# Topology

Thursday, November 6, 2008 - 8:45am - 9:30am

William Noid (The Pennsylvania State University)

Coarse-grained (CG) models provide a promising computational tool for investigating slow complex processes that cannot be adequately studied using more detailed models. However, unless the CG model is consistent with an accurate high-resolution model, the results of CG modeling may be misleading. The present talk describes a statistical mechanical framework that provides a rigorous “multiscale bridge” connecting models with different resolution.

Wednesday, October 29, 2008 - 3:00pm - 3:50pm

Gunnar Carlsson (Stanford University)

The nature and quantity of data arising out of scientific applications requires novel methods, both for exploratory analysis as well as analysis of significance and validation. One set of new methods relies on ideas and methods from topology. The study of data sets requires an extension of standard methods, which we refer to as persistent topology. We will survey this area, and show that algebraic methods can be applied both to the exploratory and the validation side of investigations, and show some examples.

Wednesday, March 5, 2014 - 11:30am - 12:20pm

Michael Farber (University of Warwick)

I will discuss several probabilistic models producing simplicial complexes, manifolds and discrete

groups. Random simplicial complexes are high dimensional analogues of random graphs and can be

used for studying the behaviour of large systems or networks depending on many random

parameters. We are interested in properties of random spaces which are satisfies with probability

tending to one. Using probabilistic models one may also test probabilistically the validity of open

groups. Random simplicial complexes are high dimensional analogues of random graphs and can be

used for studying the behaviour of large systems or networks depending on many random

parameters. We are interested in properties of random spaces which are satisfies with probability

tending to one. Using probabilistic models one may also test probabilistically the validity of open

Thursday, March 6, 2014 - 10:15am - 11:05am

Elizabeth Munch (University of Minnesota, Twin Cities)

In order to understand the properties of a real-valued function on a topological space, we can study the Reeb graph of that function. The Reeb graph is a construction which summarizes the connectivity of the level sets. Since it is efficient to compute and is a useful descriptor for the function, it has found its place in many applications. As with many other constructions in computational topology, we are interested in how to deal with this construction in the context of noise.

Tuesday, March 4, 2014 - 2:00pm - 2:50pm

Michael Robinson (American University)

Recently, sheaves have become useful for addressing problems in signal processing. Morphisms between sheaves provide a handy formal construct for understanding the relationship between measurements, intermediate data, and processed outputs. The resulting topological filters generalize the linear filters that engineers use extensively, but also describe novel, nonlinear filters. Because they are built from sheaves, the local structure of these filters can be tailored easily and may provide a solid theoretical grounding for nonlinear matched filters.

Monday, March 3, 2014 - 11:30am - 12:20pm

Alberto Speranzon (United Technologies Corporation)

In this talk we will describe methodologies to localize both a single and a team of vehicles navigating in a complex environment without GPS. During the first part of the talk, we will consider the situation when vehicles (or a single vehicle navigating in an environment with multiple beacons) can measure their relative (inter-vehicle) distances. In this case, the problem can be posed as a distributed graph embedding problem.

Monday, December 9, 2013 - 10:15am - 11:05am

Raul Rabadan (Columbia University)

The tree structure is currently the accepted paradigm to represent evolutionary relationships between organisms, species or other taxa. However, horizontal, or reticulate, genomic exchanges are pervasive in nature and confound characterization of phylogenetic trees. Drawing from algebraic topology, we present a unique evolutionary framework that comprehensively captures both clonal and reticulate evolution. We show that whereas clonal evolution can be summarized as a tree, reticulate evolution exhibits nontrivial topology of dimension greater than zero.

Thursday, October 31, 2013 - 9:00am - 9:50am

John Klein (Wayne State University)

Abstract: An area of interest in statistical mechanics is the study of

statistical distributions of stochastic currents generated in graphs.

It turns out that this problem amounts to the study of periodic families

of Markov processes that evolve according to the master equation.

Physicists have observed that, for almost every generated current,

quantization occurs in the adiabatic and low temperature limits.

My main goal in this talk will be to explain how this story can be understood using

statistical distributions of stochastic currents generated in graphs.

It turns out that this problem amounts to the study of periodic families

of Markov processes that evolve according to the master equation.

Physicists have observed that, for almost every generated current,

quantization occurs in the adiabatic and low temperature limits.

My main goal in this talk will be to explain how this story can be understood using

Friday, October 4, 2013 - 1:30pm - 2:45pm

Jonathan Taylor (Stanford University)

In the first lecture, we will provide an overview of the various ways that topological information

is used in signal detection problems in functional MRI (fMRI) and other

imaging applications. The principal tool used involves computing the expected

number of critical points of various types of a smooth random field under

some predetermined null hypothesis. We will describe roughly

how some of these calculations can be carried out

using the so-called Gaussian Kinematic Formula.

is used in signal detection problems in functional MRI (fMRI) and other

imaging applications. The principal tool used involves computing the expected

number of critical points of various types of a smooth random field under

some predetermined null hypothesis. We will describe roughly

how some of these calculations can be carried out

using the so-called Gaussian Kinematic Formula.

Monday, October 7, 2013 - 3:15pm - 4:05pm

Yuriy Mileyko (University of Hawaii at Manoa)

Problems in applied topology often require computation of an optimal (in some sense) representative among shapes satisfying particular topological constraints. In many cases this is a very challenging, if not infeasible task. Interestingly, such complex problem often can be tackled using biologically inspired algorithms. For example, slime mold can be used to construct optimal networks, ants can teach us how to find shortest paths, and simple evolutionary principles can help us find global optima.