Research Interests

Adler studies harmonic analysis on reductive p-adic groups, a subject that is closely connected to number theory.  He also occasionally directs undergraduate research projects.

Honors, Awards, and Fellowships

• Buchtel College of Arts and Sciences Early Career Research Award, The University of Akron, April, 2004.
• Member, Institute for Mathematical Sciences, Singapore, August, 2002.

Professional Presentations

   
• “Supercuspidal characters and coadjoint orbits,” Mini-colloque: Harmonic analysis on p-adic groups, Institut Henri Poincaré, Paris, June, 2000.
   
• “Murnaghan-Kirillov Theory for supercuspidal representations,” Harmonische Analysis und Darstellungstheorie topologischer Gruppen, Mathematisches Forschungsinstitut Oberwolfach, Germany, July, 2000.
   
• “Supercuspidal character germs for classical groups”, PIMS Workshop on Representations of Reductive p-adic Groups, Banff, Canada, February, 2002.
   
• “Base change, with an example”, Institute for Mathematical Sciences, Singapore, 31 July, 2002.
   
• “Categorical properties of supercuspidal representations,” Purdue University, November, 2003.
   
• “The local character expansion near a tame, semisimple element”, Recent Trends in Endoscopy and Representation Theory (a conference in honor of E.-W. Zink), Berlin, October 20, 2005.
   
• “Connectedness of centralizers: work of Steinberg”, Harish-Chandra Research Institute, Allahabad, India, November 9–10, 2005.
   
• “Representations of nilpotent groups”, Colloquium: Tata Institute of Fundamental Research, Mumbai, India, November 17, 2005.
   
• “Multiplicity one upon restriction”, Fields Institute Workshop on the Representation Theory of Reductive Algebraic Groups, University of Ottawa, January 21, 2007.
   
• “Towards a lifting of representations of finite reductive groups”, Special Session on p-adic Groups and Automorphic Forms, AMS Meeting #1041, Vancouver, Canada, October 4–5, 2008.
   
• "Tamely ramified tori", Special Session on harmonic analysis and representations of reductive p-adic groups, Joint Mathematics Meetings, San Francisco, January 16, 2010.
   
• "Towards a lifting of representations of reductive p-adic groups", Lie Theory Seminar, University of Minnesota, April 8, 2011.
   
• "Liftings and Deligne-Lusztig theory", Conference on Representation Theory of Finite Groups of Lie Type: Deligne-Lusztig Theory, Tata Institute of Fundamental Research, Mumbai, India, December 16, 2011.

Selected Publications

   
• Refined anisotropic K-types and supercuspidal representations, Pacific J. Math., 185 (1998), no. 1, 1–32.
   
• Some applications of Bruhat-Tits theory to harmonic analysis on the Lie algebra of a reductive p-adic group (with Stephen DeBacker), Mich. J. Math. 50 (2002), No. 2, 263–286.
   
• Murnaghan-Kirillov theory for supercuspidal representations of tame general linear groups, (with Stephen DeBacker), J. Reine Angew. Math. 575 (2004), 1–35.
   
• Injectivity, projectivity, and supercuspidal representations, (with Alan Roche), J. London Math. Soc. (2) 70 (2004), no. 2, 356–368.
   
• Depth-zero base change for unramified U(2, 1), (with Joshua Lansky), J. Number Theory 114 (2005), no. 2, pp. 324–360. Printer’s error corrected in vol. 121 (2006), no. 1, 186.
   
• On certain multiplicity one theorems (with Dipendra Prasad), Israel J. Math, 153 (2006), 221–245.
   
• The local character expansion near a tame, semisimple element (with Jonathan Korman), Amer. J. Math., 129 (2007), no. 2, 381–403.
   
• Good product expansions for tame elements of p-adic groups (with Loren Spice), Int. Math. Res. Pap. vol. 2008, 95 pages.
   
• Reading encrypted diplomatic correspondence: An undergraduate research project, (with Ryan Fuoss, Michael Levin, and Amanda Youell), Cryptologia,32 (2008), Issue 1, pp. 1–12.
   
• Supercuspidal characters of reductive p-adic groups (with Loren Spice), Amer. J. Math.  131 (2009), pp. 1137-1210. E-print available at arXiv:0707.3313.   
   
• Depth-zero base change for ramified U(2,1)  (with Joshua  Lansky), Trans. Amer. Math. Soc., 362 (2010), pp. 5569-5599. E-print  available at arXiv:0807.1528.   
   
• Supercuspidal characters of SL(2) over a p-adic field (with Stephen DeBacker, Paul J. Sally, Jr., and Loren Spice), in Harmonic analysis on reductive p-adic groups (Robert S. Doran, Paul J. Sally, Jr., and Loren Spice, eds.), Contemp. Math., vol. 543, pp. 19-69, American Mathematical Society, 2011.  Eprint available at arXiv:1012.5548.

Grants and Sponsored Research

   
• National Science Foundation, "Characters, liftings, and types", 9/2009-8/2012, $521,588 (jointly with Professors Hakim and Lansky).
   
• National Security Agency, "Aspects of harmonic analysis on p-adic groups", 2007–09.
   
• National Science Foundation, "Algebra, Number Theory, and Applications: a research experience for undergraduates", 2005–08.
   
• National Security Agency, "Depth-zero base change", 2005–07.
   
• Office of Naval Research, "N03-T004: Optimizing Human Resource Management Models", July, 2003.
   
• National Security Agency, "Representations of p-adic groups", 2002–04.
   
• American Mathematical Society, Mathematical Challenges Travel Grant, August, 2000.

Work In Progress

   
• Liftings of representations of finite groups (with Joshua Lansky)
   
• Self-similar periodic tilings of nilpotent Lie groups (with Judith Palagallo et al.)
   
• Tamely ramified tori in p-adic groups (with Stephen DeBacker)
   
• Characters of p-adic division algebras (with Loren Spice)