Yasin Karacan

Research interests

I work in symplectic and contact topology, with a focus on Legendrian submanifolds of 1-jet spaces and the invariants that distinguish them. Two of the main tools in this area are Legendrian contact homology, which is defined through holomorphic curves, and generating families, which are Morse-theoretic. Although these two theories have very different origins, they are expected to capture the same information about a Legendrian, and much of my work is motivated by making this relationship precise beyond the low-dimensional cases where it is currently understood.

My dissertation studies a combinatorial structure attached to the front of a Legendrian, called a Morse complex n-family (MCnF). Versions of this structure were introduced by Henry and Rutherford for Legendrian knots and extended by Rutherford and Sullivan to Legendrian surfaces; my work defines it in arbitrary dimension for Legendrians whose fronts have only simple singularities, a class that is necessary for a finite combinatorial description and is well understood through the work of Álvarez-Gavela. I then use these structures to relate generating families to the algebraic invariants of a Legendrian, with the broader aim of building a concrete, computable bridge between the two theories in all dimensions.

Preprints and work in progress

• MCnF implies Augmentability (in preparation)

Seminar talks

• Morse Complex n-Families — Five College Geometry and Topology Seminar, UMass Amherst, October 2026 (upcoming)
• Morse Theory — Reading Seminar in Geometry and Topology, UMass Amherst, September 2025
• Spectral Invariants in Contact Geometry — Graduate Student Seminar, UMass Amherst, November 2022
• Introduction to h-principles — Floer Homotopy Theory Seminar, UMass Amherst, April 2022
• Persistent Homology — Graduate Student Seminar, Wesleyan University, March 2022
• Morse Theory — Floer Homotopy Theory Seminar, UMass Amherst, February 2022

Selected conferences attended

• Contact geometry, general relativity, and thermodynamics — Simons Center for Geometry and Physics, 2026
• Symplectic topology from the K-theoretic viewpoint — Woods Hole, 2025
• From smooth to C⁰ symplectic geometry — CIRM, Marseille, 2023
• 28th Gökova Geometry–Topology Conference — Gökova, 2023
• AMS Sectional Meeting, Special Session on Lagrangian and Legendrian Submanifolds — Amherst, 2022
• Frontiers of Quantitative Symplectic and Contact Geometry — Institut Mittag-Leffler, 2022
• Symplectic Topology, Contact Topology and Interactions — Institut Henri Poincaré, 2021

I have taught at UMass Amherst since 2018. In April 2026 I received the Richard Ellis Memorial Scholarship, awarded on a holistic evaluation of teaching skills, care for the departmental community, and contributions to student well-being.

Instructor of record

Teaching assistant

Linear Algebra, Calculus of One Variable, Multivariable Calculus, Precalculus and Trigonometry, and Fundamental Concepts of Mathematics (2018–2026). Earlier, teaching assistant at Middle East Technical University (2016–2018).

Mentoring

• Directed Reading Program, UMass Amherst — co-founder and organizer (2023). The program pairs undergraduates with graduate mentors for semester-long independent reading projects.
• REU mentor, UMass Amherst (Summer 2022) — mentee: Bingheng Yang. Over three months of weekly meetings and written problem sets we worked up from differential forms to contact geometry, and then to the conormal Legendrian knot of a plane curve and what it detects about immersed curves. Bingheng presented the project at the REU’s final presentations in August.
• Topology Qualifying Exam Review Sessions — led the topology review each summer, 2021–2023.

Course evaluations are available on request.

I am the first person in my family to graduate from high school, and much of my work outside the classroom is an attempt to make mathematics reachable for students who arrive without anyone to tell them how any of it works.

Outreach and service

• Undergraduate Math Club, UMass Amherst — organizer, 2025–2026.
• Talks for undergraduates — The Monty Hall Problem: Probability vs. Intuition, Undergraduate Math Club, October 2025; Poincaré–Hopf Index Theorem and its Applications, UMass Math Club, February 2020.
• ILKYAR Summer Science School, Turkey — volunteer, summers 2017 and 2018. ILKYAR brings middle-school students from rural village and regional boarding schools (YİBOs) onto a university campus for a week, where they rotate through departments doing hands-on work with faculty and graduate students. I ran the mathematics session.
• UMass Math Festival — Euler cycles and the Königsberg bridge problem, 2022. Activity worksheet, run with Daniel Gallagher.
• “The What Is…” Graduate Student Seminar (TWIGS) — co-organizer with Hyemin Gu, 2022. A seminar series designed to help early graduate students explore research areas; the Fall 2022 talks ranged from symplectic topology to mathematical oncology.
• Mentoring Committee, UMass Amherst — co-organizer and panelist for a graduate student panel on qualifying and oral exams, choosing an advisor, and managing teaching alongside research; also held drop-in office hours for junior graduate students (2022–2023).