During each Annual Thematic Program, several seminars are offered. Talks will
include graduate-level lectures as well as seminars covering various topics
related to the theme.
Representing Large and Complex Data
Gunnar Carlsson, Stanford University
September 18, 2013 10:30 AM - 12:00 PM
Lind 305 [Map]Abstract forthcoming.
The Structure of Persistence Modules
Vin de Silva, Pomona College
September 18, 2013 1:30 PM - 2:30 PM
Lind 305 [Map]Abstract forthcoming.
Persistent Homology and Applications
Gunnar Carlsson, Stanford University
September 24, 2013 3:30 PM - 5:00 PM
Keller Hall 3-180 [Map]Abstract forthcoming.
Topological Methods in the Analysis of Biological Data
F. Javier Arsuaga, San Francisco State University
September 25, 2013 1:30 PM - 2:30 PM
Keller Hall 3-180 [Map]Abstract
In this introductory talk I will explain some of the topological tools that we are using to analyze data obtained in molecular biology and genomics experiments. The first part of my talk will be devoted to present well established applications of knot theory to the analysis of 3D DNA structure. In the second half of my talk, I will introduce our homology methods for the analysis of copy number and gene expression data in cancer.
Persistent Holes in the Universe
Pratyush Pranav, Rijksuniversiteit te Groningen
October 14, 2013 11:00 AM - 12:00 PM
Lind 305 [Map]Abstract
I will present a new topological formalism that, for the first time,
describes topology as a multi-scale concept. This has a direct and strong
relevance to the topological analysis of structure formation process in
the cosmos, given that this proceeds in a hierarchical fashion. Rooted in
algebraic topology, the concepts I will describe stem from (persistence)
homology and Morse theory. Although the mathematical theory behind the
concepts have been known for over a century, only recently have they
become of practical relevance, due to breakthroughs in computational
topology over the last decade.
Taking gaussian random fields and the web-like spatial distribution of
matter in the Universe as running examples, I will demonstrate that the
formalism allows us to describe topology at a level of detail that far
supersedes those provided by the standard topological descriptors like
Euler characteristic and genus. In this context, I will introduce
persistence and ratio landscapes as an empirical statistical description
of persistence homology. They prove to be powerful tools to discriminate
between the spatial structure emanating in different cosmologies. Most
promising is our recent realization that it provides for a novel way of
hunting for primordial non-gaussianities.
In a related aspect, I will present a software for interactive
visualization of the hierarchical topological structures. It exploits the
geometric aspects of Morse theory, to detect, describe and quantify the
filamentary patterns of the Cosmic web. The software is a promising tool
applicable to the analysis of structural patterns in a wide range of
astronomical areas of interest -- the detection and characterization of
stellar streams at galactic scales being a promising candidate.
Beyond the Borsuk-Ulam/Dold theorem (in Combinatorial Geometry)
Pavle Blagojević, Freie Universität Berlin
October 16, 2013 1:30 PM - 2:30 PM
Lind 305 [Map]Abstract
In this talk we present an evolution of equivariant topology methods in Combinatorial Geometry.
We start with
(a) the Topological Radon's theorem, an application of the Borsuk-Ulam theorem,
and proceed, via non-planarity of K_{3,3}, to
(b) the Topological Tverberg and the Weak Colored Tverberg theorem for primes,
which are applications of Dold's theorem, to continue with
(c) the Topological Tverberg for prime powers, an application beyond Dold's theorem based on
the connectivity and localization theorem for elementary abelian groups,
to finally ask:
"What needs to be done in the case of Barany-Larman conjecture and Nandakumar & Ramana-Rao
problem when all the previously known methods fail?"
Climate Seminar: Budyko's Model as a Dynamical System
Richard McGehee, University of Minnesota, Twin Cities
October 22, 2013 11:15 AM - 12:15 PM
Lind 305 [Map]Abstract forthcoming.
Read more...
A Primer on Configuration Spaces: Definitions, Examples, Fibrations, and Monodromy
Frederick Cohen, University of Rochester
October 23, 2013 10:30 AM - 12:00 PM
Lind 305 [Map]Abstract forthcoming.
Towards a Topos Bais for Persistent Homology
Mikael Vejdemo-Johansson, Royal Institute of Technology (KTH)
October 23, 2013 1:30 PM - 2:30 PM
Lind 305 [Map]Abstract
We describe a topos of sheaves with the property that classical persistent homology of a filtered complex (should) be the internal simplicial homology functor of the logic specified by the base space of the sheaves. All relevant background to understand definitions and their ramifications will be provided in the talk.
Applications at the Interface of Polyhedral Products and Configurations Spaces
Frederick Cohen, University of Rochester
October 25, 2013 10:30 AM - 12:00 PM
Lind 305 [Map]Abstract forthcoming.
Celestial Influences on Glacial Cycles
Richard McGehee, University of Minnesota, Twin Cities
October 29, 2013 11:15 AM - 12:15 PM
Lind 305 [Map]Abstract forthcoming.
Read more...
Some Properties of Clustering Methods
Facundo Mémoli, The Ohio State University
November 6, 2013 1:30 PM - 2:30 PM
Lind 305 [Map]Abstract
I'll describe joint work with Gunnar Carlsson about clustering methods. I will concentrate on the characterization, stability, and convergence of hierarchical clustering methods operating on metric spaces.Some properties of clustering methods
Persistent Homology for Metric Measures Spaces, Hypothesis Testing, and Confidence Intervals
Andrew Blumberg, University of Texas, Austin
November 13, 2013 10:30 AM - 12:00 PM
Lind 305 [Map]Abstract forthcoming.
Asymptotic topology and group actions
Greg Bell, University of North Carolina, Greensboro
November 13, 2013 1:30 PM - 2:30 PM
Lind 305 [Map]Abstract
The large-scale approach to metric spaces was popularized by Gromov who applied these ideas to finitely generated groups. Although this "asymptotic topology" is interesting in its own right, a great deal of interest in large-scale dimension followed from Guoliang Yu’s result that groups with finite asymptotic dimension satisfy the Novikov higher signature conjecture. In this talk we’ll define some dimension-theoretic notions in asymptotic topology and discuss the role geometric group theory plays in proving large-scale analogs of classical results.
Quantitative Homotopy Theory in Topological Data Analysis
Andrew Blumberg, University of Texas, Austin
November 18, 2013 10:30 AM - 12:00 PM
Keller Hall 3-180 [Map]Abstract forthcoming.
Limit Theory for Random Cech Complexes
Yogeshwaran Dhandapani, Technion-Israel Institute of Technology
November 20, 2013 1:30 PM - 2:30 PM
Keller Hall 3-180 [Map]Abstract
In this talk, I shall try to describe some asymptotic results for random Cech complexes. These shall concern thresholds for vanishing and formation of Homology groups as well as identify the limiting distribution of the Betti numbers under different regimes. I shall try to emphasize the usefulness of various stochastic geometric tools such as martingale techniques, stabilization theory, Palm formula et al in the context of random topology. Though most of the talk shall be restricted to the case of Poisson or i.i.d. point process, we shall briefly discuss extensions to general point processes too.
Persistence Modules and Category Theory
Vin de Silva, Pomona College
December 4, 2013 10:30 AM - 12:00 PM
Lind 305 [Map]Abstract forthcoming.
Scheduling spaces in concurrency via directed topology
Martin Raussen, Aalborg University
December 4, 2013 1:30 PM - 2:30 PM
Lind 305 [Map]Abstract
Concurrency theory in Computer Science studies the effects that arise when
several processors run simultaneously sharing common resources. It
attempts to advise methods to deal with the "state space explosion
problem", sometimes using models with a combinatorial/topological flavor.
It is a common feature of these models that a schedule corresponds to a
directed path (d-path), and that d-homotopies (preserving the directions)
result in computations with the same result.
I shall discuss particular classical examples of directed spaces, a class
of Higher Dimensional Automata (HDA). For such a state space, I shall
describe a (nerve lemma) method that determines the homotopy type of the
space of traces (schedules) as a prodsimplicial complex - with products of
simplices as building blocks. A description of that complex opens up for
(machine) calculations of homology groups and other topological invariants
of the trace space. The determination of the path components of trace
space is particularly important for applications.
Unfortunately, the resulting prodsimplicial complexes grow still quickly
in both dimension and the number of cells. I shall sketch ongoing work
with K. Ziemiański (Warsaw) that tries to find smaller homotopy equivalent
simplicial complexes via suitable homotopy decompositions of trace spaces.
Persistence Modules and Category Theory
Vin de Silva, Pomona College
December 6, 2013 10:30 AM - 12:00 PM
Lind 305 [Map]Abstract forthcoming.
The Variational Bicomplex
Irina Kogan, North Carolina State University
January 29, 2014 1:30 PM - 2:30 PM
Lind 305 [Map]Abstract
Introduction of the variational bicomplex can be motivated by drawing
an analogy with vector calculus. It is well known that one can
reformulate vector calculus in terms of differential forms, and use
the exterior derivative to represent gradient, divergence, and curl
operators. Thus vector calculus can be efficiently expressed by the de
Rham complex. Similarly to vector calculus, many important aspects of
variational calculus, such as Euler-Lagrange operator, Helmholtz
operator, or Noether correspondence, can be formulated in terms of
complexes of differential forms. This leads to notion of the
variational bicomplex (and related notions of the variational complex,
the variational spectral sequence). These constructions originated
with the work of Dedecker (1957) and a large body of literature has
appeared afterwards. The purpose of my talk is to define the
variational bicomplex, explain how it encodes various aspects of
variational calculus and what role is played by its cohomology.
Towards Utilizing Topological Data Analysis for Studying Machining Models
Firas Khasawneh, State University of New York Institute of Technology
February 19, 2014 1:30 PM - 2:30 PM
Lind 305 [Map]Abstract
Delay differential equations (DDEs) appear in many models in science and engineering either as an intrinsic component or as a modeling decision. Machining dynamics are an important example of processes that include delays as an intrinsic part of the system. The infinite dimensionality of DDEs significantly complicates the resulting analysis from both an analytical and numerical perspective. Recent developments in the theory of DDEs have helped with obtaining more accurate mathematical models for capturing the process dynamics. However, even though machining processes are known to be stochastic, the majority of existing machining models are deterministic. Further, data analysis methods for DDEs are either few or non-existant largely due to the non-Markovian nature of these systems. This talk briefly discusses the challenges associated with DDEs and how they are incorporated into machining dynamics models. We also discuss some new results which suggest that topological data analysis, specifically persistent homology, can be a very useful tool for analyzing numerical and experimental time series corresponding to time delay systems.
Lipschitz Extensions and Higher-Order Metric Certificates
Vin de Silva, Pomona College
February 26, 2014 1:30 PM - 2:30 PM
Lind 409 [Map]Abstract
The traditional TDA (topological data analysis) pipeline converts data to a filtered simplicial complex, and the filtered simplicial complexes to a persistence diagram (PD). The PD can then be studied for information about the input data. This process can be recursed: given lots of data sets, each can be turned into a PD, and this collection of PDs can be viewed as a data set in diagram space. One can then build Cech or Vietoris-Rips complexes from this data.
Homology-vanishing Theorems for Random Simplicial Complexes
Matthew Kahle, The Ohio State University
March 12, 2014 1:30 PM - 2:30 PM
Lind 305 [Map]Abstract
Linial-Meshulam random 2-complexes are analogues of Erdős-Rényi random
graphs, and their topological properties have been the subject of
extensive study. We now know a few methods for proving homology-vanishing
theorems in settings like this: (1) cocycle counting, i.e. combinatorial
methods, (2) spectral methods, using concentration results for spectral
gaps of certain random matrices, and (3) what might be called random
linear algebra. Each of these has its advantages, and I will briefly
overview each. The main new result I'll discuss by the end of the talk is
using the third method to describe the threshold for vanishing of homology
with integer coefficients, a problem which was very resistant to earlier
methods. This is joint work with Christopher Hoffman and Elliot Paquette.
Multiparameter Persistent Homology for Shape Comparison: Continuous
versus Discrete
Tomasz Kaczynski, University of Sherbrooke
March 19, 2014 1:30 PM - 2:30 PM
Lind 305 [Map]Abstract
The theory of multiparameter persistent homology was initially
developed in the discrete setting of filtered simplicial complexes.
Stability of persistence was proved for topological spaces filtered by
continuous vector-valued functions. Our aim is to provide a formal
bridge between the continuous setting, where stability properties hold,
and the discrete setting, where actual computations are carried out. The
existence of this bridge is not obvious for two reasons. One is related
to the Sard’s lemma, namely, we do not have the isolation of critical
values for generic vector functions. The other one is due to the
phenomenon of structural gap between the two settings which appears in
the multi-parameter case when using the standard piecewise linear
interpolation of the discrete model.
Necktie knots, formal languages and network security
Mikael Vejdemo-Johansson, Royal Institute of Technology (KTH)
March 26, 2014 1:30 PM - 2:30 PM
Lind 305 [Map]Abstract
A chore for some, space for personal expression for others, the necktie knot used to have very few speci?c knots in widespread use. In their 1999 paper, Fink & Mao (Designing tie knots by random walks." Nature 398, no. 6722 (1999): 31-32) list all possible ways to tie a necktie. They limit their enumeration task by focusing on knots that present a flat front just like all the classical tie knots. This way they established a list of 85 possible tie knots.
Tie knots with intricate patterns of the necktie winding into symmetric but no longer at front displays have emerged in the past decade, introduced by the movie Matrix Reloaded and recreated hobbyists. These tie knots are tied with the narrow end of the tie, wrapping it to create patterns on the surface of the tie knot. As such these knots are not covered by the listing proposed by Fink & Mao.
With a team of collaborators I have extended the listing by Fink and Mao to cover these new tie knots. While doing this we have been able to determine the computational complexity classes of the grammar that describes tie knots.
The formal language techniques that help us analyze the tie knot grammars are used in contemporary security research: a large class of security problems online emerge from different implementations of a communications protocol disagreeing on the actual grammar used. We will talk about the connections between the language techniques for tie knots and those that help us analyze security.
http://www.newscientist.com/article/dn25019-matrix-villain-spawns-177000-ways-to-knot-a-tie.html
http://phys.org/news/2014-02-mathematicians-ways.html
Random Linear Algebra and the Vanishing Threshold for Integer Homology
Matthew Kahle, The Ohio State University
April 2, 2014 1:30 PM - 2:30 PM
Lind 305 [Map]Abstract
There are a few different approaches now to
homology-vanishing theorems for random simplicial complexes: (1)
cocycle counting, (2) spectral gap methods, and a new approach, (3)
random linear algebra. In this talk I will talk about the third
approach, and new joint work with Hoffman and Paquette, giving the
vanishing threshold for homology with integer coefficients. This was a
problem that had resisted attempts by the other methods for several
years, and the approach is also elementary. (This is a follow up to an
earlier talk, but this talk will also aim to be self contained.)
A-infinity Persistence
Aniceto Murillo, University of Málaga
April 9, 2014 1:30 PM - 2:30 PM
Lind 305 [Map]Abstract
We will introduce A-infinity persistence of a given homology filtration of topological spaces. This is a family homological invariants which provide information not readily available by the (persistent) Betti numbers of the given filtration. This may help to detect noise, not just in the simplicial structure of the filtration but in further geometrical properties in which the higher codiagonals of the A-infinity structure are translated.
Topology and coding in neural networks
Chad Giusti, University of Nebraska
April 16, 2014 1:30 PM - 2:30 PM
Lind 305 [Map]Abstract
Natural and artificial neural networks are traditionally studied using classical methods from dynamical systems and linear algebra. However, there are many problems of interest for which topological tools can offer new perspectives. Here, we survey our recent work on the relationship between the structure of a network and that of the data it encodes: first, using classical tools from combinatorial topology, we show that a simple one-layer feed-forward network can encode any prescribed simplicial complex, but that a large class of non-convex codes cannot be encoded; second, using recent notions and computer tools for computing persistent homology, we discover signatures of "geometric structure" in the correlations of activity among neurons in the hippocampal network in rats. In the latter project, we frame persistent homology of clique complexes as a tool for the study of equivalence classes of real symmetric matrices under the action of the group of monotone increasing functions, building on work of M. Kahle describing "Betti curves" of such classes to interpret the results. Time permitting, we will also touch on work in progress regarding detecting convexity of codes. This is in various parts joint work with C. Curto, V. Itskov and W. Kronholm. No background in biology or networks is assumed!
Chromatic Homology and Graph Configuration Spaces
Radmila Sazdanović, North Carolina State University
April 23, 2014 1:30 PM - 2:30 PM
Lind 305 [Map]Abstract
We will discuss the proof of the conjecture due to M. Khovanov relating the algebraic and topological categorification of the chromatic polynomial. We show that there exists a spectral sequence relating the chromatic graph homology and the homology of a graph configuration space and discuss higher differentials.
Studying Recombination Pathways Using Band Surgeries
Mariel Vazquez, San Francisco State University
May 7, 2014 1:00 PM - 2:00 PM
Lind 305 [Map]Abstract
DNA replication is the mechanism by which a cell copies its genetic code prior to dividing into two daughter cells. DNA replication must produce two identical and independent copies of the parental DNA molecule(s). However, replication of circular DNA results in two topologically linked DNA circles. In this case, cell survival relies on removing this topological obstruction. It has been shown experimentally that recombinases XerC and XerD can unlink the chromosomal links. We model recombination as a band surgery, and use the tangle method of Ernst and Sumners (1990) and other topological methods to show definitively that there is a unique shortest pathway of unlinking by Xer recombination that strictly reduces the complexity of the links at every step. We also compute the mechanism of action of the enzymes at each step along this pathway and provide a 3D interpretation of the results.
This is joint work with Koya Shimokawa, Kai Ishihara, Ian Grainge and David Sherratt.
Introduction to Gröbner Bases
Mikael Vejdemo-Johansson, Royal Institute of Technology (KTH)
May 15, 2014 10:00 AM - 11:30 AM
Lind 305 [Map]Abstract
There are several connections between algorithms for classical persistent homology, representations of persistence modules, and computational algebra. The connections to matrix algebra and linear algebra are relatively well known, and anchored in how the community works with and talks about algorithms. Far less pervasive, but a good source of intuitions and techniques, are methods originally developed to deal with more complex cases: Gröbner bases build up the core of computational commutative algebra, handling quotient rings of polynomial rings in several variables and their modules. The techniques and terminology from Gröbner basis techniques match up nicely - not only with multi-dimensional persistence, but also with techniques and algorithms for classical persistence - and can contribute intuitions and new algorithms.
In this talk, we will be looking into the basics of computational commutative algebra and find points of contact with the classical persistence algorithm. Based on this new language, we will then introduce perspectives on algebraic constructions for persistence modules that generalize and build new contexts for previous work on kernel, cokernel and image persistence.
Previous Annual Seminars