# Tutorials

It's more of a multi-media online short course than a publication, but my DARPA Tutorial on Sheaves in Data Analytics is a pretty comprehensive introduction to applied sheaf theory. It includes a number of videos as well as example software and datasets. Be sure to check it out!

# Books

I have published a book called Topological Signal Processing, which explains how topology in general, and sheaf theory in particular, can be used to develop new and powerful algorithms for signal processing. Here is the errata page for the book.

# Preprints and Technical Reports

- (with Emilie Purvine and Cliff Joslyn), "A Category Theoretical Investigation of the Type Hierarchy for Heterogeneous Sensor Integration," arXiv:1609.02883
- "Modeling wireless network routing using sheaves," arXiv:1607.06023
- (with Jimmy Palladino) "Protocol-independent critical node detection," arXiv:1607.06022
- (with Tara Shreve and David D'Auria) "Angle-valued Spectral Filters for Image Analysis," AU-CAS-MathStats Technical Report No. 2016-1
- "Tracking before detection using partially ordered sets and optimization," AU-CAS-MathStats Technical Report No. 2015-5.
- "Topological Investigation of Target/Clutter Features in Sonar Data," AU-CAS-MathStats Technical Report No. 2015-3.
- "Pseudosections of sheaves with consistency structures," AU-CAS-MathStats Technical Report No. 2015-2.
- (with Cliff Joslyn, Emilie Hogan, and Chris Capraro) "Conglomeration of heterogeneous content using local topology," which outlines an approach to heterogeneous data fusion using sheaves. AU-CAS-MathStats Technical Report 2015-1
- (with Matthew Hubler and Mark Verdi) "Experimental validation of the azimuthal high frequency cutoff of high resolution satellite radar images of the ocean," which shows that (1) a theoretical model of the azimuthal high frequency cutoff in maritime radar images is correct, but (2) that this is not the dominant factor in determining azimuthal blurring -- the resolution can be considerably better.
- (with Morgan DeHart, Matt Hubler, Mark Verdi, and Zhu Zhu) "Measuring Ocean Winds from Space Using a Radar Satellite," AU-CAS-MathStats Technical Report 2014-1.
- "Sheaf invariants for information systems," AU-CAS-MathStats Technical Report 2014-2.
- (joint with Robert Ghrist and Hank Owen) "DTIME: Discrete Topological Imaging for Multipath Environments," University of Pennsylvania ESE Technical Report
- "Instability of a parabolic equation with a quadratic nonlinearity," arXiv:0704.3989, which gives an explicit construction showing that a certain critical point whose linearization is stable is in fact unstable. This is not a new result, but the proof is a good example of the blow-up method of H. Fujita.

# Refereed Journal Articles

- (with Sean Fennell, Brian DiZio, and Jennifer Dumiak), "Geometry and topology of the space of sonar target echos,"
*Journal of the Acoustical Society of America*, volume 143, number 3, March 2018, pp. 1630-1645. (preprint version is AU-CAS-MathStats Technical Report No. 2016-2.) - "Sheaves are the canonical datastructure for information integration,"
*Information Fusion*, 36 (2017), pp. 208-224. (preprint version is arxiv:1603.01446) - "A Topological Lowpass Filter for Quasiperiodic Signals,"
*IEEE Signal Processing Letters*, vol. 23, no. 12, December 2016, pp. 1771-1775. (preprint version is arXiv:1607.06032) - "Imaging geometric graphs using internal measurements,"
*J. Differential Equations*260 (2016) 872-896. - "Knowledge-based antenna pattern interpolation,"
*IEEE Trans. Ant. Prop.*Volume 62, Issue 1, January 2014, pp. 72-79. (preprint version is arXiv:1203.2528) - "Multipath-dominant, pulsed doppler analysis of rotating blades,"
*IET Radar Sonar and Navigation*, Volume 7, Issue 3, March 2013, pp. 217-224. Preprint version is arXiv:1204.4366 - "Asynchronous logic circuits and sheaf obstructions,"
*Electronic Notes in Theoretical Computer Science*(2012), pp. 159-177. Preprint version is arXiv:1008.2729. - (with Robert Ghrist) "Topological localization via signals of opportunity,"
*IEEE Trans. Sig. Proc,*Vol. 60, No. 5, May 2012. - (with Robert Ghrist) "Euler-Bessel and Euler-Fourier transforms,"
*Inverse Problems*, Vol. 27, No. 12, 2011. Preprint version is arxiv:1011.4494. - "Classification of connecting solutions of semilinear parabolic
equations,"
*Electron. J. Diff. Eqns.*, Vol. 2011(2011), No. 61, pp. 1-10. - "A cell complex structure for the space of heteroclines for a
semilinear parabolic equation,"
*Electron. J. Diff. Eqns.*, Vol. 2009(2009), No. 16, pp. 1-17. - "Construction of eternal solutions for a semilinear parabolic
equation,"
*Electron. J. Diff. Eqns.*, Vol. 2008(2008), No. 139, pp. 1-8. - "An asymptotic-numerical approach for examining global
solutions to an ordinary differential equation,"
*Ergodic Theory and Dynamical Systems*, Vol. 29, (2009), pp. 223-253. - "IMEX method convergence for a semilinear parabolic equation,"
*J. Differential Equations*, vol. 241, no. 2, October 2007, pp 225-236; doi:10.1016/j.jde.2007.07.001. - "A wavefront launching model for predicting channel impulse
response,"
*ACES Journal*, vol. 22, no. 2, July 2007, pp 302-305.

# Book Chapters

- "Sheaf and duality methods for analyzing multi-model systems," in
*Novel Methods in Harmonic Analysis*, Pesenson, I., Le Gia, Q.T., Mayeli, A., Mhaskar, H., Zhou, D.-X. (eds.), Springer,*in press*. (preprint version is arXiv:1604.04647.) - "A sheaf-theoretic perspective on sampling", in
*Sampling Theory, a Renaissance*, Goetz Pfander (ed.), 2015. (preprint version is arxiv:1405.0324.) - (with Robert Ghrist and Justin Curry) "Euler calculus and its applications to signals and sensing," in
*Proceedings of Symposia in Applied Mathematics: Advances in Applied and Computational Topology,*Afra Zomorodian (ed.), 2012.

# Refereed Archival Conference Proceedings

- Cliff Joslyn, Brenda Praggastis, Emilie Purvine, Arun Sathanur, Michael Robinson, Stephen Ranshous, "Local Homology Dimension as a Network Science Measure", in
*SIAM Workshop on Network Science 2016*, July 15-16, 2016, Boston, pages 86-87. - (with Harish Chintakunta and Hamid Krim), "Introduction to the special session on Topological Data Analysis,"
*IEEE Conf. on Acoustics, Speech, and Signal Processing (ICASSP)*2016, Shanghai, China. - "Simplicial Complex Sampling in Inference using Exact Sequences." Data Algorithms and Problems on Graphs (DAPG), Columbia University. September 28, 2015.
- "Universal factorizations of quasiperiodic functions,"
*Sampling Theory and Applications*, Washington, DC, 2015. (preprint version is arXiv:1501.06190) - "Analyzing wireless communication network vulnerability with homological invariants,"
*IEEE Global Conference on Signal and Information Processing (GlobalSIP)*2014, Atlanta, Georgia. (preprint version is arXiv:1311.1532.) - (with Cliff Joslyn and Emilie Hogan), "Towards a topological framework for integrating semantic information sources,"
*Semantic Technologies for Intelligence, Defense, and Security (STIDS)*, 2014. - "Understanding networks and their behaviors using sheaf theory,"
*IEEE Global Conference on Signal and Information Processing (GlobalSIP)*2013, Austin, Texas. (preprint version is arXiv:1308.4621) - "The Nyquist theorem for cellular sheaves," Sampling Theory and Applications 2013, Bremen, Germany.
- "Polarizing frequency of a fluid plasma antenna element,"
*IEEE Antennas and Propagation Society Symposium*, 2004. (preprint)

# Software

- The Python sheaf library pysheaf.

# Patents

- (with James Gonnella) "Methods and Systems for Local Principal Axis Rotation Angle Transform," U.S. Patent No. 9,651,661 B2 granted 16 May 2017.
- "System and method for antenna pattern extrapolation," U.S. Patent No. 9,255,953 granted 9 February 2016.
- (with Andy Davis, Harvey Schuman, Don McPherson, John Wiley, David Hagenmayer) "Bench-Top Measurement Method, Apparatus and System for Phased Array Radar Apparatus Calibration," U.S. Patent No. 8,686,896 granted 16 August 2012.

# Selected Talks

- Invited talk, "Topological filters: a toolbox for processing dynamic signals." Workshop on Topology: Identifying order in complex systems, Institute for Advanced Study, Princeton, NJ. April 7, 2018. (slides) (video)
- Flash talk, "Sheaf methods for inference." NIST Applied Category Theory Workshop. March 15, 2018. (slides) (video)
- Invited talk, "Performance Measurement of Sonar Classifiers that use Persistent Homology." Morgan State University Mathematics Colloquium, Baltimore, MD. October 19, 2017.
- Invited talk, "Topological filters and their application to signal processing." Georgetown University Mathematics and Statistics Colloquium, Washington, DC. September 15, 2017.
- Invited talk, "Sheaves and numerical analysis." SIAM Conference on Applid Algebraic Geometry (SIAM AG) 2017, Atlanta, GA. August 3, 2017. (abstract) (slides)
- Invited talk, "The Performance of Topological Classifiers on Sonar Data." Acoustics '17, Boston, MA. June 26, 2017.
- Invited talk, "Topological Symmetries: Quasiperiodicity and its Application to Filtering and Classification Problems." SIAM Conference on Applications of Dynamical Systems (SIAM DS) 2017, Snowbird, UT. May 24, 2017. (abstract) (slides)
- Invited talk, "Topological Sensor Data Fusion." Norbert Weiner Seminar, University of Maryland, College Park, MD. March 27, 2017. (abstract)
- Invited talk, "Local Topological Analysis of Complex Systems." Joint Mathematics Meetings Special Session on Sheaves in Topological Data Analysis, Atlanta, GA. January 4, 2017. (video)
- Invited talk, "Modeling and analyzing sensor data topologically." Morgan State University Mathematics Colloquium, Baltimore, MD. October 13, 2016.
- Invited talk, "Finding cross-species orthologs with local topology." ACM-Biocomputing and Bioinformatics Workshop on Topological Data Analysis in Biomedicine, Seattle, WA. October 2, 2016. (slides) (video)
- Session organizer talk, "Introduction to the special session on topological data analysis," IEEE Conference on Acoustics, Speech, and Signal Processing (ICASSP) 2016, Shanghai, China. March 24, 2016.
- Invited talk, "Sheaf-based communication network invariants." Joint Mathematics Meetings Special Session on Computational and Applied Topology, Seattle, WA. January 9, 2016. (slides)
- Non-presenting coauthor: Emilie Purvine, Michael Robinson, and Cliff Joslyn, "Categorification in the real world." Joint Mathematics Meetings MAA Session on Mathematics Experiences and Projects in Business, Industry, and Government, Seattle, WA. January 8, 2016.
- Invited talk "Computation of the Godement Resolution for Sheaves over Finite Spaces." American University Math and Stat Colloquium, November 10, 2015. (video) (slides)
- Invited talk "Examining Sonar Targets with Topological Invariants." Norbert Weiner Seminar, University of Maryland, October 27, 2015.
- Invited talk "Quasiperiodicity: processing signals using topology." Applied Topology Seminar, University of Pennsylvania. Philadelphia, PA, April 27, 2015.
- Invited talk "Sheaf-based modeling of wireless communications". Applied Algebraic Topology Network Seminar, April 1, 2015. Online talk through WebEx.
- Invited talk "Topological analysis of wireless networks." NC State University ECE Seminar. Raleigh, NC. January 23, 2015.
- Invited talk "Morphisms of logic circuits." Algebraic Topological Methods in Computer Science (ATMCS 6), Vancouver, BC, Canada, May 30, 2014.
- Invited talk "Computing using sheaves." Pacific Northwest National Laboratory, Discrete Mathematics Seminar, May 21, 2014.
- Invited talk "A Unified Framework for Multi-INT Signal Processing." Pacific Northwest National Laboratory, Signature Detection Initiative Seminar, May 20, 2014.
- "Topological filters: theory and practice." Workshop on Topological Systems: Communication, Sensing, and Actuation. Institute for Mathematics and its Applications. March 4, 2014 (video).
- Plenary talk "Sheaf morphisms describe digital signal processors." Applied Topology 2013, Bedlewo, Poland, July 26, 2013.
- "The Nyquist theorem for cellular sheaves." Sampling Theory and Applications 2013, Bremen, Germany. July 3, 2013.
- "Sheaves for engineering problems." NC State University ECE Seminar. Raleigh, NC. January 18, 2013. (video)
- "How would you explain the Whitney embedding theorem to an engineer?" AMS Sectional Meeting, Akron, OH. October 21, 2012.
- "The Whitney embedding theorem in signal processing." MAA MathFest, Madison, WI. August 4, 2012.
- "Internal imaging of graphs and applications to urban sensing." AFRL Math Seminar, Dayton, OH. May 30, 2012.
- "Sheaf invariants for temporal logic." AMS/MAA Joint Meetings, Boston, MA. January 5, 2012.
- "Euler integral transforms and applications." Workshop on Computational Topology. Fields Institute. November 8, 2011.
- "Sounding your surroundings." IMA Summer School on Topological Analysis. University of Pennsylvania. Week of July 25, 2011.
- "Measuring Topology and Geometry by Distributed Sensing." NRO Technical Seminar, July 11, 2011. Also at Drexel University (April 14, 2011), Lehigh University (October 13, 2010), Tulane University (October 28, 2010). Demonstrated acoustic sounding equipment to supply opportunistic localization data.
- (poster) "Sheaf theoretic invariants for computational networks." NMMC 2011, May 16, 2011.
- "Logic circuits as sheaves." NSA Mathematics Colloquium. March 4, 2011. Also at Tulane University, March 25, 2011.
- (poster) "Euler characteristic integrals in signal processing." FFT 2011. February 17, 2011.
- "Forays into signal processing using the Euler characteristic integral." Princeton University, February 12, 2010.
- "Constructible sheaves and their cohomology for asynchronous logic and computation." GETCO 2010, Aalborg, Denmark. January 14, 2010. Also at DARPA SToMP meeeting, February 18, 2010.
- "Localization of Mobile Receivers using Opportunistic Signals." PASSHEMA Conference, Mansfield University. March 21, 2009.
- "Frequency assignment in land mobile radio systems." Applied Mathematics and Computational Science Seminar, University of Pennsylvania. October 10, 2008.
- "Some finite dimensionality results for the space of heteroclinic orbits of a semilinear parabolic equation." Dynamics Seminar, Cornell University. February 1, 2008.
- "Dynamics of a semilinear parabolic equation." Joint Mathematics Meetings, San Diego, California. January 6, 2008.
- "Instability of an equilibrium of a nonlinear parabolic equation with a negative definite linearization." Applied Mathematics Seminar, Université de Provence. March 13, 2007.
- "Towards a characterization of the dynamics of a semilinear parabolic differential equation." Mathematical Sciences Seminar, Cornell University. November 8, 2006.
- "Tug-of-war: how nonlinearity and the Laplacian interact." Olivetti Club, Cornell University. April 25, 2006.
- "Do electrons really spin?" Mathematical Sciences Seminar, Cornell University. February 8, 2006.
- "Studying the bifurcation behavior of a nonlinear PDE." Mathematical Sciences Seminar, Cornell University. October 5, 2005.
- "Polarizing frequency of a fluid plasma antenna element." Mathematical Sciences Seminar, Cornell University. November 8, 2004.

# Student Publications

- Sean Fennell, "Classification Performance of Support Vector Machines on Topologically Filtered Synthetic Aperture Sonar Signals," Master's thesis, American University, May 2017.
- Samara Fantie, "Implementation and Testing of Sheafification of Goodwin Models," Master's thesis, American University, May 2017.
- Brian DiZio, "Sonar Signature Enhancement Using Quasi-Periodic Factorization Techniques," Master's thesis, American University, May 2017.
- Philip Coyle, "An Extended Goodwin Model with Horizontal Trade: A Sheaf Theoretic Approach," Master's project, American University, December 2016.
- Matt Hubler, "Validation of Maritime Spectral Features," Master's thesis, American University, May 2014.
- Sam Krupa, "Numerical Analysis of Target Enumeration via Euler Characteristic Integrals: 2 Dimensional Disk Supports," undergraduate research project, University of Pennsylvania, 2012.