My research is mainly about an area of mathematical physics called supersymmetry, using diagrams called Adinkras. But my earlier work was in the differential topology of four dimensional manifolds, and has published in general relativity and cryptography. In general, I am open to working on whatever interesting problems come my way.
Adinkras
Supersymmetry is an idea from the physics of subatomic particles, originating in the 1970s. It posits a relationship between the kinds of particles that make up matter (fermions) and the kinds of particles that describe forces between these particles (bosons). We do not know if supersymmetry is really a feature of our universe, but if it is, it would help explain a few things that are currently puzzling. It is also a requirement for string theory, so if you are a fan of string theory, you will have to accept supersymmetry. In supersymmetric theories, there are often many different particles related by a number of equations. In 2004, M. Faux and S. J. Gates introduced diagrams called Adinkras that summarize these equations in a way that is geometrically more intuitive. Kevin Iga has been working on these Adinkras ever since. It turns out there are surprising relationships to error correcting codes, cubical cohomology, Clifford algebras, Riemann surfaces, as well as generalizations of sandpile problems from graph theory.
I am currently cowriting a textbook on Adinkras, which will probably be the first textbook on the subject.
Topology of Four dimensional Manifolds
In geometry, we study lines and planes, or their more curved versions curves and surfaces. A manifold is a generalization of a curve or surface. A curve is a one dimensional manifold, and a surface is a two dimensional manifold. There are manifolds of higher dimensions, though we cannot typically visualize these in our ordinary three dimensional world. Manifolds can come about when describing physical systems that have a lot of moving parts that are constrained in some way, say, like a robotic arm with multiple joints. Topology seeks to classify manifolds according to the overall way they are connected, while ignoring the details of size and angle. So from a topological point of view, a large sphere and a small sphere are the same shape, and if we distort it by flattening it or stretching it, topology views it as still the same shape. But the surface of a donut is not the same shape, topologically speaking.
The topological classification of manifolds of dimensions 0, 1, and 2 is fairly well understood, and paradoxically, manifolds in dimensions 5 and higher are also understood as well, at least up to some difficult algebraic questions. The reason is that there are tricks that help us simplify a manifold, but these tricks require 5 dimensions to have enough room. Dimensions 3 and 4, however, remain difficult. In the 1980s, S. Donaldson introduced a new tool to study 4 dimensional manifolds: the instanton equations. These are partial differential equations that make sense on a 4 dimensional manifold, and while it is often difficult to write down the specific solutions, the mathematics of the set of solutions turns out to lead to important theorems about 4 dimensional manifolds. It was strange why these specific equations would be so helpful, but a hint, due to E. Witten, was that these could be understood in terms of supersymmetry.
In 1994, the work of N. Seiberg and E. Witten found a relationship among various supersymmetric theories, called S-duality, and when applied to the instanton equations, gave rise to new equations we now call the Seibeg-Witten equations. These can be used like the instanton equations, but the details are much simpler to use.
In my Ph.D. thesis, I investigated the Seiberg-Witten equations on four dimensional manifolds that look like a long cylinder on an arbitrary three dimensional manifold. As is the case with instantons, this leads to a kind of Morse theory called Floer homology.
Christianity and Mathematics
While this is not a subject I have published in, the philosophy of mathematics and how it relates to my Christian faith is a topic I find interesting. Like many practicing mathematicians, I believe that the abstract things studied by mathematics really do exist and their existence is independent of human minds, or even the physical universe. This view is associated with Platonism, at least as it relates to the philosophy of mathematics, because it categorizes mathematical objects in a way similar to how Plato understood forms. Some reasons for this include the sense, when I am doing mathematics, that one is discovering things that would have been true even without my influence, and indeed, I can be surprised by a result coming from my own work. In addition, there is a kind of agreement that takes place between researchers that do not know of each others’ work, that suggests that mathematical truths are “universal”. The universe itself follows laws of physics that can be written in mathematical language, and often, mathematics that was studied for its own sake later finds surprising applications to understanding the universe. That suggests that even the existence of the universe is predicated on the existence of the mathematics underlying its laws.
All of this is in tension with a view of the universe that only recognizes the existence of physical things. Mathematical objects, if they exist independently of the physical universe, are not physical things. As a Christian, I likewise accept the existence of God, who I believe is not a physical thing, and whose existence is prior to the existence of the physical universe.
This raises the question as to whether mathematics or God is more fundamental. I believe God is more fundamental than mathematics. Just as many find inspiration from the beauty found in nature: a sublime sunset, a field of flowers, a terrifying storm, or the vastness of galaxy clusters, and believe that only a truly awesome God can be behind all of this, likewise I get the same sense from the beauty of the analysis of complex functions, the intricacy and variety in knot theory, the symmetries in the monster group, and the towers of cardinalities of infinite sets found in mathematics. This suggests that mathematics is a creation of God. It is hard to imagine that mathematics could be any different if God chose to create it differently. Perhaps it could not be. That does not contradict the dependence of mathematics on God. Our sense of causality is tied up with notions of time, but both God and mathematics are beyond time, so we should already expect difficulties here.
Published papers
E. Goins, K. Iga, J. Kostiuk, K. Stiffler, The signed monodromy group of an Adinkra, Ann. Inst. Henri Poincaré, Comb. Phys. Interact. 10 (2023), no. 1, pp. 1–30, arXiv:1909.02609, DOI 10.4171/AIHPD/132
K. Iga, C. Klivans, J. Kostiuk, Chi Ho Yuen, Eigenvalues and Critical Groups of Adinkras, Advances in Applied Mathematics 143, (2023) 102450. arXiv:2202.02821
K. Iga, Adinkras: Graphs of Clifford Algebra Representations, Supersymmetry, and Codes, Adv. Appl. Clifford Algebras 31, 76 (2021). arXiv:2110.01665 https://doi.org/10.1007/s00006-021-01181-0
S.J. Gates, K.M. Iga, L. Kang, V. Korotkikh, K. Stiffler, Generating All 36,864 Four-Color Adinkras via Signed Permutations and Organizing into $\ell$- and $\tilde{\ell}$-Equivalence Classes, Symmetry 2019, 11(1), 120; arXiv:1712.07826 https://doi.org/10.3390/sym11010120.
C.F. Doran, K. Iga, J. Kostiuk, S. Mendez--Diez, Geometrization of N-extended 1-dimensional supersymmetry algebras, II, Advances in Theoretical and Mathematical Physics, Vol. 22, No. 3 (2018), pp. 565-613. arXiv:1610.09983
C.F. Doran, K.M. Iga, G.D. Landweber, An Application of Cubical Cohomology to Adinkras, AIHPD, European Mathematical Society, Vol. 4, No. 3, (2017), pp. 387--415, arXiv:1207.6806.
C.F. Doran, M.G. Faux, S.J. Gates, Jr., T. Hübsch, K.M. Iga, G.D. Landweber, Off-shell supersymmetry and filtered Clifford supermodules, Algebras and Representation Theory, April 2018, Volume 21, Issue 2, pp 375–397, arXiv:math-ph/0603012 DOI 10.1007/s10468-017-9718-8.
K. Iga and Y.X. Zhang, Structural Theory and Classification of 2D Adinkras, Advances in High Energy Physics, vol. 2016, Article ID 3980613, 12 pages, 2016. arXiv:1508.00491, doi:10.1155/2016/3980613.
C.F. Doran, K. Iga, G. Landweber, S. Mendez--Diez, Geometrization of N-Extended 1-Dimensional Supersymmetry Algebras, Advances in Theoretical and Mathematical Physics, Vol. 19, No. 5 (2015), pp. 1043-1113. arXiv:1311.3736
N. Fazio, K. Iga, A. Nicolosi, L. Perret, W. Skeith, Hardness of Learning Problems over Burnside Groups of Exponent 3, Designs, Codes and Cryptography: Volume 75, Issue 1 (2015), Page 59-70, DOI 10.1007/s10623-013-9892-6.
C.F. Doran, T. Hübsch, K. Iga, G. Landweber, On General Off-Shell Representations of World Line (1D) Supersymmetry, Symmetry 2014, 6(1), 67--88 (2014). arXiv:1310.3258
C.F. Doran, M.G. Faux, S.J. Gates, Jr., T. Hübsch, K.M. Iga, G.D. Landweber, R. Miller, Codes and Supersymmetry in One Dimension, Adv. Theor. Math. Phys. 15 (2011), 1909-1970. arXiv:1108.4124
M. Faux, K. Iga, G. Landweber, Dimensional Enhancement via Supersymmetry, Advances in Mathematical Physics, (2011) Article ID 259089, 45 pages, doi:10.1155/2011/259089. arXiv:0907.3605
C.F. Doran, M.G. Faux, S.J. Gates, Jr., T. Hübsch, K.M. Iga, G.D. Landweber, A Superfield for Every Dash-Chromotopology, Int.J.Mod.Phys., A24 (2009), 5681--5695.
C.F. Doran, M.G. Faux, S.J. Gates, Jr., T. Hübsch, K.M. Iga, G.D. Landweber, R. Miller, Topology Types of Adinkras and the Corresponding Representations of N-Extended Supersymmetry, J. Phys. A42 (2009) 065402 (12pp). arXiv:0806.0050
C.F. Doran, M.G. Faux, S.J. Gates Jr, T. Hübsch, K.M. Iga, G.D. Landweber, Relating Doubly-Even Error-Correcting Codes, Graphs, and Irreducible Representations of N-Extended Supersymmetry, in Discrete and Computational Mathematics, eds. F. Liu et al., (Nova Science Pub., Inc., Hauppage, 2008); arXiv:0806.0051.
C.F. Doran, M.G. Faux, S.J. Gates, Jr., T. Hübsch, K.M. Iga, G.D. Landweber, Adinkras and the Dynamics of Superspace Prepotentials, Adv. S. Th. Phys. 2 (3) (2008), 113--164. arXiv:hep-th/0605269
C.F. Doran, M.G. Faux, S.J. Gates, Jr., T. Hübsch, K.M. Iga, G.D. Landweber, A Counter-Example to a Putative Classification of 1-Dimensional, N-extended Supermultiplets, Adv. S. Th. Phys. 2 (3) (2008), 99--111. arXiv:hep-th/0611060
C.F. Doran, M.G. Faux, S.J. Gates, Jr., T. Hübsch, K.M. Iga, G.D. Landweber, On Graph-Theoretic Identifications of Adinkras, Supersymmetry Representations and Superfields, Int. J. Mod. Phys., A22 (2007), 869--930. arXiv:math-ph/0512016
K. Iga, R. Maddox, Pebble Sets in Convex Polygons, Journal of Discrete and Computational Geometry, Vol. 38 (2007), 4, 680--700.
K. Iga, A Truck Driver's Straw Problem and Cantor Sets, College Mathematics Journal, 39 (2008), 4, 280--290.
K. Iga, K. Killpatrick, Truck Drivers, Straws, and Sharing a glass of water, College Mathematics Journal, March 2006, vol. 37, no. 2, 82--92.
K. Iga, A Dynamical Systems proof of Fermat's Little theorem, Mathematics Magazine, Vol. 76, No. 1 (2003), 48--51.
H. Bray, K. Iga, Superharmonic functions in R^n and the Penrose Inequality in General Relativity, Communications in Analysis and Geometry, Vol. 10, No. 5, (2002), 999--1016.
K. Iga, What do Topologists want from Seiberg--Witten theory?, International Journal of Modern Physics A, 17, No. 30 (2002), 4463--4514.