Math Research

I enjoy working on problems that involve different branches of mathematics interacting in interesting ways. Problems of this sort often have connections to physics. Sometimes the problem itself is physically motivated, but othertimes the problem is purely mathematical, involving objects from physics.

Thus far, my research program has two components. The first component involves working towards understanding families of Calabi-Yau manifolds, while the second is about the mathematics of adinkras, which are graphs that are used to describe certain theories of supersymmetry.

In all of the research I do, emphasis is placed on exhibiting observed phenomena explicitly, often by using computer software packages such as Sage, Maple, and Mathematica.

Motion of the singular fibres in the internal K3-surface fibration structure in the quintic mirror family of Calabi-Yau threefolds.

Geometry

Publications:

C. Doran, J. Kostiuk, "Geometric Variations of Local Systems and Elliptic Surfaces", arXiv:1911.00402, 2019 (submitted for peer-review)
C. Doran, J. Kostiuk, F. You, "The Doran-Harder-Thompson Conjecture for Toric Complete Intersections", arXiv:1910.11955, 2019 (submitted for peer-review)
J. Kostiuk, "Geometric Variations of Local Systems" Ph.D. Thesis, University of Alberta ProQuest Dissertations Publishing, 2018
J. Kostiuk, "Heegner Points, Hilbert’s Twelfth Problem, and the Birch and Swinnerton- Dyer Conjecture" M.Sc Thesis, University of Alberta ProQuest Dissertations Publishing, 2013

Adinkras

Publications:

E. Goins, K. Iga, J. Kostiuk, K. Stiffler, "The Signed Monodromy Group of an Adinkra", arXiv:1909.02609, 2019 (accepted by AIHPD)
I. Friend, J. Kostiuk, Y. Zhang, "Enumerative Gadget Phenomena for (4,1) Adinkras", arXiv:1810.05545 , 2018 (accepted by AIHPD)
C. Doran, K. Iga, J. Kostiuk, S. Méndez-Diez, "Geometrization of N-Extended 1-Dimensional Supersymmetry Algebras II" Advances in Theoretical and Mathematical Physics, 22(3): 565-613 2018
C. Doran, K. Iga, J. Kostiuk, G. Landweber, S. Méndez-Diez, "Geometrization of N-Extended 1-Dimensional Supersymmetry Algebras I" Advances in Theoretical and Mathematical Physics, 19(5):1909--1970, 2015

An adinkra is a graphical tool that encodes information about certain supersymmetry algebras.

Presentations

"Adinkras: Snapshots of Supersymmetry", Pomona College, ANTC Seminar, September 2019
"Adinkras: Snapshots of Supersymmetry", Brown University, Undergraduate Colloquium, September 2019
"Intro to Adinkras", Brown University Algebra & Combinatorics Day, May 2019
"Geometric Variations of Local Systems", Brown University Algebraic Geometry Seminar, February 2019
"...and there were graphs...", Brown University special Adinkras session, November 2017
"Geometrization of Adinkras", Mathematical Congress of the Americas, Montréal, July 2017
"Lecture on Geometrization of Adinkras" Brown University Math/Phys Hangout, December 2016;

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