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.

1. Towards a refinement of the Bloch-Kato conjecture, joint with Jan Minac, Cihan Okay, Andy Schultz, and  Charlotte Ure. https://arxiv.org/abs/2405.13223[PDF]
2. 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)https://arxiv.org/abs/2308.13428 [PDF]
3. Complexities of falling freely, joint with Anindya Sen (under preparation)
4. Hyperplane arrangements over the ring of integers mod n, joint with Ehiareshan (under preparation)

---

Publications

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

1. Puzzling Rectangles, joint with Manjunath Krishnapur, Resonance (1997)
2. Refining thick subcategory theorems, Fundamenta Mathematicae 189 (2006) 61-97.   arXiv:math/0508101 [pdf]
3. Thick subcategories in stable homotopy theory  Oberwolfach Report 8 (2006) 12-20. arXiv:math/0607245 [pdf]
4. 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]
5. Krull-Schmidt decompositions for thick subcategories,  Journal of Pure and Applied Algebra 210 (2007) 11 – 27. arXiv:math/0507181 [pdf, ps, other]
6. Abelian subcategories closed under extensions: K-theory and decompositions, Communications in Algebra 35 (2007)  807 – 819. arXiv:math/0507320 [pdf]
7. 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]
8. 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]
9. Ghosts in modular representation theory, joint with  J. Daniel Christensen and Jan Minac, Advances in Mathematics 217 (2008), 2782 – 2799 arXiv:math/0609699 [pdf]
10. Classifying subcategories of modules over a PID, JP Journal of Algebra, Number Theory and Applications 2 (2009)  211  –  220. arXiv:math/0607300 [pdf]
11. 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]
12. 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]
13. 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]
14.  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]
15. 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]
16. 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]
17. 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]
18. 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]
19. What is special about the divisors of 24? Math. Mag. Vol. 85, No. 5 (2012). arXiv:1104.5052 [pdf]
20. Representations of The miraculous Klein group, joint with Jan Minac, Ramanujan Mathematics Society Newsletter 22 (2012), No. 1, 135-145. arXiv:1209.4074 [pdf]
21. What is special about the divisors of 12?  joint with Michael Mayers Math. Mag. Vol. 86, No. 2 (2013). arXiv:1212.3347 [pdf]
22. 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]
23. Fuchs’ problem for indecomposable abelian groups,   joint with  Keir Lockridge, Journal of Algebra 438 (2015)  325 – 336.  arXiv:1505.03508 [pdf]
24. 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]
25. Fields with indecomposable multiplicative groups?  joint with  Keir Lockridge, Expositiones Mathematicae 34, (2016) no 2. 237-242. arXiv:1407.3481 [pdf]
26. 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]
27. 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]
28. 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]
29. Fuchs’ problem for dihedral groups, joint with  Keir Lockridge, Journal of Pure and Applied Algebra 221 (2017)  971 – 982. arXiv:1607.00687 [pdf]
30.  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]
31. How many units can a commutative ring have? Joint with Lockridge, Amer. Math. Monthly 124 (2017), no. 10, 960-965. arXiv:1701.02341 [pdf]
32. Fuchs’ problem for p-groups, joint with Keir Lockridge,  Journal of Pure and Applied Algebra (2019), no. 11, 4652 -4666  arXiv:1901.10081 [pdf]
33. Measuring Mountains on the Moon, Math Horizons 26 (2019), no. 4, 24-25.  arXiv:1905.08191 [pdf]
34. Packing Moons Inside the Earth,  Math Horizons 27 (2020), no. 4, 18-20 arXiv:2006.00603 [pdf]
35. 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]
36. The 7-day week [pdf],  Math Horizons 29 (2022), no. 1, 20-22.
37. Baking Star Pi, to appear in Recreational Mathematics 
38. 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.)
39. 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.)
40. 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] 
41. Generalized sine functions, complexified, joint with Pisheng Ding, Math. Mag. 96 (2023), no 5, 498-507  arxiv.org/abs/2201.07414  [pdf]
42. 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] 
43. 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]
44. 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. 
45. Fuchs’ problem for endomorphisms of abelian groups,  joint with Keir Lockridge, arXiv.org/abs/2308.11815  [pdf]   Journal of  Algebra, 635 (2023), 671–688. 
46. Calculating the Lagrange Point (L2) for the JWST, [PDF] to appear in Mathematical Gazette.
47. Fuchs’ problem for endomorphisms of nonabelian groups, https://arxiv.org/abs/2408.08195 [PDF] joint with Keir Lockridge to appear in Journal of  Algebra.

---

Entries in OEIS

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

1. A018253 : The divisors of 24
2. A282572 : Odd numbers that occur as the cardinality of the group of units in a commutative ring
3. A084828 : Maximum number of spheres of radius one that can be packed in a sphere of radius n
4. A296241 : Finite number of units in a commutative ring;
5. 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.

1. Faculty Research Award, 2023-2024, Illinois State University. 
2. Simons Collaboration Grant for Mathematicians, 2017-2023
3.  Young Investigator Grant, National Security Agency (NSA), 2013-2015 
4.  University Research Initiative Award, Illinois State University, 2011 
5.  Received funding from the Midwest Topology Network, National Science Foundation (NSF), 2008-2011.
6.  McFarlan Fellowship, Mathematics department, University of Washington, 2005.
7. Hewitt Academic Excellence Award, University of Washington, 2001
8.  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.

---

---

---

 My Research Profiles