Martin Frankland's home page

Research interests

My research is in algebraic topology and homotopy theory, with a focus on realization problems. My interests include homotopy operations (such as Π-algebras), algebraic models for homotopy n-types, cohomology operations, power operations, and examples from chromatic and motivic homotopy theory.

I am also interested in obstruction theory, Quillen cohomology, simplicial methods, and tools from homological and homotopical algebra.

Here is my research statement.

Publications

(with Ioannis Dokas and Sacha Ikonicoff) Quillen (co)homology of divided power algebras over an operad. Preprint.
(with Sebastian Martensen and Marius Thaule) Toda brackets in n-angulated categories. Preprint.
(with Don Stanley) Multiparameter persistence modules in the large scale. Preprint.
(with Dan Christensen) On good morphisms of exact triangles. J. Pure Appl. Algebra (2021), doi:10.1016/j.jpaa.2021.106846. Preprint.
(with Hans-Joachim Baues) The DG-category of secondary cohomology operations. Appl. Categ. Structures 28 (2020), no. 6, 877-905. Preprint.
(with Markus Spitzweck) Towards the dual motivic Steenrod algebra in positive characteristic. Preprint.
(with Hans-Joachim Baues) Eilenberg-MacLane mapping algebras and higher distributivity up to homotopy. New York J. Math. 23 (2017), 1539-1580. Preprint.
(with Dan Christensen) Higher Toda brackets and the Adams spectral sequence in triangulated categories. Algebr. Geom. Topol. 17-5 (2017), 2687-2735. Preprint.
(with Hans-Joachim Baues) 2-track algebras and the Adams spectral sequence. J. Homotopy Relat. Struct. 11 (2016), no. 4, 679-713. Preprint.
(with Tobias Barthel) Completed power operations for Morava E-theory. Algebr. Geom. Topol. 15-4 (2015), 2065-2131. Preprint.
Behavior of Quillen (co)homology with respect to adjunctions. Homology Homotopy Appl. 17 (2015), no. 1, 67-109. Preprint.
(with Hans-Joachim Baues) The realizability of operations on homotopy groups concentrated in two degrees. J. Homotopy Relat. Struct. 10 (2015), no. 4, 843-873. Preprint.
Moduli spaces of 2-stage Postnikov systems. Topology Appl. 158 (2011), 1296-1306. Preprint.
Théorème de Künneth en homologie de Morse. Ann. Sci. Math. Québec 31 (2007), no. 1, 31-39.

Here is my list of publications as a pdf file.

Notes and expository writings

Is calculus secretly algebraic topology? Presentation for Pi Day.
Cohomology of the Hawaiian earring. A short survey.
A user's guide: Completed power operations for Morava E-theory. A behind-the-scenes look at the paper.
Equivalent statements of the telescope conjecture. Notes from a talk given in the UIUC topics seminar in algebraic topology.
Fibered category of Beck modules. Notes from a talk given at the Graduate Student Topology and Geometry Conference 2010.
Kervaire invariant and Whitehead square. Notes from a talk given in the summer seminar on the Kervaire invariant problem.

Presentations

Prairie Mathematics Colloquium, November 7, 2024.
Slides: Quillen cohomology of divided power algebras over an operad.
New Directions in Group Theory and Triangulated Categories, October 17, 2024.
Slides: Toda brackets in n-angulated categories.
Foundational Methods in Computer Science Workshop, University of Calgary, July 10, 2024.
NTNU Trondheim Topology Seminar, June 20, 2024.
CMS Summer Meeting, University of Saskatchewan, June 1, 2024.
Slides: Multiparameter persistence modules in the large scale.
Mini-course on Persistent Homology, University of Saskatchewan, May 31, 2024.
Alberta Topology Seminar, July 8, 2023.
University of Haifa Topology and Geometry Seminar, December 20, 2022.
Slides: Multiparameter persistence modules in the large scale.
Canadian Geometry-Topology Seminar, UQAM, November 13, 2022.
Slides: Multiparameter persistence modules in the large scale.
NTNU Trondheim Topology Seminar, October 10, 2022.
Slides: Multiparameter persistence modules in the large scale.
Foundational Methods in Computer Science Workshop, University of Calgary, June 26, 2022.
Slides: Modules over bialgebroids and Beck modules.
CMS Summer Meeting, Memorial University, June 4, 2022.
Slides: Modules over bialgebroids and Beck modules.

Slides:

Multiparameter persistence modules in the large scale

New Directions in Group Theory and Triangulated Categories, May 3, 2022.
Slides: On good morphisms of exact triangles.
Mathematical Congress of the Americas, July 21, 2021.
Slides: The homotopy theory of simplicial Beck modules.
Princeton University Algebraic Topology Seminar, March 25, 2021.
Slides: The monoidal fibered category of Beck modules.
University of Calgary Peripatetic Seminar, March 19, 2021.
Slides: The monoidal fibered category of Beck modules.
University of Melbourne Topology Seminar, October 27, 2020.
Slides: On good morphisms of exact triangles.
Fields Institute Toric Topology Seminar, May 7, 2020.
Slides: On good morphisms of exact triangles.
Ohio State University Homotopy Theory Seminar, February 20, 2020.
Slides: On good morphisms of exact triangles.
NTNU Trondheim Topology Seminar, November 7, 2019.
Equivariant Topology and Derived Algebra, NTNU Trondheim, July 29, 2019.
Slides: On good morphisms of exact triangles.
Universität Osnabrück Oberseminar Topologie, July 24, 2019.
Electronic Computational Homotopy Theory Seminar, February 28, 2019.
Slides: The DG-category of secondary cohomology operations.
University of Western Ontario Geometry & Topology Seminar, February 20, 2019.
CMS Winter Meeting, Vancouver, December 9, 2018.
Slides: Towards the dual motivic Steenrod algebra in positive characteristic.
University of British Columbia Topology Seminar, December 6, 2018.
University of Washington Topology Seminar, November 8, 2018.
Adam Mickiewicz University in Poznań Geometry and Topology Seminar, July 3, 2018.
Topology Meeting, University of Oslo, May 30, 2018.
Workshop on Homotopy Theory, Universität Regensburg, May 5, 2018.
University of North Carolina Wilmington, March 15, 2018.
University of Louisiana at Lafayette Mathematics Colloquium, March 12, 2018.
Georgia Southern University, March 9, 2018.
University of Regina, March 6, 2018.
EPFL Topology Seminar, October 17, 2017.
Homotopy Theory: Tools and Applications, Urbana-Champaign, July 19, 2017.
Slides: Eilenberg-MacLane mapping algebras and higher distributivity up to homotopy.
Adam Mickiewicz University in Poznań Geometry and Topology Seminar, June 2, 2017.
Slides: Realization problems in algebraic topology.
Universität Regensburg Arbeitsgruppenseminar Prof. Bunke, December 1, 2016.
Colloque du GDR "Topologie Algébrique et Applications", Amiens, October 14, 2016.
Conference on Combinatorial Structures in Geometry, Osnabrück, August 12, 2016.
University of Copenhagen Algebra/Topology Seminar, June 16, 2016.
Operations in Highly Structured Homology Theories, Banff, May 26, 2016.
Slides: The mod p motivic Steenrod algebra in characteristic p.
Westfälische Wilhelms-Universität Münster Oberseminar Topologie, April 11, 2016.
University of Edinburgh Topology Seminar, March 25, 2016.
Slides: Higher Toda brackets and the Adams spectral sequence.
Newcastle University Algebra & Geometry Seminar, March 22, 2016.
24th NRW Topology Meeting, Bochum, November 13, 2015.
Slides: The algebra of tertiary cohomology operations.
MPIM Oberseminar, October 8, 2015.
Wayne State University Topology Seminar, November 4, 2014.
Wayne State University Mathematics Colloquium, November 3, 2014.
Slides: Realization problems in algebraic topology.
Joint Mathematics Meetings, Baltimore, January 17, 2014.
Slides: Completed power operations for Morava E-theory.
Ohio State University Topology Seminar, March 5, 2013.
Purdue University Topology Seminar, November 29, 2012.
Ruhr-Universität Bochum Oberseminar Topologie, November 20, 2012.
University of Haifa Geometry and Topology Seminar, July 26, 2012.
CMS Summer Meeting, University of Regina, June 2, 2012.
Slides: Non-realizable 2-stage Π-algebras.
Joint Mathematics Meetings, Boston, January 7, 2012.
Slides: Non-realizable 2-stage Π-algebras.
Ruhr-Universität Bochum Oberseminar Topologie, November 22, 2011.
CMS Summer Meeting, University of Alberta, June 5, 2011.
Slides: Moduli spaces of 2-stage Postnikov systems.
Johns Hopkins University Topology Seminar, April 11, 2011.
University of Georgia Topology Seminar, February 7, 2011.
Ohio State University Topology Seminar, February 3, 2011.
Northwestern University Topology Seminar, November 8, 2010.
University of Chicago Algebraic Topology Seminar, October 19, 2010.
UIUC Topology Seminar, September 7, 2010.
Université Lille 1 Séminaire de topologie, March 26, 2010.
Heinrich-Heine-Universität Düsseldorf Oberseminar Topologie, January 20, 2010.
Ruhr-Universität Bochum Oberseminar Topologie, January 19, 2010.
Wesleyan University Topology Seminar, December 9, 2009.
University of Chicago Algebraic Topology Seminar, November 17, 2009.
Northwestern University Topology Seminar, November 16, 2009.
Penn State Altoona Homotopy Theory Mini Conference, October 23, 2009.
Slides: Quillen cohomology of Π-algebras.