Home + Contact Info

Welcome to my homepage!

My research thesis adviser was Karen E. Smith (Karen E. Smith ). I do research in asymptotic-, combinatorial-/convex-, and homological commutative algebra as they interface with algebraic geometry, combinatorics, and representation theory.

My position as an NSF Postdoctoral Researcher + Van Vleck Visiting Professor based at the University of Wisconsin - Madison has concluded. My NSF sponsoring scientist based in Madison was Daniel Erman.

Starting in Fall 2022, I will be working at Google Park View in Seattle, WA as a Visiting Scholar -- looking forward to it!

Some Notes Related to these outcomes:

1. The official citation of record on nsf.gov is Linked Here (DMS-1902748).

2. My NSF Postdoc Job Materials can be found Here and Here

2. I am in process of brainstorming/logging ideas for postdoctoral math projects (collaborative and otherwise) to pursue and turn around as publications in the 2021-2024 academic year window. If you'd care to inquire or pitch stuff, feel free to reach out -- I'll aim to respond carefully in kind!

Link to Most Current Author Profile on AMS MathSciNet (Author ID = MR 1031098): Linked Here

My ORCID is 0000-0002-8517-8603

Publications / Accepted Submissions

(in Reverse Chronological Order of Receipt of Acceptance to Journal)

1. H. Banos, N. Bushek, R. Davidson, E. Gross, P.E. Harris, R. Krone, C. Long, A. Stewart, R.M. Walker. "Phylogenetic Trees: a package for Macaulay2." Accepted to Journal of Software for Algebra and Geometry. J. Softw. Alg. Geom. 11 (2021) 1-7 (arXiv/1611.05805)

2. R.M. Walker. "Uniform Symbolic Topologies in Non-Regular Rings." This is my Ph.D. Dissertation, to be linked on Karen E. Smith's webpage (her Ph.D. Students section). Linked Google Drive Upload

3. I. Swanson and R.M. Walker. "Tensor-Multinomial Sums of Ideals: Primary Decompositions and Persistence of Associated Primes." Accepted to Proceedings of the AMS as of 03/21/2019, with all relevant ZIP files sent along to the AMS folks 04/04/2919 so all set there. (arXiv/1806.03545; the Proc. AMS Online Published Version ; my JMM 2019 Slides expositing this joint work :) !)

4. R.M. Walker. "Uniform Symbolic Topologies in Normal Toric Rings." Journal of Algebra 511 (2018), pp. 292-298. Link to Science Direct (https://doi.org/10.1016/j.jalgebra.2018.05.038). (arXiv/1706.06576) The results advertised in this paper were the original motivation for my thesis back in Fall 2014! After a sinuous journey, I did pin down my thesis problem -- eventually!

5. H. Banos, N. Bushek, R. Davidson, E. Gross, P.E. Harris, R. Krone, C. Long, A. Stewart, R.M. Walker. "Dimensions of Group-based Phylogenetic Mixtures." Accepted to appear in The Bulletin of Mathematical Biology (Special Issue: Algebraic Methods in Phylogenetics). (arXiv/1711.08686, Springer Link (Bull Math Biol (2018). https://doi.org/10.1007/s11538-018-0489-0 ), Link to Full-Text View-Only PDF of the Published Version) This work is based on participation in the 2016 AMS MRC workshop in Algebraic Statistics. I was part of the Phylogenetic Algebraic Geometry group led by Elizabeth Gross and Colby Long, who were each advised/mentored by Seth Sullivant, formerly a student of Bernd Sturmfels.

6. T. Hammonds, J. Johnson, A. Patini, and R.M. Walker. "Counting Roots of Polynomials over Z/ p^2 Z." Houston Journal of Mathematics (2018), Vol. 44, no. 4, pp. 1111-1119. (This paper is based on research conducted in the 2017 Mathematical Sciences Research Institute Undergraduate Program, or MSRI-UP for short!) (arXiv/1708.04713) (Video Link) Note that this work has been extended to full generality in a joint paper (arXiv: 1711.01355) by Qi Cheng, Shuhong Gao, J. Maurice Rojas, and Daqing Wan. A more recent Summer 2018 entry in this vein by Rojas and REU collaborators is the following preprint (arXiv: 1808.10531)!

7. R.M. Walker. "Uniform Harbourne-Huneke Bounds via Flat Extensions." Journal of Algebra 516 (2018), pp. 125-148. Link to Science Direct (https://www.sciencedirect.com/science/article/pii/S0021869318305106) (arXiv/1608.02320)

8. R.M. Walker. "Uniform Symbolic Topologies via Multinomial Expansions." Proc. Amer. Math. Soc. 146 (2018), pp. 3735-3746. (arXiv/1703.04530, AMS Proceedings Link) This contains the most powerful results of my dissertation.

9. E. Canton, D. Hernandez, K. Schwede, E. Witt, A. De Stefani, J. Jeffries, Z. Kadyrsizova, R.M. Walker, G. Whelan. "On the behavior of singularities at the F-pure threshold." Illinois Journal of Mathematics. Volume 60, No. 3-4, Fall/Winter 2016, pp. 669-685. I am a co-author of the appendix to this paper based on participation in the 2015 AMS MRC workshop in Commutative Algebra. I was part of the Positive Characteristic group led by Karl Schwede. (arXiv/1508.05427)

10. R.M. Walker. "Rational Singularities and Uniform Symbolic Topologies." Illinois Journal of Mathematics. Volume 60, No. 2, Summer 2016, pp. 541-550. (arXiv/1510.02993, 10/1/15 talk notes, 1/8/16 JMM Talk)

11. D.J. Bruce, M. Logue, and R. Walker. "Monomial Valuations, Cusp Singularities, and Continued Fractions." J. Comm. Algebra (2015) Volume 7, Number 4 (2015), 495-522. (arXiv/1311.6493, MAA MathFest Beamer Talk)

12. A. Schultz and R. Walker. "A generalization of the Gaussian formula and a q-analog of Fleck's congruence." J. Number Theory 133 (2013), no. 11, 3717-3738. (arXiv/1202.0199)

Preprints on arXiv

In (Various Stages of) Preparation

1. Ethan Harpal Chauhan and R.M. Walker. "Matroid Configurations in $\mathbb{P}^n$, Polynomial Algebraic Matroids, and ASP Ideals." I've been traveling during Winter 2019 giving seminar talks on this work-in-progress. I have given some preliminary report talk(s) at the UM-Ann Arbor Combinatorics Seminar, the AMS Sectional meeting in Boston, the UIUC Algebra-Geometry-Combinatorics Seminar, and the 2018 Temple University Grad Student Conference in Algebra, Geometry, and Topology. A Hyperlink to Temple U Beamer TeX Slides Here Click here for paper preprint so far

2. Ethan Harpal Chauhan and R.M. Walker. "The Flip of a Square Matrix, or Upside Down with Determinants." See Here and Here

3. Ethan Harpal Chauhan and R.M. Walker. "Quick Proofs of Ring-Theoretic and Other Facts." Link to Overleaf Project

4. R.M. Walker. "Polynomial Equation Systems with nonzero coefficients are dense." Click Here for current progress

5. R.M. Walker. "On a combinatorial inequality or two." Click Here

6. R.M. Walker. "A Quick Fact about Polynomial Regression" Click Here

7. R.M. Walker. "All Graphs of Polynomial Functions are Smooth." Click Here

8. R.M. Walker. "A Quick Fact about Ehrhart numbers" Click Here

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Sarcophaugustus Poetry (by Apollo Marais X, my pen name)

For a Google Doc, Click Here

Sarcophaugustus is a Book of Poems I am writing about, e.g., my experiences throwing down with cancer and other topics too. Feel free to comment in the document, that's allowed!

Apollo Marais X's Self-Help Guru Book

For a Google Doc you can comment in, Click Here

In addition to Sarcophaugustus, I am writing this book (a pamphlet so far).

PHIL 640 Final Paper (On the Sentiments)

For my final paper click here

Undergrad-Graduate General Audience Talks (2018)

Occasionally, I may offer to pair formal Research Seminar visits at other U.S. math departments with giving these General Audience Talks with an aim of doing "recruitment ambassadorship" for the UM-Ann Arbor math graduate program. If you'd be interested in having me speak with undergrads Q&A style ahead of the 2022-24 grad school application cycle, and can work with me to defray travel/housing costs or present me with an honorarium, feel free to reach out!

1. R.M. Walker. M.A.D. Tapas. Notes from a talk on 04/25/18 in UIUC Math 496 course (Honors Seminar: Introduction to Mathematical Research), taught by Professor Bruce Reznick. Rough Spiel: We record a "fast exponentiation" fact from p-adic analysis and extend it using undergrad-level algebra. We indicate how this fact can interact and blend with modular arithmetic divisibility tests. Click Here I have also given this talk at Oberlin College for an honorarium.

2. R.M. Walker. Beamer Talk on 04/25/18 for UIUC MATRIX undergraduate math club, combining slides from my two talks in the UM-Ann Arbor undergraduate math club in 2017. Click Here