Talks

Upcoming talks, mini-symposia, and conferences

• Speaking at the Special Session on Topological Data Analysis, Statistics and Applications, AMS Southeastern Sectional Meeting, Auburn University, Mar 15-17, 2019.
• Speaking at the Arches Topology Conference, Hurricane, Utah, April 24-27, 2019
• Speaking at the NSF-CBMS Conference and Software Day on Topological Methods in Machine Learning and Artificial Intelligence, College of Charleston, South Carolina, May 18, 2019.
• Visiting Oberwolfach for a Research in Pairs visit with Florian Frick and Žiga Virk, June 2-14, 2019.
• Giving a tutorial on persistent homology at the Summer School on Data Science for Dynamical Systems, Lorentz Center, Leiden University, Netherlands, July 2, 2019.
• With Maria D'Orsogna, Rachel Neville, Jose Perea, Chad Topaz, co-organizing an ICERM Topical Workshop on Applied mathematical modeling with topological techniques, Aug 5-9, 2019.

Research talks

• Metric reconstruction via optimal transport. [Abstract, Slides, Notes]
  
  Given a metric space X and a distance threshold r > 0, the Vietoris-Rips simplicial complex has as its simplices the finite subsets of X of diameter less than r. A theorem of Jean-Claude Hausmann states that if X is a Riemannian manifold and r is sufficiently small, then the Vietoris-Rips complex is homotopy equivalent to the original manifold. Janko Latschev proves an analogous theorem for sufficiently dense samplings. Little is known about the behavior of Vietoris-Rips complexes for larger values of r, even though these complexes arise naturally when using persistent homology. We describe how as r increases, the Vietoris-Rips complex of the circle obtains the homotopy types of the circle, the 3-sphere, the 5-sphere, the 7-sphere, ..., until finally it is contractible. These homotopy types are related to cyclic and centrally symmetric polytopes and orbitopes. We argue that infinite Vietoris-Rips complexes should be equipped with a different topology: an optimal transport or Wasserstein metric thickening the metric on X. With this new metric, we describe the first change in homotopy type (as r increases) of Vietoris-Rips complexes of higher-dimensional spheres. Joint work with Michal Adamaszek and Florian Frick.

  • AMS Special Session on Topological Data Analysis: Theory and Applications, Joint Mathematics Meetings, Baltimore, Jan 2019. [Slides with an application to Borsuk-Ulam theorems]
  • Topology Seminar, Texas State University, Nov 2018. [Abstract]
  • Upstate New York Topology Seminar (UNYTS), University of Albany, Nov 2018. [Notes]
  • Lafayette-Lehigh Geometry-Topology Seminar, Apr 2018. [Notes]
  • AATRN Online Seminar, Oct 2017. [Video]
  • 58th Cascade Topology Seminar, University of British Columbia, May 2017.
  • Florida International University Winter Conference on Geometry, Topology, and Applications, Jan 2017. [Abstract, Old Notes]
• An introduction to applied and computational topology.
  • The Symposium of Physics and Mathematics FCFM-IFM at the University of Michoacan, Morelia, Mexico, Nov 2018. [Abstract, Slides]
  • Department Colloquium, Texas State University, Nov 2018. [Abstract, Slides]
  • Data Science Seminar, University of Tennessee, Nov 2018. [Slides]
  • Undergraduate colloquium at Lafayette College, Apr 2018. [Abstract, Slides]
  • Florida International University Winter Conference on Geometry, Topology, and Applications, Jan 2017. [Abstract, Slides]
  • CSU Computer Science Colloquium, Oct 2016. [Abstract, Slides, Video]
• Survey: From persistent homology to machine learning feature vectors. TRIPODS Summer Bootcamp on Topology and Machine Learning, ICERM, Aug 2018. [Slides]
• Connections between Čech complexes and Carathéodory orbitopes. Special Session on Algebraic, Geometric, and Topological Methods in Combinatorics at the AMS Sectional Meeting, Northeastern University, Boston, Apr 2018. [Slides]
• The theory of Vietoris-Rips complexes.
  • AMS special session on Topological Data Analysis, Joint Mathematics Meetings in San Diego, Jan 2018. [Abstract, Slides]
  • Applied Topology Seminar, Brown University, March 2017. [Abstract, Notes]
    
    Given a metric space X and a distance threshold r > 0, the Vietoris-Rips simplicial complex has as its simplices the finite subsets of X of diameter less than r. A theorem of Jean-Claude Hausmann states that if X is a Riemannian manifold and r is sufficiently small, then the Vietoris-Rips complex is homotopy equivalent to the original manifold. Janko Latschev proves an analogous theorem for sufficiently dense samplings. Little is known about the behavior of Vietoris-Rips complexes for larger values of r, even though these complexes arise naturally in applications of using persistent homology. We describe how as r increases, the Vietoris-Rips complex of the circle obtains the homotopy types of the circle, the 3-sphere, the 5-sphere, the 7-sphere, ..., until finally it is contractible. These homotopy types are related to cyclic and centrally symmetric polytopes and orbitopes. Interestingly, certain Vietoris-Rips complexes of ellipses contain ephemeral summands in their persistent homology modules. We argue that infinite Vietoris-Rips complexes should be equipped with a different topology: an optimal transport or Wasserstein metric thickening the metric on X. With this new metric, we describe the first change in homotopy type (as r increases) of Vietoris-Rips complexes of higher-dimensional spheres. Joint work with Michal Adamaszek, Florian Frick, and Samadwara Reddy.

  • Mini-symposium on Applied and Computational Topology at the SIAM Central States Section Meeting, Oct 2016. [Slides]
• Vietoris-Rips complexes of circles, ellipses, and higher-dimensional spheres.
  • Topology, Geometry, and Data Analysis seminar at The Ohio State University, Feb 2017. [Abstract, Notes]
    
    Given a metric space X and a distance threshold r > 0, the Vietoris-Rips simplicial complex has as its simplices the finite subsets of X of diameter less than r. A theorem of Jean-Claude Hausmann states that if X is a Riemannian manifold and r is sufficiently small, then the Vietoris-Rips complex is homotopy equivalent to the original manifold. Janko Latschev proves an analogous theorem for sufficiently dense samplings. Little is known about the behavior of Vietoris-Rips complexes for larger values of r, even though these complexes arise naturally in applied topology and persistent homology. We describe how as r increases, the Vietoris-Rips complex of the circle obtains the homotopy types of the circle, the 3-sphere, the 5-sphere, the 7-sphere, ..., until finally it is contractible. These homotopy types are connected to cyclic and centrally symmetric polytopes and orbitopes. Interestingly, Latschev's result fails for the ellipse with larger r values, and certain Vietoris-Rips complexes of ellipses contain ephemeral summands in their persistent homology modules. We argue that infinite Vietoris-Rips complexes should be equipped with a different topology: an optimal transport or Wasserstein metric thickening the metric on X. Using this new metric, we describe the first change in homotopy type (as r increases) of Vietoris-Rips complexes of higher-dimensional spheres. Joint work with Michał Adamaszek, Florian Frick, and Samadwara Reddy.

  • Joint Mathematics Meetings, AMS Special Session on Applied and Computational Topology, Jan 2016. [Abstract, Slides]
• Cyclic polytopes and nerve complexes. Rocky Mountain Algebraic Combinatorics Seminar, Oct 2016. [Abstract, Notes]
• What is topology, and how is it applied to data analysis? Front Range Computational & Systems Biology Symposium, July 2016.
• Random cyclic dynamical systems. Rocky Mountain Algebraic Combinatorics Seminar, Sept 2015. [Abstract]
• The Vietoris-Rips complexes of a circle. [Abstract, Slides, Poster]
  
  Given a metric space X and a distance threshold r > 0, the Vietoris-Rips simplicial complex has as its simplices the finite subsets of X of diameter less than r. A theorem of Jean-Claude Hausmann states that if X is a Riemannian manifold and r is sufficiently small, then the Vietoris-Rips complex is homotopy equivalent to the original manifold. Little is known about the behavior of Vietoris-Rips complexes for larger values of r, even though these complexes arise naturally in applications using persistent homology. We show that as r increases, the Vietoris-Rips complex of the circle obtains the homotopy types of the circle, the 3-sphere, the 5-sphere, the 7-sphere, ..., until finally it is contractible. As our main tool we introduce a directed graph invariant, the winding fraction, which in some sense is dual to the circular chromatic number. Using the winding fraction we classify the homotopy types of the Vietoris-Rips complex of an arbitrary (possibly infinite) subset of the circle, and we study the expected homotopy type of the Vietoris-Rips complex of a uniformly random sample from the circle. Moreover, we show that as the distance parameter increases, the ambient Čech complex of the circle also obtains the homotopy types of the circle, the 3-sphere, the 5-sphere, the 7-sphere, ..., until finally it is contractible. Joint with Michał Adamaszek.
  • University of Rochester Data Science Colloquium, Apr 2015.
  • Applied Algebraic Topology Research Network, Online Seminar Series, Mar 2015. [Video]
  • Colloquium at Colorado State University, Jan 2015.
  • Geometry and Topology Seminar at Tulane University, Nov 2014.
  • Applied Topology Seminar at the University of Pennsylvania, Nov 2014.
  • Graduate and Faculty Seminar at Duke University, Oct 2014.
  • Colloquium at UNC Greensboro, Oct 2014.
  • Geometry and Topology Seminar at North Carolina State University, Sept 2014.
  • IMA Postdoc Seminar, May 2014.
  • IMA Postdoc Seminar, Oct 2013.
• Evasion paths in mobile sensor networks. [Abstract, Slides, Poster, Multimedia]
  
  Suppose that ball-shaped sensors wander in a bounded domain. A sensor doesn't know its location but does know when it overlaps a nearby sensor. We say that an evasion path exists in this sensor network if a moving intruder can avoid detection. In "Coordinate-free coverage in sensor networks with controlled boundaries via homology", Vin de Silva and Robert Ghrist give a necessary condition, depending only on the time-varying connectivity data of the sensors, for an evasion path to exist. Using zigzag persistent homology, we provide an equivalent condition that moreover can be computed in a streaming fashion. However, no method with time-varying connectivity data as input can give necessary and sufficient conditions for the existence of an evasion path. Indeed, we show that the existence of an evasion path depends not only on the fibrewise homotopy type of the region covered by sensors but also on its embedding in spacetime. For planar sensors that also measure weak rotation and distance information, we provide necessary and sufficient conditions for the existence of an evasion path. Joint with Gunnar Carlsson.
  • Williams College Colloquium, Mar 2018.
  • Topological data analysis of exclusion zones, Edinburgh, Scotland, Dec 2017.
  • CSU Pattern Analysis Lab, Oct 2015.
  • Duke Graduate & Faculty Seminar, Feb 2015.
  • IMA Workshop on Topological Systems: Communication, Sensing, and Actuation, Mar 2014. [Video]
  • Rocky Mountain Algebraic Combinatorics Seminar, Nov 2013.
  • SIAM Conference on Applied Algebraic Geometry, Aug 2013.
  • Applied Topology in Będlewo, July 2013.
  • MSRI Workshop on Algebraic Topology, June 2013. [Video]
  • Ayasdi Topology Day, June 2013.
  • Stanford CompTop Seminar, May 2013.
  • Special Session on Applied and Computational Topology at MAA MathFest, Aug 2012.
  • Algebraic Topology: Applications and New Directions, July 2012.
  • Minisymposium on Applied Algebraic Topology at SIAM Annual Meetings, July 2012. [Video]
  • Schloss Dagstuhl Seminar on Applications of Combinatorial Topology to Computer Science, Mar 2012.
  • AMS Special Session on Computational and Applied Topology, Joint Meetings, Jan 2012.
  • SIAM Conference on Applied Algebraic Geometry, Oct 2011.
• Nudged elastic band in topological data analysis. [Abstract]
  
  We use the nudged elastic band method from computational chemistry to analyze high-dimensional data. Our approach is inspired by Morse theory, and as output we produce an increasing sequence of small cell complexes modeling the dense regions of the data. We test the method on data sets arising in social networks and in image processing. Furthermore, we apply the method to identify new topological structure in a data set of optical flow patches. Joint with Atanas Atanasov and Gunnar Carlsson.
  • SIAM Conference on Applied Algebraic Geometry, Oct 2011. [Slides]
  • Stanford CompTop Seminar, Jan 2010. [Slides]
  • AMS-SIAM Special Session on Applications of Algebraic Geometry, Joint Meetings, Jan 2010.
• Topological data analysis: Understanding optical flow. IMA Short Course on Applied Algebraic Topology, June 2009. [Slides]

Software talks and demonstrations

• Introduction to Javaplex software for persistent homology. [Abstract]
  
  This exercise session will be a hands-on introduction to persistent homology in topological data analysis. We will use the Javaplex software package, which comes equipped with a tutorial with examples, exercises, and solutions. Two of the real datasets included in the tutorial are the three-circle model for optical image patches, and the conformation space of the cyclo-octane molecule (which is a Klein bottle glued to a sphere along two circles of singularities).
  • Applied Algebraic Topology Research Network, Student Online Seminar Series, Oct 2015.
  • Young Topologists' Meeting, July 2015.
  • AMS Mathematical Research Community on Computational and Applied Topology, June 2011.
• Introduction to JPlex software for persistent homology.
  • AMS Short Course on Computational Topology, Joint Meetings, Jan 2011. [Slides]
  • CSRI Workshop on Combinatorial Algebraic Topology, Aug 2009. [Slides]
  • IMA Short Course on Applied Algebraic Topology, June 2009. [Notes]

Departmental talks and Expository talks

• Lovász' proof of the Kneser conjecture. CSU Topology Seminar, November 2018. [Notes]
• Using homotopy colimits to understand infinite simplicial complexes. CSU Topology Seminar, May 2018. [Abstract, Notes]
  
  Last week we gave an introduction to homotopy colimits. Building on this, we will describe how to use homotopy colimits to understand infinite simplicial complexes. The main tools we will describe are the Projection Lemma, the Quasifibration Lemma, and the Cofinality Theorem from Section 3 of Homotopy colimits - Comparison lemmas for combinatorial applications by Welker, Ziegler, and Zivaljevic. The running example we will use is the Vietoris-Rips simplicial complex of the circle.

• An introduction to homotopy colimits. CSU Topology Seminar, April 2018. [Abstract, Notes]
  
  Though colimits exist in the category of topological spaces, they are often not well-behaved from the perspective of homotopy theory. Homotopy colimits are a tool to fix this. I will give examples of homotopy colimits and describe some of their useful properties. Time permitting, I may describe where homotopy colimits have appeared in my research, or describe their relationship to simplicial sets, or describe their relationship to model categories.

• Vietoris-Rips complexes of the circle. CSU Topology Seminar, Oct 2017.
• Sperner's lemma and fair division. CSU Math Club, Oct 2017. [Abstract, Notes]
  
  Suppose three roommates need to pay $3,000 dollars of rent per month for their three-bedroom apartment. The three bedrooms are not equivalent: some are bigger, some are smaller, some are closer to the kitchen, etc, and so it's not necessarily fair to have each room cost $1,000. How should the roommates fairly divide the rent between the rooms so that the total rent for the apartment is $3,000, and how should they decide who gets which room? I will describe how Sperner's lemma, a mathematical lemma related to combinatorics and topology, can be used to find a fair division of rent. This topic has appeared in a New York Times article and in a paper by Francis Su.

• An introduction to Vietoris-Rips complexes. CSU Topology Seminar, Sep 2017.
• Introduction to discrete Morse theory. University of Minnesota Student Topology Seminar, Dec 2013. [Notes]
• Applied topology. Stanford University Mathematics Camp Guest Lecture Series, July 2013. [Slides]
• Applied topology. Stanford University Mathematics Camp Guest Lecture Series, July 2011.
• Coverage problems in sensor networks. Stanford Undergraduate Mathematical Organization Speaker Series, May 2010. [Abstract]

Conferences organized

• With Jeffrey Brock, Melissa McGuirl, Bjorn Sandstede, and Yitzchak Solomon, co-organized an ICERM TRIPODS Summer Bootcamp on Topology and Machine Learning, August 6-10, 2018. Software tutorial available. [Photo]

Mini-symposia organized

• With Mehmet Aktas, Wenwen Li, Murad Ozaydin, co-organized a minisymposium on Applied and Computational Topology at the SIAM Central States Section Meeting, University of Oklahoma, Oct 5-7, 2018
• With Ellen Gasparovic and Katharine Turner, co-organized the 7th Annual Minisymposium on Computational Topology at Computational Geometry Week, Budapest, June 2018.
• With Mikael Vejdemo-Johansson, co-organized an AMS special session on Topological Data Analysis at the Joint Mathematics Meetings in San Diego, Jan 2018.
• With Florian Frick, co-organized a session on Symmetric Simplicial Complexes and Polytopes at the SIAM Conference on Applied Algebraic Geometry, Georgia Institute of Technology, Jul-Aug 2017.
• With Bala Krishnamoorthy, co-organized a session on Recent Advances in Applied Algebraic Topology at the AMS Western Sectional Meeting, Washington State University, Apr 2017.
• With Patrick Shipman, co-organized a minisymposium on Applied and Computational Topology at the SIAM Central States Section Meeting, University of Arkansas at Little Rock, Oct 2016.

Poster presentations

• Metric reconstruction via optimal transport. SIAM Conference on Applied Algebraic Geometry, July 2017. [Poster]
• The Vietoris-Rips complexes of a circle. ICERM Workshop on Topology and Geometry in a Discrete Setting, Nov 2016. [Poster]
• The Vietoris-Rips complex of the circle. IMA Workshop on Topology and Geometry of Networks and Discrete Metric Spaces, Apr 2014. [Poster]
• Vietoris-Rips and restricted Čech complexes of circular points. IMA Workshop on Topological Systems: Communication, Sensing, and Actuation, Mar 2014. [Poster]
• Evasion paths in mobile sensor networks. [Poster]
  • IMA Workshop on Modern Applications of Homology and Cohomology, Oct 2013.
  • IMA Workshop on Topological Data Analysis, Oct 2013.
• Mobile sensors and pursuit-evasion: Can directed algebraic toplogy help? Aalborg GETCO Workshop, Jan 2010. [Poster]