Martin Helmer

I am a Royal Society Wolfson Fellow and Lecturer in Applied Algebra in the Department of Mathematics at Swansea University.

From 2022--2024 I was an Assistant Professor in the Department of Mathematics at North Carolina State University. Before this I was an MSI Fellow in the Mathematical Sciences Institute at The Australian National University from 2019--2021, a Postdoctoral Fellow in the Department of Mathematical Sciences at the University of Copenhagen (mentored by Elisenda Feliu) from 2017--2019, and an NSERC Postdoctoral Fellow in the Department of Mathematics at the University of California, Berkeley (mentored by Bernd Sturmfels) from 2015--2017.

Contact

Email: martin.helmer at swansea.ac.uk

Address:
Room 314, The Computational Foundry,
Swansea University Bay Campus,
Fabian Way, Swansea, SA1 8EN, Wales, UK.

Research Interests

My primary interest is to construct computational and theoretic tools to aid in understanding algebraic geometry and its applications. Broadly I am interested in algebraic geometry (theoretical, computational, and applied) and in related topics in computer algebra. I am particularly interested in practical methods in intersection theory, polynomial system solving, combinatorial algebraic geometry and real algebraic geometry. Recently I have been interested in applying ideas from algebraic geometry to study chemical reaction networks.

Preprints

Boulos El Hilany, Martin Helmer, Elias Tsigaridas. Stratification of Projection Maps From Toric Varieties . Arxiv: 2408.08991.

Martin Helmer and Rafael Mohr. A New Algorithm for Whitney Stratification of Varieties . Arxiv: 2406.17122.

Martin Helmer, Georgios Papathanasiou, Felix Tellander. Landau Singularities from Whitney Stratifications . Arxiv: 2402.14787.

Martin Helmer and Vidit Nanda. Effective Whitney Stratification of Real Algebraic Varieties . Arxiv: 2307.05427.

Publications

Christoph Dlapa, Martin Helmer, Georgios Papathanasiou, Felix Tellander. Symbol Alphabets from the Landau Singular Locus . Journal of High Energy Physics, Volume 2023, article number 161, 2023. arxiv link.
Martin Helmer and Elias Tsigaridas. Segre-Driven Radicality Testing . Journal of Symbolic Computation. DOI: 10.1016/j.jsc.2023.102262, 2023. arxiv link.
Martin Helmer and Vidit Nanda. Complex Links and Hilbert-Samuel Multiplicities. SIAM Journal on Applied Algebra and Geometry, DOI: 10.1137/22M1475533, 2023. arxiv link.
Martin Helmer and Felix Tellander. Cohen-Macaulay Property of Feynman Integrals . Communications in Mathematical Physics, DOI: 10.1007/s00220-022-04569-6, 2022. arxiv link.
Martin Helmer and Vidit Nanda. Conormal Spaces and Whitney Stratifications. Foundations of Computational Mathematics, DOI: 10.1007/s10208-022-09574-8, 2022. arxiv link.
Yacin Ameur, Martin Helmer, Felix Tellander. On the Uniqueness Problem for Quadrature Domains. Computational Methods and Function Theory, DOI: 10.1007/s40315-021-00373-w, 2021. arxiv link.
Michael F Adamer, Martin Helmer. Families of toric chemical reaction networks. Journal of Mathematical Chemistry, 2020. DOI: 10.1007/s10910-020-01162-x. arxiv link.
Jonathan D. Hauenstein, Martin Helmer. Probabilistic Saturations and Alt’s Problem. Experimental Mathematics, 2020. DOI: 10.1080/10586458.2020.1740835. arxiv link..
Corey Harris and Martin Helmer. Segre Class Computation and Practical Applications. Mathematics of Computation, 89.321, pages 465-491, 2020. arxiv link.
James Fullwood and Martin Helmer. On the Euler characteristic of a relative hypersurface. Journal of Mathematical Physics, 60, 052302, 2019. arxiv link.
Elisenda Feliu and Martin Helmer. Multistationarity and Bistability for Fewnomial Chemical Reaction Networks. Bulletin of Mathematical Biology, DOI: 10.1007/s11538-018-00555-z, 2018. arxiv link.
Michael F Adamer, Martin Helmer. Complexity of Model Testing for Dynamical Systems with Toric Steady States. Advances in Applied Mathematics Volume 110, September 2019, Pages 42-75. arxiv link.
Martin Helmer, Bernt Ivar Utstøl Nødland. Polar degrees and closest points in codimension two. Journal of Algebra and Its Applications (2017): 1950095 Software page. arxiv link.
Martin Helmer and Bernd Sturmfels. Nearest Points on Toric Varieties. Mathematica Scandinavica, Volume 122.2, page 213-238, 2018. Software page. arxiv link.
Carlos Améndola, Nathan Bliss, Isaac Burke, Courtney R. Gibbons, Martin Helmer, Serkan Hoşten, Evan D. Nash, Jose Israel Rodriguez, Daniel Smolkin. The Maximum Likelihood Degree of Toric Varieties. Journal of Symbolic Computation, 2018. arxiv link.
Martin Helmer. Computing characteristic classes of subschemes of smooth toric varieties. Journal of Algebra, Volume 476, 2017, Pages 548-582, ISSN 0021-8693. arxiv link.
Martin Helmer (with Appendix by Martin Helmer and Éric Schost). A Direct Algorithm to Compute the Topological Euler Characteristic and Chern-Schwartz-MacPherson Class of Projective Complete Intersection Varieties. Theoretical Computer Science, Volume 681, 2017, Pages 54-74, ISSN 0304-3975. arxiv link.
Martin Helmer. Computing the Chern-Schwartz-MacPherson Class of Complete Simplical Toric Varieties. Springer Proceedings in Mathematics & Statistics book series (PROMS), volume 198 (ACA 2015), pp 207-217, July 2017. arxiv link.
Martin Helmer. Algorithms to compute the topological Euler characteristic, Chern-Schwartz-Macpherson class and Segre class of projective varieties. Journal of Symbolic Computation, 73, page 120-138, 2016. arxiv link.
Martin Helmer. An algorithm to compute certain Euler characteristics and Chern-Schwartz-MacPherson classes. In Proceedings of the 2014 Symposium on Symbolic-Numeric Computation, pp. 130-131. ACM, 2014.
PL Buono, Martin Helmer, and Jeroen SW Lamb. On the zero set of G-equivariant maps. In Mathematical Proceedings of the Cambridge Philosophical Society, volume 147, page 735. Cambridge Univ. Press, 2009. arxiv link.
Ramiro Liscano, John Khalil Jacoub, Anand Dersingh, Jinfu Zheng, Martin Helmer, Charles Elliott, and Ali Najafizadeh. Network performance of a wireless sensor network for temperature monitoring in vineyards. In Proceedings of the 8th ACM Symposium on Performance evaluation of wireless ad hoc, sensor, and ubiquitous networks, pp. 125-130. ACM, 2011.

Software

WhitneyStratifications: A Macaulay2 software package which computes Whitney stratifications of complex algebraic varieites. Current version: 2.03. This package may be download here: Download. It is also a Macaulay2 built-in package in versions 1.2.1 and above and can be loaded with: needsPackage "WhitneyStratifications"; note the current built-in package has not yet updated to the new version and more functions are avalible in the version linked to here.
SegreClasses: A Macaulay2 software package which tests containment of varieties and computes algebraic multiplicity using new Segre-driven methods. The package also computes Fulton-MacPherson intersection products and Segre classes of subschemes of products of projective spaces. This package is avalible as a built-in package in Macaulay2 version 1.13 and above, you may also download it here: Download.
ToricInvariants: A Macaulay2 software package to compute the Euclidean Distance Degree, the polar degrees, and the Chern-Mather class of a projective toric variety using combinatorial methods (combines features of the EDPolytope and EDPolytopeCD2 packages listed below).
EDPolytopeCD2: A Macaulay2 software package to compute the Euclidean Distance Degree, the polar degrees, and the Chern-Mather class of a projective toric variety using combinatorial methods, optimized for the case where the projective toric variety has codimension two. A version of this package is available in Macaulay2 version 1.13 and above under the name ToricInvariants.
EDPolytope: A Macaulay2 software package to compute the Euclidean Distance Degree, the polar degrees, and the Chern-Mather class of a projective toric variety using combinatorial methods. A version of this package is available in Macaulay2 version 1.13 and above under the name ToricInvariants.
CharacteristicClasses: A Macaulay2 built-in software package coauthored with Christine Jost. The package computes the Segre class, Chern-Schwartz-MacPherson class and Euler characteristic of a given subscheme of certain smooth toric varieties.
char-class-calc: A Sage program to compute the Segre class, Chern-Schwartz-MacPherson class and Euler characteristic of a projective variety.

These and other software projects can also be found on my github page https://github.com/Martin-Helmer.

Teaching

Conference Talks

Core Computational Methods in Nonlinear Algebra at ICERM. Sept 18, 2018. Brown University, Providence, USA. Segre-driven methods to compute algebraic multiplicity and test ideal membership. Slides. Video.
Open Source Computer Algebra Research Conference at the Max Planck Institute for Mathematics in the Sciences. Dec 12, 2017. Leipzig, Germany. Effective Intersection Theory.
SIAM Conference on Applied Algebraic Geometry. August 3, 2017. Georgia Institute of Technology, Atlanta, USA. Nearest Points on Toric Varieties. Slides.
Joint Math Meetings: Algebraic Statistics Special Session. January 5, 2017. Atlanta, USA. Topological Invariants and the Maximum Likelihood Degree of a Toric Variety. Slides.
Fields Institute Thematic Program on Computer Algebra. Workshop on Algebra, Geometry and Proofs in Symbolic Computation. December 7, 2015. Fields Institute, Toronto, Canada. Algorithms to Compute Characteristic Classes of Subschemes of Certain Toric Varieties. Slides. Video of the talk.
SIAM Conference on Applied Algebraic Geometry. August 3-7, 2015. National Institute of Mathematical Sciences, Daejeon, Korea. Talk Slides (Titled "Algorithms to Compute Chern-Schwartz-Macpherson and Segre Classes and the Euler Characteristic") . Conference webpage.
ACA 2015 Conference (Applications of Computer Algebra). July 20-23, 2015 Kalamata, Greece. Talk Slides (Titled "Computing the Chern-Schwartz-MacPherson Class of a Complete Simplicial Toric Variety") . Conference webpage.
SNC 2014 Conference (Symbolic-Numeric Computation Conference). July 28-31, 2014 East China Normal University. Shanghai, China. Talk Slides (Titled "An Algorithm to Compute Certain Euler Characteristics and Chern-Schwartz-MacPherson Classes") . Conference webpage.
ACA 2014 Conference (Applications of Computer Algebra). July 9-12, 2014 Fordham University. New York, New York. Talk Slides (Titled "Algorithms to Compute Chern-Schwartz-Macpherson and Segre Classes and the Euler Characteristic") . Conference webpage.
ECCAD 2014, (April 26, 2014, Duke University, Durham, NC). Poster .

Education

Ph.D Applied Mathematics, University of Western Ontario, Canada. (2011-2015)
- Advisor: Éric Schost
- Thesis: Algorithms to Compute Characteristic Classes
M.Sc Mathematics, Queen's University, Canada. (2009-2011)
B.Sc Applied Mathematics, University of Ontario Institute of Technology, Canada. (2005-2009)

Curriculum Vitae

A pdf of my C.V. can be found here.