2014 Fall Theme Semester on Discrete Networks: Geometry, Dynamics and Applications

Department of Mathematics, University of Pittsburgh
September - December 2014

The goal of the Semester is to expose graduate students, visitors and faculty to a variety of research directions involving discrete geometrical structures, networks, and dynamics of such. In a nutshell, these directions can be summarized as:
  • Morphogenesis of Discrete Structures: How do discrete structures achieve their shapes? How do they themselves evolve over time?
  • Dynamics on Networks: How do network features influence emergent dynamics of connected collections of dynamic nodes?
  • Dynamic Networks: What happens when the nodes have time-dependent states and the network itself is also evolving?

    The semester will consist of concentrated activities, including five mini-courses at graduate level, delivered by: I. Belykh, D. Burago, R. Connelly, L. Deville and M. Newman. The Semester will also include a workshop "Advances in Discrete Networks" in December 12-14, 2014.


    Sep 19, 2014
    Professor Assaf Naor
    Princeton University

    Vertical versus horizontal Poincare inequalities

    Lecture : Fri 3:30pm, 704 Thackeray Hall

    Sep 25 - Sep 26, 2014
    Professor Mark Newman
    Department of Physics and
    Center for the Study of Complex Systems
    University of Michigan

    Epidemics, Erdos numbers, and the Internet:
    The form and function of networks

    Lecture 1 : Th 10am, 104 Lawrence Hall
    Lecture 2 : Th 3pm, 11 Thaw Hall
    Lecture 3 : Fri 10am, 104 Lawrence Hall
    Lecture 4 : Fri 3:30pm, 704 Thackeray Hall

    Oct 13 - Oct 17, 2014
    Professor Robert Connelly
    Department of Mathematics
    Cornell University

    Basic rigidity in three flavors

    Lecture 1 : Mon 3pm, 107 Lawrence Hall
    Lecture 2 : Tue 3pm, 106 Allen Hall
    Lecture 3 : Wed 3pm, G36 Bendum Hall
    Lecture 4 : Th 3pm, 11 Thaw Hall

    Oct 27 - Oct 28, 2014
    Professor Dmitri Burago
    Department of Mathematics
    Penn State University

    Several mathematical stories around discretization
    in Geometry, Dynamics, and PDEs on manifolds

    Lecture 1: Mon 10am, Public Health A 522
    Lecture 2: Mon 3pm, Public Health A 522
    Lecture 3: Tue 10am, Benedum G27
    Lecture 4: Tue 3pm, Allen 106

    The book: A course in metric geometry

    Nov 10 - Nov 14, 2014
    Professor Lee Deville
    Department of Mathematics and
    Illinois Institute for Universal Biology
    University of Illinois at Urbana-Champaign

    Dynamical Systems defined on Networks:
    Symmetry, Stability, and Stochasticity

    Lecture 1: Mon 3pm, 107 Lawrence Hall
    Lecture 2: Tue 3pm, 106 Allen Hall
    Lecture 3: Wed 3pm, G36 Bendum Hall
    Lecture 4: Th 3pm, 11 Thaw Hall

    Preparatory material: Paper 1 , Paper 2

    Dec 8 - Dec 12, 2014
    Professor Igor Belykh
    Department of Mathematics & Statistics
    Georgia State University

    Static and evolving dynamical networks:
    an interplay between the dynamics and graph topology

    Lecture 1: Mon 3pm, Benedum G30
    Lecture 2: Tue 3pm, Benedum G30
    Lecture 3: Wed 3pm, Benedum G30
    Lecture 4: Th 3pm, Benedum G30

    Preparatory material: Paper 1 , Paper 2
    Paper 3 , Paper 4 , Paper 5 , Paper 6

    Funding for the Semester has been provided by the Mathematics Research Center at the University of Pittsburgh .

    The organizing committee consists of: Brent Doiron , Bard Ermentrout , Marta Lewicka and Jonathan Rubin .


    The goal of the Semester is to expose graduate students, visitors and faculty to a variety of research directions involving discrete geometrical structures, networks, and dynamics of such. Recently, there has been a lot of interest in the morphogenesis (shape formation) of semi-rigid structures, particularly low-dimensional ones such as filaments, laminae and their assemblies, which arise routinely in biological systems and their artificial mimics. At the continuum level, differential growth in a body leads to residual strains that result in changes of its shape. Eventually, the growth patterns are expected to be, in turn, regulated by these strains, so that this principle might well be the basis for the physical self-organization of biological tissues. While such questions lie at the interface of biology, chemistry and physics, fundamentally they also have a deeply geometric and analytical character, and may be seen as a variation on a classical theme in differential geometry - that of embedding a shape with a given metric in a space of possibly different dimension. Similar questions can, naturally, also be posed on networks, via the discrete differential geometry approach.

    Networks of nodes arise in other contexts where an intrinsic time-evolving activity pattern is associated with each node and communication between nodes influences the patterns that are generated. Biological examples include models of disease spread, in which nodes are members of a susceptible population and the dynamics of each node characterizes the evolution of any disease-related quantities that population member carries, and models of neuronal communication, where the membrane potential and concentrations of various ions associated with each neuron change over time and neurons communicate electrically or chemically based on certain connectivity patterns. In such applications, it is generally of interest to describe the relation between emergent dynamics of a distributed network, such as network-wide patterns of infection or action potential generation in the examples above, and the pattern of connections between network elements.

    Finally, in the case of dynamic networks, the connections between dynamic nodes may themselves evolve. Certainly, contact networks of individuals in a society change over time, while at a more microscopic level, cellular networks change their interactions either to achieve a new state or to maintain an existing state (i.e., homeostasis); for example, learning in neural networks occurs through changes in synaptic connectivities or weights, known collectively as synaptic plasticity. The forward and inverse problems become more complicated on such dynamic networks. Here, it is also important to confront control questions, such as how to evolve a network to achieve a desired state, based on evolution laws prescribed by the setting in which the network arises.

    Previous Theme Semesters at the University of Pittsburgh: