^0 bounds on the locations of functions, and such arise naturally in the study of universal covers. I will illustrate this by giving applications to the Novikov conjecture and some connections to problems that also arise in theoretical computer science.

  • Disordered solids and the dynamics of manifolds of bounded geometry

    The final lecture is in some ways rather speculative. At least, it aims to show how a slight variant of a construction that occurs in the study of the physics of quasicrystals unifies several constructions in geometry and topology. It turns out that every manifold of bounded geometry naturally fits into a space of "fluctuations" of itself, that has a "prefoliated structure" giving it a type of dynamical nature. It reflects both the interaction of the local with the global and the recurrence that infinite objects with bounded complexity necessarily have.

Lecture one is based on ideas of Gromov, and also joint work with Nabutovsky and with Ferry. Lecture two grew out of work with Niyogi and Smale, and also thinking about results of Edlesbrunner and Harer, Carlsson, and others. Lecture 3 is based on work of many mathematicians in both topology and non-commutative geometry. I will try to give some credits during the talk. The last lecture is based on joint work with Bellissard and Ulgen-Ylidirim. These lectures are independent of one another, but also have interconnections. It should be enjoyed as a series of short stories with overlapping characters, rather than as a novel.

Registration

There is no registration fee for the lectures and everyone is welcome to attend. If you are planning to participate please send an e-mail message to rako(at)tulane.edu with your name and affiliation, an estimated time of arrival and departure, which will allow us to prepare a name tag for you and an information packet . Note that there is no financial support available for the attendees, thus all attendees will have to cover their expenses. For recommended hotels please see here.


Organizers