Research Projects

• Sheaves, Cosheaves and Applications I am extending the theory of cosheaves with an eye towards applications to Morse theory, persistent homology, network coding, sensor networks, Euler calculus, et al. This is part of my PhD thesis. Here are some papers that have arisen out of this project:
  • My manuscript, Sheaves, Cosheaves and Applications, is meant to provide an integrated one-stop-shop for learning applied (co)sheaf theory. It is assumed that the reader has some familiarity with homology and cohomology. The arXiv version was submitted on March 13, 2013:http://arxiv.org/abs/1303.3255
  • Abstract Existence of Cosheafification is a short technical note regarding the existence of cosheafification. It is intended for mathematicians mostly. Last update: March 19, 2013.
  • In Sheaves, Co-Sheaves, and Verdier Duality I obtained a covariant equivalence between the bounded derived category of cellular sheaves and the bounded derived category of cellular cosheaves, establishing a conjecture of Bob MacPherson. As a corollary, Verdier duality for both sheaves and cosheaves is obtained. When I proved it in March 2012, I was not aware of the work of Schneiders ("Verdier Duality on the Building") who had already obtained the result. Last update: March 16, 2012.
  • Level-set persistence implies sub-level set persistence. Towards a sheaf-theoretic proof of the pyramid theorem.
  • Interleavings and Obstructions
• Categorified Sensor Networks: I model sensor networks that sense properties living in a vector space. In the scattered disk model of an ad-hoc sensor network this leads naturally to constructible sheaves. This provides a model for multi-modal sensing. In the continuum limit model a vector bundle is used.
• Pursuit-Evasion in Mobile Sensor Networks: In collaboration with Robert Ghrist, we use constructible (co)sheaves to study the intruder problem in mobile sensor networks.

---

Papers

• "Discrete Morse theory for computing cellular sheaf cohomology" with Robert Ghrist and Vidit Nanda has been submitted for publication.
• "Sheaves, Cosheaves and Applications" is a pre-print.
• "Euler Calculus with Applications to Signal Processing" with Robert Ghrist and Michael Robinson. Submitted to AMS Proceedings.
• "Soliton Solutions of Integrable Systems and Hirota's Method." Harvard College Mathematics Review. Issue 2.

---

Talks

• "Persistent Homology via Cellular Cosheaves" at the mini-symposium on "Applied and Computational Topology" at the SIAM conference on Applied Algebraic Geometry in Fort Collins, CO, August 1-4, 2013.
• "Persistent Homology via Cellular Cosheaves" "Applied and Computational Algebraic Topology" at the University of Bremen, Germany, July 15-19, 2013.
• "Persistent Homology via Cellular Cosheaves" Symposium on Computational Geometry in Rio de Janeiro, Brasil, June 17-20, 2013.
• "Applied Sheaf Theory I,II, and III": Workshop on Applied Topology at Kyoto University, June 10-14, 2013.
• "Linear Algebra over Cell Complexes: Applications to Data, Coding and Sensor Networks." IAS-Penn-Rutgers joint workshop in "Identifying Order in Complex Systems," given at Rutgers joint with the BioMaPS institute, February 6, 2013.
• "Sheaves and Cosheaves over Posets: A Tool for Applications." Three lectures for the MacPherson Seminar. October 8, 24, November 5, 2012.
• "Cosheaves, Persistence, and Sensor Networks." AMS Sectional Meeting: Akron, Ohio. Special Session on Applied Topology, October 21, 2012
• Special Session on Applied and Computational Topology at MAA MathFest, Aug 2-4 2012.
• "Cosheaves and Dualities in Generalized Sensor Networks." Stanford Symposium - July 23-27, 2012. Algebraic Topology: Applications and New Directions. A conference to celebrate the birthdays of Gunnar Carlsson, Ralph Cohen, Ib Madsen. July 27th, 2012.
• "Cosheaves and Dualities in Generalized Sensor Networks", Applied and Computational Topology: ATMCS 5 International Centre for Mathematical Sciences (ICMS), 15 S. College St. Edinburgh, Scotland, July 6th, 2012.
• "Obstruction-Theoretic Sensing." Indiana University Graduate Student Topology Conference, March 31, 2012.
• "Obstruction-Theoretic Sensing." AMS Special Session on Computational and Applied Topology, Joint Meetings. Jan 5, 2012.
• "Sensor Networks and Constructible (Co)Sheaves." Bob MacPherson's Sheaf Theory Seminar. IAS. December 15, 2011.
• "Cellular Sheaves, Cohomology and Applications." Bob MacPherson's Sheaf Theory Seminar. IAS. December 5, 2011.
• "Sheaves Incarnate." Graduate Student Pizza Seminar. University of Pennsylvania. December 2, 2011.
• "Physical Instantiations of Mathematics." PACM Graduate Student Seminar. Princeton University. November 8, 2011.
• "Constructible Sheaves." MIT-RTG Geometry Workshop with David Nadler: Fukaya Categories via Microlocal Sheaf Theory. June 15, 2011
• "Physical Instantiations of Mathematics." Graduate Student Pizza Seminar. University of Pennsylvania. March 18, 2011.
• Three talks: "Microlocal/Singular Supports of Sheaves and Morse Theory I,II,III." Jonathan Block's Research Group Seminar. University of Pennsylvania. February 2-11, 2011.
• "Intro to Morse/Conley/Floer Theory." Geometry Topology Reading Seminar. University of Pennsylvania. October 27, 2009.
• "Morse Theory and Classifying Spaces." Graduate Student Pizza Seminar. University of Pennsylvania. November 6, 2009.
• "Soliton Solutions of Integrable Systems and Hirota's Method." Graduate Student Pizza Seminar. University of Pennsylvania. February 18, 2009

Demos

• Experimental Verification of Signals Embedding Theorem via wireless and acoustic data. AMS MRC on Applied and Computational Topology. Snowbird, UT June 21-28 2011.

Conferences/Workshops Attended

• Joint Mathematics Meetings. Boston, MA. 2012.
• Workshop on Topology: Identifying Order in Complex Systems. University of Pennsylvania. November 2, 2011.
• Workshop on Topology: Identifying Order in Complex Systems. Rutgers University. October 5, 2011.
• Applied and Computational Topology. IMA Summer School. University of Pennsylvania. July 25 - August 12, 2011.
• AMS MRC on Applied and Computational Topology. Snowbird, UT. June 21-28, 2011.
• MIT-RTG Geometry Workshop: Fukaya Categories via Microlocal Sheaf Theory. Breckenridge, CO. June 13-20, 2011.
• Workshop on Topology: Identifying Order in Complex Systems. University of Pennsylvania. April 6, 2011.
• Workshop on Topology: Identifying Order in Complex Systems. IAS. March 2, 2011.
• Workshop on Topology: Identifying Order in Complex Systems. Rutgers University. February 2, 2011.
• Joint Mathematics Meetings. New Orleans, LA. 2011.
• Workshop on Topology: Identifying Order in Complex Systems. IAS. December 1, 2010.
• Workshop on Topology: Identifying Order in Complex Systems. University of Pennsylvania. November 3, 2010.
• Workshop on Topology: Identifying Order in Complex Systems. Rutgers University. October 6, 2010.
• Algebra and Topology: Methods, Computation, and Science (ATMCS IV). Meunster, Germany. June 21-26, 2010.
• Young Topologists Conference. New Orleans, LA. November 13-15, 2009.