## 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. https://arxiv.org/abs/2405.13223[PDF]
- 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
- Complexities of falling freely, joint with Anindya Sen (under preparation)
- 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

**Puzzling Rectangles,**joint with Manjunath Krishnapur, Resonance (1997)**Refining thick subcategory theorems,***Fundamenta Mathematicae***189**(2006) 61-97.**arXiv:math/0508101****[****pdf****]****Thick subcategories in stable homotopy theory***Oberwolfach Report***8**(2006) 12-20.**arXiv:math/0607245****[****pdf****]****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****]****Krull-Schmidt decompositions for thick subcategories,***Journal of Pure and Applied Algebra***210**(2007) 11 – 27.**arXiv:math/0507181****[****pdf****,****ps****,****other****]****Abelian subcategories closed under extensions: K-theory and decompositions,***Communications in Algebra***35**(2007) 807 – 819.**arXiv:math/0507320****[****pdf****]****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****]****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****]****Ghosts in modular representation theory,**joint with J. Daniel Christensen and Jan Minac,*Advances in Mathematics***217**(2008), 2782 – 2799**arXiv:math/0609699****[****pdf****]****Classifying subcategories of modules over a PID,***JP Journal of Algebra, Number Theory and Applications***2**(2009) 211 – 220.**arXiv:math/0607300****[****pdf****]****Freyd’s generating hypothesis with almost split sequences,**joint with*Proc. Amer. Math. Soc.***137**(2009), 2575 – 2580.**arXiv:0806.2165****[****pdf****]****Absolute Galois groups viewed from small quotients and the Bloch-Kato conjecture,**joint with*Geometry and Topology monographs***16**(2009) 31-47.**arXiv:0902.0992****[****pdf****]****Reciprocity laws for representations of finite groups,**joint with*Annales des sciences mathematiques du Quebec***34**(2010), no 1, 37-61.**arXiv:0911.3830****[****pdf****]****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****]****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****]****Counting irreducible polynomials over finite fields using the inclusion-exclusion principle,**joint with*Math. Mag***84**(2011), No. 5, 369-371.**arXiv:1001.0409****[****pdf****]****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****]****Freyd’s generating hypothesis for groups with periodic cohomology,**joint with*Canadian Mathematical Bulletin***55**(2012), 48-59.**arXiv:0710.3356****[****pdf****]****What is special about the divisors of 24?***Math. Mag.*Vol. 85, No. 5 (2012).**arXiv:1104.5052****[****pdf****]****Representations of The miraculous Klein group,**joint with Jan Minac,*Ramanujan Mathematics Society Newsletter***22**(2012), No. 1, 135-145.**arXiv:1209.4074****[****pdf****]****What is special about the divisors of 12?**joint with Michael Mayers*Math. Mag.*Vol. 86, No. 2 (2013).**arXiv:1212.3347****[****pdf****]****Detecting Fast solvability of equations via small powerful Galois groups,**joint with*Transactions of the AMS***367**(2015), no. 12, 8439 – 8464.**arXiv:1310.7623****[****pdf****]****Fuchs’ problem for indecomposable abelian groups,****,***Journal of Algebra***438**(2015) 325 – 336.**arXiv:1505.03508****[****pdf****]****Characterizations of Mersenne and 2-rooted primes,**joint with**,***Finite Fields and their Applications***35**(2015) 330 – 351.**arXiv:1404.4096****[****pdf****]****Fields with indecomposable multiplicative groups?**joint with**,***Expositiones Mathematicae***34**, (2016) no 2. 237-242.**arXiv:1407.3481****[****pdf****]****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****]****When is a subgroup of a ring an ideal?**joint with*Involve***9**(2016) no. 3, 503-516,**arXiv:1506.05513****[****pdf****]****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****]****Fuchs’ problem for dihedral groups,**joint with Keir Lockridge,*Journal of Pure and Applied Algebra***221**(2017) 971 – 982.**arXiv:1607.00687****[****pdf****]****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****]****How many units can a commutative ring have?**Joint with Lockridge,*Amer. Math. Monthly*124 (2017), no. 10, 960-965.**arXiv:1701.02341****[****pdf****]****Fuchs’ problem for**p**-groups,**joint with Keir Lockridge,*Journal of Pure and Applied Algebra*(2019), no. 11, 4652 -4666**arXiv:1901.10081****[****pdf****]****Measuring Mountains on the Moon,***Math Horizons*26 (2019), no. 4, 24-25.**arXiv:1905.08191****[****pdf****]****Packing Moons Inside the Earth,***Math Horizons*27 (2020), no. 4, 18-20**arXiv:2006.00603****[****pdf****]****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****]****The 7-day week**[pdf],*Math Horizons*29 (2022), no. 1, 20-22.**Baking Star Pi,**to appear in*Recreational Mathematics***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.)***Rings with an elementary abelian p-group of units,**joint with Jeremy Corry, Elizabeth Grimm, and Andrew Hatfield,*J. Commut. Algebra***15***,*no. 4, 469–480.*.***arxiv.org/abs/2112.03853***(This paper is the outcome of a 400-level graduate research course taught at ISU.)***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]****Generalized sine functions, complexified,**joint with Pisheng Ding,*Math. Mag.***96****arxiv.org/abs/2201.07414****[pdf]****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]****Is there an infinite field whose multiplicative group is indecomposable?**Joint with Keir Lockridge,**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.**Fuchs’ problem for endomorphisms of abelian groups**, joint with Keir Lockridge, arXiv.org/abs/2308.11815 [pdf]*Journal of Algebra,*635 (2023), 671–688.**Calculating the Lagrange Point (L2) for the JWST**,**[PDF]**to appear in*Mathematical Gazette.***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)

- 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 **f**or 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.)
- Okay, Cihan (Bilkent University)
- Pasini, Federico W.
- 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)
- Deepayan Sarkar (Indian Statistical Institute)
- Anindya Sen (University of Otaga)

## 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), South Africa (Johannesburg and Cape Town), Hong Kong, and Singapore (NUS) .