mim2 (at) illinois.edu
mim2 (at) math.uiuc.edu Mailing Address: 250 Altgeld Hall, 1409 West Green Street,
Urbana, IL 61801 Campus Mail: Department of Mathematics, MC-382 Seminars:AG & Related Seminars
Doctor of Philosophy (Ph.D.) in Mathematics at the University of Illinois at Urbana-Champaign
Master of Arts (M.A.) in Mathematics at the University of Georgia
Master of Philosophy (M.Phil.) in Mathematics at the University of Birmingham, England
My research interests primarily lie in representation theory. More specifically, I have studied
generalized Grothendieck-Springer resolutions (i.e., universal quiver flags) through quiver representations, the nilpotent cone,
almost-commuting varieties, and generalizations and applications of invariant theory.
Quiver flag varieties and filtered quiver varieties (these are isomorphic to fibers of natural projection maps from generalized Grothendieck-Springer resolution to each factor)
are closely related to
KLR-algebras, quiver Hecke algebras, q-Schur algebras and categorification of quantum groups; my research interests are studying representations of these algebras using combinatorial and geometric techniques.
Integrable systems. I am also interested in t-deformations of certain solutions to various integrable systems.
Currently, my collaborators (D. Addabbo and B. Turmunkh) and I are systematically studying and relating t-deformation of q-characters of quantum affine algebras (which are related to deformed T-systems constructed by Nakajima using quiver varieties)
to quantum cluster algebras (which are a different realization of deformed Q-systems). We are interested in combinatorial, algebraic, and geometric constructions of these solutions.
Applications. Third aspect of my research interests lies in applications of representation theory, e.g., fast Fourier transforms (FFT) for homogeneous spaces of groups (algebras, monoids, etc), connections between FFT and (binary) codes, high dimensional data sets of images and shapes.
Doctoral dissertation: main results include the study of semi-invariants of certain family of filtered quiver representations;
the construction of filtered quiver representations is given in my thesis,
which was written under Tom Nevins' supervision.
On fast Fourier transform for partially ordered subgroup chains of the symmetric group, with A. Wu. In progress.
On classification of iterated wreath products of finite groups, with A. Wu. In preparation.
Generalized iterated wreath products of symmetric groups and generalized rooted trees correspondence, with A. Wu, preprint, 2014.
Generalized iterated wreath products of cyclic groups and rooted trees correspondence, with A. Wu, preprint, 2014.
On Nakajima's deformation of q-characters (tentative title), with D. Addabbo and B. Turmunkh. In preparation.
On semi-invariants of filtered representations of quivers and the cotangent bundle of the enhanced Grothendieck-Springer resolution, IDEALS,
Suggestions to study affine and GIT quotients of the extended Grothendieck-Springer resolution,
Semi-invariants of filtered quiver representations with at most two pathways,
The regularity of the cotangent bundle of the Grothendieck-Springer resolution, paper, 2013.
The regular semisimple locus of the affine quotient of the cotangent bundle of the Grothendieck-Springer resolution,
On Kostant's theorem for Lie algebra cohomology, (with UGA VIGRE Algebra Group), Contemporary Mathematics, 478 (2009),
Nonstandard approach to Hausdorff measure theory and an analysis of some sets of dimension less than 1, British Library,
Examples relating to Green's conjecture in low characteristics and genera, with T. Estu, J. Manning, Z. Michaels, W. Rulla, J. Pasko, N. Wijesinghe,
accepted upon revision, Experimental Mathematics, 2004.
January 2013, Mathematical Sciences Research Institute (MSRI), Berkeley, CA.
University of Illinois at Urbana-Champaign.
Campus Research Board Research Assistantship, Summer 2013, Fall 2013, Spring 2014.
NSA Research Assistantship, Fall 2012, Spring 2013, Summer 2013.
B.G. Trijtzinsky Fellowship, Spring 2012.
E.T. Parker Fellowship, Spring 2011.
University Fellowship, Spring 2011, Spring 2012.
NSF REGS2 Research Fellowship, Summer 2012, Spring 2013.
NSF REGS Research Fellowship, Summer 2009, Summer 2010, Summer 2011.
Conference Travel Award, Summer 2008, Spring 2009, Fall 2010, Spring 2011, Summer 2011, Spring 2013, Spring 2014, Summer 2014, Fall 2014.
AMS/MRC/NSF Travel Grant, Fall 2014, Spring 2015.
List of Teachers Ranked as Excellent, Fall 2007, Fall 2008, Spring 2009, Fall 2009, Spring 2011.
Teacher Scholar Certificate, April 2010.
Graduate Teaching Certificate, April 2010.
University of Georgia.
Summer Research Award, with the Department of Physics.
REU Summer Research Award, with the Department of Mathematics.
VIGRE Summer Research Award, with the Department of Mathematics.
Hope Scholarship, every semester.
Pell Grant, every semester.
July 2014, Modern applications of representation theory, Filtered quiver representations, research presentation. University of Chicago, Chicago, IL.
June 2014, Representation theory in data analysis, group presentation. Mathematics Research Communities, American Mathematical Society, Salt Lake City, UT.
May 2014, On constructing invariants for generalized quiver varieties with applications to shape deformation in target and activity recognition. Arlington and Alexandria, VA. [abstract]
May 2014, Algecom10, Viewing deformation of objects as equivalence classes of orbits on representation spaces and generalized quiver varieties, poster presentation. University of Illinois, Urbana, IL.
April 2014, Topology and Algebra Research Symposium, On filtered representations of quivers with at most two pathways and on the generalized Grothendieck-Springer resolution. University of Califoria, Berkeley, CA.
April 2014, Topology and Geometry Conference, Parabolic equivariant geometry in representation theory. University of Texas, Austin, TX.
December 2013, Applications of filtered quiver varieties in representation theory. Lexington, MA.
November 2013, The geometry of filtered quiver varieties. Baltimore, MD.
April 2013, Graduate Research Conference in Algebra and Representation Theory, Declined, schedule conflict. Kansas State University, Manhattan, KS.
January 2013, Noncommutative Algebraic Geometry and Representation Theory, Invariants and semi-invariants of arbitrary filtered quiver varieties, poster presentation. MSRI, Berkeley, CA.
April 2011, Topology and Algebra Research Symposium, Matrix varieties: an analogue of the isospectral Hilbert scheme. University of Iowa, Iowa City, IA.
October 2010, Midwest Algebraic Geometry Graduate Conference, An analogue of the Hilbert scheme of n points on a complex plane. University of Wisconsin-Madison, WI.
September 2008, Fano manifolds and Lagrangian fibration of symplectic manifolds, An introduction to algebraic surfaces. Lukecin, Poland.
July 2008, Summer Program: Geometrical Singularities and Singular Geometries, Singularities in Calabi-Yau varieties. IMA, University of Minnesota, MN.
University of Illinois at Urbana-Champaign.
May 2014, Graduate student algebraic geometry, Kazhdan-Lusztig polynomials, quiver Hecke algebras, and categorification. [abstract]
April 2014, Dissertation defense, On semi-invariants of filtered representations of quivers and the cotangent bundle of the Grothendieck-Springer resolution.
October 2013, AWM-Illinois, The evolution of invariant theory: from classics to modern.
October 2013, Graduate student algebraic geometry, The geometry of filtered quiver varieties.
December 2012, Algebraic geometry, Invariants and semi-invariants of filtered quiver representations.
November 2012, Pi Mu Epsilon: Illinois Honors Society, Quotients of curves and surfaces and higher dimensional analogues.
November 2012, AWM-Illinois, Moduli problems through representations of quivers.
January 2012, Graduate geometry & topology, The Hamiltonian reduction of a certain affine variety.
December 2011, AWM-Illinois, Geometric invariant theory.
March 2011, Graduate geometry & topology, On a certain family of matrices in the context of algebraic geometry.
Math 231, Calculus II, discussion section, Spring 2011.
Math 125, Linear Algebra, instructor for a large lecture, Fall 2010. Responsible for all aspects of the course.
Math 415, Linear Algebra, grader, Spring 2010.
Math 241, Merit Calculus III, discussion sections, Spring 2009, Fall 2009.
Math 347, Problem Solving and Proofs, grader and led discussions, Summer 2008.
Math 231, Calculus II, tutor for 4 days a week, Spring 2008.
Math 125, Linear Algebra, grader, Spring 2008.
Math 234, Business Calculus I, discussion sections, Fall 2007.
Tutored ECE 310/410 (Digital Signal Processing), Math 115, Math 220, Math 221, Math 231, Math 241,
Math 362/ECE 313 (Probability with Engineering Applications), Math 415, Phys 508 (Math Methods in Physics), and high school (proof based) geometry and AP BC Calculus.
* ECE denotes Electrical and Computer Engineering.
University of Birmingham, United Kingdom. [During the 15 month research program.]
MSM1Da and MSM1Db, Applicable Mathematics A and B, course assistant.
Drop-in tutoring for all courses, two days a week.
MSM1Xa, The Impact of Mathematics A, course assistant.
MSM2Ca, Linear Algebra A, course assistant.
MSM1Ab, Mathematical Physics B, course assistant.
MSM2Bb, Real and Complex Variable Theory B, course assistant.
MSM2O2b, Statistics II B, course assistant.
* A denotes the first (Fall) semester sequence while B denotes the second (Spring) semester sequence.
University of Georgia.
Math 5210/7210, Euclidean and non-Euclidean Geometry, course assistant.
Phys 1112, Introduction to Electricity and Magnetism II, lab assistant.
Math 2700, Partial Differential Equations, grader.