Research

Overview

My research interests lie in algebra, representation theory, and number theory. Specifically, I worked on an analog of Frey's generating hypothesis and ghost maps in the stable module category of group algebras, refinements of chromatic towers in the stable categories, Fuchs's problem on classifying groups which arise as a unit group of rings, quotients of the absolute Galois group, and refinement of the Bloch-Kato conjecture.

My research has been supported by several external funding agencies, including the NSF's Midwest Topology Network (MTN), Young Investigator Grant of the National Security Agency (NSA), and the Simons Foundation's Collaboration Grant for Mathematicians.

Ph.D. Thesis

Refinements of chromatic towers and Krull-Schmidt decompositions in stable homotopy category, Ph.D. thesis, University of Washington, 2005 arXiv:math/0607726 [pdf]

Advisor: John Palmieri (Click here for my genealogy)

Preprints

Some of these were submitted and others are in preparation. Contact me if you are interested in seeing a preprint.

Towards a refinement of the Bloch-Kato conjecture, joint with Jan Minac, Cihan Okay, Andy Schultz, and Charlotte Ure (under preparation)
Hyperplane arrangements over the ring of integers mod n, joint with Ehiareshan (under preparation)
On the arithmetic of the join rings over finite fields, joint with Jonathan L. Merzel, Ján Mináč, Tung T. Nguyen, Federico W. Pasini, and Nguyen Duy Tân (submitted)
Fuchs' problem for endomorphisms of nonabelian groups, joint with Keir Lockridge (submitted)

Publications

You can view or download my papers from the PDF links below. All papers appear in reverse chronological order. You can also read reviews of these papers in MathSciNet or Zentralblatt Math

1. Calculating the Lagrange Point (L2) for the JWST, [PDF] to appear in Mathematical Gazette.
2. Fuchs' problem for endomorphisms of abelian groups, joint with Keir Lockridge, arXiv.org/abs/2308.11815 [pdf] Journal of Algebra, 635 (2023), 671–688.
3. On the joins of group rings, joint with Jonathan L. Merzel, Ján Mináč, Lyle Muller, Tung T. Nguyen, Federico W. Pasini, and Nguyen Duy Tân arXiv:2208.07413 [PDF], J. Pure Appl. Algebra 227 (2023), no. 9, Paper No. 107377, 33 pp.
4. Is there an infinite field whose multiplicative group is indecomposable? Joint with Keir Lockridge, Indian J. Pure Appl. Math. 54 (2023), no. 2, 398–403. arxiv.org/abs/2204.10146 [pdf]
5. A clock model for planetary conjunctions, Mathematical Gazette .107, no. 570 (November 2023): 422 - 429. https://doi.org/10.1017/mag.2023.94 arxiv.org/abs/2203.12823 [pdf]
6. Generalized sine functions, complexified, joint with Pisheng Ding, Math. Mag. 96 (2023), no 5, 498-507 arxiv.org/abs/2201.07414 [pdf]
7. Zero-sum free tuples and hyperplane arrangements, joint with Papa Sissokho, Integers 22 (2022), Paper No. A12, 25 pp. arxiv.org/abs/2201.01714 [pdf]
8. Rings with an elementary abelian p-group of units, joint with Jeremy Corry, Elizabeth Grimm, and Andrew Hatfield, J. Commut. Algebra 15 (2023), no. 4, 469–480. .arxiv.org/abs/2112.03853 (This paper is the outcome of a 400-level graduate research course taught at ISU.)
9. Trigonometric functions in the p-norm, joint with Andrew Hatfield, Riley Klette, Christopher Moore, and Beth Warden, Mathematics Exchange ,Vol 16, No. 1 (2022) arXiv:2109.14036 [pdf] (This paper is the outcome of a 200-level undergraduate research course taught at ISU.)
10. Baking Star Pi, to appear in Recreational Mathematics
11. The 7-day week [pdf], Math Horizons 29 (2022), no. 1, 20-22.
12. Gaussian binomial coefficients in group theory, field theory, and topology, joint with Keir Lockridge, Amer. Math. Monthly 129 (2022) no. 5, 466-473, arXiv:2108.10956 [pdf]
13. Packing Moons Inside the Earth, Math Horizons 27 (2020), no. 4, 18-20 arXiv:2006.00603 [pdf]
14. Measuring Mountains on the Moon, Math Horizons 26 (2019), no. 4, 24-25. arXiv:1905.08191 [pdf]
15. Fuchs' problem for p-groups, joint with Keir Lockridge, Journal of Pure and Applied Algebra (2019), no. 11, 4652 -4666 arXiv:1901.10081 [pdf]
16. How many units can a commutative ring have? Joint with Lockridge, Amer. Math. Monthly 124 (2017), no. 10, 960-965. arXiv:1701.02341 [pdf]
17. Witt's cancellation theorem seen as a cancellation, joint with Dan McQuillan and Jan Minac, Expositiones Mathematicae 35 (2017), no. 3, 300-314. arXiv:1106.2595 [pdf]
18. Fuchs' problem for dihedral groups, joint with Keir Lockridge, Journal of Pure and Applied Algebra 221 (2017) 971 - 982. arXiv:1607.00687 [pdf]
19. Ghosts and Strong Ghosts in the Stable Module Category, joint with Jon F. Carlson and Jan Minac, Canadian Mathematical Bulletin, 59 (2016), 682-692. arXiv:1509.02845 [pdf]
20. When is a subgroup of a ring an ideal? joint with Christina L. Henry, Involve 9 (2016) no. 3, 503-516, arXiv:1506.05513 [pdf]
21. Galois p-groups and Galois modules, joint with Jan Minac and Andrew Schultz, Rocky Mountain Journal of Mathematics 46 (2016), 1405-1446. arXiv:1411.6495 [pdf]
22. Fields with indecomposable multiplicative groups? joint with Keir Lockridge, Expositiones Mathematicae 34, (2016) no 2. 237-242. arXiv:1407.3481 [pdf]
23. Characterizations of Mersenne and 2-rooted primes, joint with Keir Lockridge and Gaywalee Yamskulna, Finite Fields and their Applications 35 (2015) 330 - 351. arXiv:1404.4096 [pdf]
24. Fuchs' problem for indecomposable abelian groups, joint with Keir Lockridge, Journal of Algebra 438 (2015) 325 - 336. arXiv:1505.03508 [pdf]
25. Detecting Fast solvability of equations via small powerful Galois groups, joint with Jan Minac and Claudio Quadrelli, Transactions of the AMS 367 (2015), no. 12, 8439 - 8464. arXiv:1310.7623 [pdf]
26. What is special about the divisors of 12? joint with Michael Mayers Math. Mag. Vol. 86, No. 2 (2013). arXiv:1212.3347 [pdf]
27. Representations of The miraculous Klein group, joint with Jan Minac, Ramanujan Mathematics Society Newsletter 22 (2012), No. 1, 135-145. arXiv:1209.4074 [pdf]
28. What is special about the divisors of 24? Math. Mag. Vol. 85, No. 5 (2012). arXiv:1104.5052 [pdf]
29. Freyd's generating hypothesis for groups with periodic cohomology, joint with J. Daniel Christensen and Ján Mináč, Canadian Mathematical Bulletin 55 (2012), 48-59. arXiv:0710.3356 [pdf]
30. Quotients of absolute Galois groups which determine the entire Galois cohomology, joint with Ido Efrat and Ján Mináč, Math. Annalen 352 (2012), 205-221. arXiv:0905.1364 [pdf]
31. Counting irreducible polynomials over finite fields using the inclusion-exclusion principle, joint with Jan Minac, Math. Mag 84 (2011), No. 5, 369-371. arXiv:1001.0409 [pdf]
32. Finite generation of Tate cohomology, joint with Jon F. Carlson and Jan Minac, Representation theory (AMS Journal) 15 (2011) 244 - 257. arXiv:0804.4246 [pdf]
33. A small quotient of the big absolute Galois group (joint work with Ido Efrat and Ján Mináč), Oberwolfach Report 32 (2010) 14-17. arXiv:1101.5738 [pdf]
34. Reciprocity laws for representations of finite groups, joint with Jan Minac and Clive Reis, Annales des sciences mathematiques du Quebec 34 (2010), no 1, 37-61. arXiv:0911.3830 [pdf]
35. Absolute Galois groups viewed from small quotients and the Bloch-Kato conjecture, joint with Ján Mináč, Geometry and Topology monographs 16 (2009) 31-47. arXiv:0902.0992 [pdf]
36. Freyd's generating hypothesis with almost split sequences, joint with Jon F. Carlson and Jan Minac, Proc. Amer. Math. Soc. 137 (2009), 2575 - 2580. arXiv:0806.2165 [pdf]
37. Classifying subcategories of modules over a PID, JP Journal of Algebra, Number Theory and Applications 2 (2009) 211 - 220. arXiv:math/0607300 [pdf]
38. Ghosts in modular representation theory, joint with J. Daniel Christensen and Jan Minac, Advances in Mathematics 217 (2008), 2782 - 2799 arXiv:math/0609699 [pdf]
39. Auslander-Reiten sequences for homotopists and arithmeticians, joint with, Annales des sciences mathÃematiques du QuÃebec 32 (2008), no 2, 139 - 157, arXiv:0811.0561 [pdf]
40. Groups which do not admit ghosts, joint with J. Daniel Christensen and Jan Minac, Proc. Amer. Math. Soc. 136 (2008), 1171 - 1179. arXiv:math/0610423 [pdf]
41. Abelian subcategories closed under extensions: K-theory and decompositions, Communications in Algebra 35 (2007) 807 - 819. arXiv:math/0507320 [pdf]
42. Krull-Schmidt decompositions for thick subcategories, Journal of Pure and Applied Algebra 210 (2007) 11 - 27. arXiv:math/0507181 [pdf, ps, other]
43. The generating hypothesis for the stable module category of a p-group, joint with David J. Benson, J. Daniel Christensen, and Jan Minac, Journal of Algebra 310 (2007) 428 - 433. arXiv:math/0611403 [pdf]
44. Thick subcategories in stable homotopy theory Oberwolfach Report 8 (2006) 12-20. arXiv:math/0607245 [pdf]
45. Refining thick subcategory theorems, Fundamenta Mathematicae 189 (2006) 61-97. arXiv:math/0508101 [pdf]
46. Refinements of chromatic towers and Krull-Schmidt decompositions in stable homotopy category, Ph.D. thesis, University of Washington, 2005 arXiv:math/0607726 [pdf]
47. Puzzling Rectangles, joint with Manjunath, Resonance (1997)

Entries in OEIS

The following sequences that appeared in my research were noted in OEIS (Online Encyclopedia of Integer Sequences)

A018253 : The divisors of 24
A282572 : Odd numbers that occur as the cardinality of the group of units in a commutative ring
A084828 : Maximum number of spheres of radius one that can be packed in a sphere of radius n
A296241 : Finite number of units in a commutative ring;
Not added yet: The number of irreducible zero-sum-free sequences of length d over the ring of integers mod n.

Awards and Grants for Research

I am very grateful for all the internal and external funding agencies which supported my research over the years.

Faculty Research Award, 2023-2024, Illinois State University.
Simons Collaboration Grant for Mathematicians, 2017-2023
Young Investigator Grant, National Security Agency (NSA), 2013-2015
University Research Initiative Award, Illinois State University, 2011
Received funding from the Midwest Topology Network, National Science Foundation (NSF), 2008-2011.
McFarlan Fellowship, Mathematics department, University of Washington, 2005.
Hewitt Academic Excellence Award, University of Washington, 2001
Internal grants at Illinois State University: New faculty initiative grant (NFIG), Pre-Tenure faculty initiative grant (PFIG), Summer Faculty Fellowship (SFF), International Travel grant, and Summer research grants.

Coauthors

I had the pleasure and honor to collaborate with the following people. My current Erdos number is 3: Erdos ---> Conway ---> Dave Benson ---> Me. You can read about Erdos number here.

Benson, David John (University of Aberdeen )
Carlson, Jon F. (University of Georgia)
Christensen, J. Daniel (University of Western Ontario)
Corry, Jeremy (Illinois State University)
Ding, Pisheng (Illinois State University)
Efrat, Ido (Ben-Gurion University of the Negev)
Grimm, Elizabeth (Illinois State University)
Hatfield, Andrew (Illinois State University)
Henry, Christina L. (Pinnacle Insurance)
Klette, Riley (Illinois State University)
Lockridge, Keir H. (Gettysburg College)
Muller, Lyle (University of Western Ontario)
Mayers, Michael (Illinois Central College)
Merzel, Jonathan (Soka University of America)
McQuillan, Daniel John (Norwich University)
Mináč, Ján (University of Western Ontario)
Moore, Christopher (Illinois State University)
Nguyen, Tung T. (University of Western Ontario and Onepick Inc.)
Pasini, Federico W. (Huron University College)
Quadrelli, Claudio, (University of Milano Bicocca)
Reis, Clive M. (Victoria, BC, Canada)
Sissokho, Papa (Illinois State University)
Schultz, Andrew (Wellesley College)
Tân, Nguyen Duy (Hanoi University of Science and Technology)
Warden, Elizabeth (Illinois State University)
Yamskulna, Gaywalee (Illinois State University)
Charlotte Ure (Illinois State University)

Travels

I visited the following countries, primarily for talks, mathematics conferences, and research collaborations, but also for some vacation: India (ISI, IISc, IIT, IMSc, CMI, TIFR), Bhutan (CSO), USA (IMA, MSRI, plus too many to list), Canada (UWO, Fields Institute, PIMS, Banff), Germany (Bonn, Oberwolfach), Sweden (Mittag-Leffler), France (Paris), Italy (Florence and Venice), Switzerland (Zurich), Czech Republic (Academy of Sciences), China (Chern Institute), Brazil (ICM), Vietnam (ICIS), Panama, Japan (Univ. of Tokyo), England (Oxford and Cambridge), Scotland (Glasgow), Wales (Cardiff), and South Africa (Johannesburg and Cape Town).

My Research Profiles

arXiv

Research Gate

Google Scholar