Homepage of Ezra J. Aylaian
I am a second year math PhD student at Duke University. Prior, I was an undergraduate at the University of Maryland, where my advisor was Boyu Zhang. I am interested in a variety of topics, including differential geometry, knot theory, and applications of mathematics to biology and epistemology.
My email address is ezra.aylaian + @ + duke + . + edu.
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 (Slides)
- 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
- Floer Homology of Double Branched Covers of Alternating Links (Knot Floer Homology Seminar, NC State), December 11, 2025
- Squeezing and Non-Squeezing in 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 Graduatate 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
- The Tale of a Tricky Triangle: An Introduction to Moduli Spaces (The Math Club, UMD), April 17, 2023
- Integration on Manifolds and the de Rham Cohomology (Directed Reading Program Talks, UMD), December 8, 2022
- The History of Geometry: The Changing Nature of Math's Second Oldest Field (The Math Club, UMD), October 17, 2022
- The Poincaré and Super-Poincaré Algebras (Research Interaction Team on Geometry and Physics, UMD), October 13, 2022
- A Chic Card Trick (The Math Club, UMD), October 3, 2022
- The Model Geometries (Geometry and Topology Undergraduate Summer Workshop, Notre Dame), August 4, 2022
Teaching
- In Fall 2025, I was a Lab TA for Duke's Math 111L, Laboratory Calculus I.
- In Spring 2025, I was a Help Room TA for Duke's Math 112L, Laboratory Calculus II.
- In Fall 2024, I mentored two undergraduate students through Duke Math's Directed Reading Program as they learned about metric spaces. At the end of the semester, they gave a presentation expositing a metric space proof due to Laidacker and Poole of the existence of a minimal convex worm blanket.
- In Fall 2022, I taught UMD's Math 299R with Sam Lidz, which explored the way the philosophy of math interacts with math. We studied platonism, formalism, intuitionism, constructivism, finitism, and structuralism, the arguments for and against each, and the mathematical systems arising from them. A news article was written about Math 299R and another student-run math course.
Service
- Duke American Mathematical Society Treasurer, Fall 2025 - Present
- Duke Math Grad-Fac Seminar Organizer, Spring 2025 - Present
- Duke University Libraries' Graduate and Professional Student Advisory Board, Fall 2024 - Present
- Triangle Competition in Mathematical Modeling Judge, Fall 2024
- UMD Math Department Climate Committee Undergraduate Representative, Fall 2023 - Spring 2024
- Treasurer of The Math Club at UMD, Fall 2022 - Spring 2023
Honors
Links