Homepage of Ezra Jordan Aylaian

Math PhD Student at Duke University
Email: ezra.aylaian [at] duke [dot] edu

I am a second year math PhD student at Duke University. Prior, I was an undergraduate at the University of Maryland (UMD), where my advisor was Boyu Zhang. I am interested in a variety of topics, including geometric topology, homotopy theory, and applications.

This semester, I am organizing the Topology and Hydrodynamics Seminar.

Research Papers

Tile Numbers of Knot Corner Mosaics, Geombinatorics, vol. 34, no. 4, April 2025, pp. 130-43. arXiv:2311.12258 [math.GT]

Research Talks

Partitions and Foliations of Punctured Space (Topological and Geometric Structures in Low Dimensions, SLMath), July 31, 2025
Partitions and Foliations of Punctured Space (LG&TBQ2, Centre de Recherches Mathématiques), June 3, 2025
Partitions and Foliations of Punctured Space (Triangle Area Graduate Mathematics Conference, Duke), February 22, 2025
Tile Numbers of Knot Corner Mosaics (Triangle Area Graduate Mathematics Conference, NC State), November 2, 2024
Tile Numbers of Knot Corner Mosaics (Poster) (Nebraska Conference for Undergraduate Women in Mathematics, UN Lincoln), January 27, 2024
Hierarchical Temporal Memory for Air and Missile Defense Applications (Military Operations Research Society Emerging Techniques Forum, JHUAPL), December 5, 2023
Knot Corner Mosaics and Tile Numbers (Towson Undergraduate Mathematics Research Conference, Towson), April 22, 2023
Three Theorems About Knot Corner Mosaics and Their Tile Numbers (Student Geometry and Topology Seminar, UMD), February 17, 2023
Investigations of Hierarchical Temporal Memory (Nebraska Conference for Undergraduate Women in Mathematics, UN Lincoln), January 22, 2023

Expository Talks

The Volume-Preserving Diffeomorphism Group (Topology and Hydrodynamics Seminar, Duke), February 6, 2026
Structural Stability and Anosov Flows (Topology and Hydrodynamics Seminar, Duke), January 23, 2026
Floer Homology of Double Branched Covers of Alternating Links (Knot Floer Homology Seminar, NC State), December 11, 2025
Squeezing and Non-Squeezing in Real and p-Adic Symplectic Geometry (Grapevine Conference, Duke), December 9, 2025
Manolescu and Sarkar's Knot Floer Spectra (Math 690-10 Stable Homotopy Theory, Duke), December 5, 2025
C. T. C. Wall's Amazing Theorem on Non-Smoothable 8-Manifolds (Triangle Area Graduate Mathematics Conference, NC State), November 15, 2025
Foliations on Surfaces (Fluids Reading Group, Duke), November 11, 2025
Multiply-Pointed Heegaard Diagrams for Links (Knot Floer Homology Seminar, NC State), September 25, 2025
Infinity-Categories (Special Innovative Seminar, Duke), September 13, 2025
Lozenge Tilings (Statistical Mechanics Reading Group, Duke), July 7, 2025
The Isomorphism Between de Rham Cohomology and Cech Cohomology (Math 730 Algebraic Topology, UMD), November 28, 2023
Introduction to Knot Theory (Girls Talk Math, UMD), June 22, 2023
Existence and Uniqueness of Stiefel-Whitney Classes (Student Geometry and Topology Seminar, UMD), May 12, 2023
Sphere Eversion and the Whitney-Graustein Theorem (Math 848L Geometric Structures, UMD), May 9, 2023

Teaching

Fall 2025: Lab TA for Duke's Math 111L, Laboratory Calculus I.
Spring 2025: Help Room TA for Duke's Math 112L, Laboratory Calculus II.
Fall 2024: Mentored two undergraduate students through Duke Math's Directed Reading Program on metric spaces. At the end, the students gave an expository presentation on Laidacker and Poole's metric space proof of the existence of a minimal convex worm blanket.
Fall 2022: Taught UMD's Math 299R with Sam Lidz, which explored the way the philosophy of math interacts with math. Math 299R studied platonism, formalism, intuitionism, constructivism, finitism, and structuralism, the arguments for and against each, and the mathematical systems arising from them. Math 299R was featured in a news article.

Service

Organizer of the Duke Topology and Hydrodynamics Seminar, Spring 2026
Treasurer of the Duke American Mathematical Society, Fall 2025 - Present
Board Member on the Duke University Libraries' Graduate and Professional Student Advisory Board, Fall 2024 - Present
Organizer for the Duke Math Grad-Fac Seminar, Fall 2024 - Spring 2025
Undergraduate Representative on the UMD Math Department Climate Committee, Fall 2023 - Spring 2024

Honors

Sloan Affiliate
UMD Math High Honors, May 2024
Milton Abramowitz Award, May 2024
Milton Abramowitz Award, May 2023