*Welcome to my homepage!*

My Ph.D. degree conferral receipt date was May 3, 2019. My research thesis adviser was

*(Karen E. Smith). I do research in asymptotic-, combinatorial-, and homological commutative algebra as they interface with algebraic geometry.***Karen E. Smith** Recently, I served as an NSF-funded graduate student research assistant, namely under Karen's grant DMS-1501625 "

**".***Frobenius in Geometry and Combinatorics* Right now, I am an

**NSF Postdoctoral Researcher + Van Vleck Visiting Professor based at the University of Wisconsin - Madison**. My NSF sponsoring scientist based in Madison is Daniel Erman. Some Notes Related to these outcomes:

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

2. I am so "(Navier-)stoked" for this next postdoctoral phase of my career as a mathematician :) !

3. I am in process of

**brainstorming/logging ideas for postdoctoral math projects (collaborative and otherwise)**to pursue and turn around as publications in the 2019-2023 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*Publications / Accepted Submissions*

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

- 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
- 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 :) !) - 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! - 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. - 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)! - 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) For context, while published last chronologically this work is a successor to item (7), and precedes/motivates items (1) and (5) as far as my thesis work. - R.M. Walker. "Uniform Symbolic Topologies via Multinomial Expansions." Proc. Amer. Math. Soc. 146 (2018), pp. 3735-3746.
- 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)
- 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)
- 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)
- 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*

- 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." Submitted to
*Journal of Software for Algebra and Geometry.*(arXiv/1611.05805)

*In (Various Stages of) Preparation*

- R.M. Walker. "Symbolic Generic Initial Systems of A.S.P. Ideals (previously: of Hypersurface Configurations)." 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
- R.M. Walker. "TBD." (Maybe I'll pleasantly surprise myself as far as productivity going into Summer 2019!...)
- R.M. Walker. "TBD." (Maybe I'll pleasantly surprise myself as far as productivity going into Summer 2019!...)

*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 2018-19 grad school application cycle, and can work with me to defray travel/housing costs, feel free to reach out!

- R.M. Walker.
. Notes from a talk on 04/25 in UIUC Math 496 course (Honors Seminar: Introduction to Mathematical Research), taught by Professor Bruce Reznick.**M.A.D. Tapas***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 - R.M. Walker. Beamer Talk on 04/25 for UIUC MATRIX undergraduate math club, combining slides from my two talks in the UM-Ann Arbor undergraduate math club in 2017. Click Here

215 Van Vleck Hall Department of Mathematics 480 Lincoln Drive Madison, WI 53706 An instance of "Geometric Modeling" using Algebraic Geometry: A "Yin-Yang" symbol with cubic rotisserie skewer, using WolframAlpha. | The above photo appears under "Rising Stars," on the Mathematically Gifted and Black website. MG&B 2018 Rising Stars Link 2015 MathFest, with my adviser |