Mathematician

Qiuyu Ren

Clay Research Fellow · Low-dimensional topology

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I am a Clay Research Fellow. During the 2026–2027 academic year, I am at Stanford. I obtained my PhD in 2026 from UC Berkeley, where my advisor was Ian Agol.

My research interests lie in low-dimensional topology. I work on Khovanov homology, smooth 4-manifolds, 3-manifolds, surfaces, and knots.

Curriculum vitae (PDF, 2026/08/15)

Ongoing and Upcoming Events

• Will be a participant in the Quantum Topology, Character Varieties and Low-Dimensional Geometry semester program at IPAM, Los Angeles, Fall 2026. Will give a lecture in the tutorials.
• Will give a talk in the 13th Southern California Topology Colloquium (SCTC), Caltech, October 17, 2026.

Publications and Preprints

1. Multisections and bridge positions in arbitrary dimensions, with Sylvain Courte, Delphine Moussard, and Xiaozhou Zhou
2. Insensitivity of Khovanov homology under rim surgery, with Fang-Rong Zhan
3. Families of cosmetic surgeries
4. Ribbon concordance of fibered knots and compressions of surface homeomorphisms, with Ian Agol
5. Khovanov skein lasagna modules with $1$-dimensional inputs, with Ian Sullivan, Paul Wedrich, Michael Willis, and Melissa Zhang
6. Intrinsic Khovanov homology in $\mathbb{RP}^3$, with Hongjian Yang
7. Adjunction inequality for spatially refined $s$-invariants, Pac. J. Math. 344 (2026) 145-150
8. Cosmetic surgery on satellite knots, Bull. Lond. Math. Soc. 57 (2025) 3934-3940
9. Khovanov homology and exotic $4$-manifolds, with Michael Willis, to appear in Ann. of Math.
10. Toroidal Hitomezashi Patterns, with Shengtong Zhang, Discrete Math. 348 (2025) 114231
11. Slice genus bound in $DTS^2$ from $s$-invariant, Algebr. Geom. Topol. 24 (2024) 4115-4125
12. Lee filtration structure of torus links, Geom. Topol. 28 (2024) 3935-3960
13. A succinct proof of Defant and Kravitz's Theorem on the length of Hitomezashi loops, with Shengtong Zhang, Ann. Comb. 29 (2025) 117-122
14. Spectral asymptotics for kinetic Brownian motion on Riemannian manifolds, with Zhongkai Tao, Math. Ann. 393 (2025) 1175–1194
15. Spectral asymptotics for kinetic Brownian motion on locally symmetric spaces, with Zhongkai Tao

My PhD thesis, Khovanov homology and smooth 4-manifolds, contains small mathematical and expositional improvements over [4], [7], and [11] above.

Expository Writing

The following are earlier expository papers and course notes; some may not reflect my current perspective.

• Toric varieties in algebraic and symplectic geometry (final paper for MATH 256A, UCB Fall 2021)
• Spectral Properties of the Laplacian on Compact Lie Groups (one of the projects for 18.821, MIT Fall 2020; see the end to see my part of writing)
• Quantization commutes with reduction (UChicago REU 2020)
• Equivariant de Rham theory (notes for an independent reading course, MIT Spring 2020)
• Nonnegatively curved sl_2(R)-connections on surfaces, with Zhuofan Xie (undergraduate research supervised by Professor Paul Seidel, MIT Spring 2020)
• Complex Analytic Proofs of Bernstein's Theorem (final paper for 18.994, MIT Spring 2020; I wrote Sections 4 and 6)
• Conjugacy of Cartan Subalgebras (final paper for 18.745, MIT Fall 2019)
• On the union of essentially distinct δ-tubes (MIT SPUR Summer 2019, mentored by Yuqiu Fu, problem proposed by Professor Larry Guth; my paper won the Hartley Rogers Jr. Prize)
• Linear fractional transformations on the Berkovich projective line (final paper for an independent study course in p-adic analysis, MIT Spring 2019)

Past Events

• Gave a talk in the Trisectors Workshop 2026, Western Washington University, July 2026.
• Participated and gave a talk in the PCMI 2026 Research Program, Park City, June-July 2026.
• Visited the University of Hamburg, Germany, May-June 2026.
• Gave a Chern-Vorlesung (Chern lecture) in the Kolloquium über Reine Mathematik (colloquium in pure math) at the University of Hamburg, Germany, June 23, 2026.
• Participated in the Matemale Spring School 2026: Knotted 3-balls in the 4-sphere, Matemale, France, June 2026.
• Gave a talk in the Research Seminar on Quantum Topology and Categorification (QTCat), University of Hamburg, Germany, May 2026.
• Participated in the Topological and Geometric Structures in Low Dimensions program at SLMath, Berkeley, Spring 2026.
• Gave a talk in the online International Young Seminar on Bounded Cohomology and Simplicial Volume, May 2026.
• Participated in the Simons Collaboration on New Structures in Low-Dimensional Topology Annual Meeting, New York, March 2026.
• Gave talks in the Online Geometry and Physics Seminar at the IASM, Zhejiang University, March 2026.
• Visited the University of Seville, Spain, January 2026.
• Co-mentored the 2nd Scottish Talbot on Algebra and Topology workshop on skein lasagna modules, Crianlarich, Scotland, January 2026.
• Gave a talk in the Southern University of Science and Technology (SUSTech) geometry and topology seminar, December 2025.
• Gave a talk in the 2025 Virginia Topology Conference - New developments in 4-manifold topology, December 2025.
• Gave a talk in the Ohio State University topology seminar, December 2025.
• Gave a talk in the online quantum topology seminar organized by L. Kauffman, December 2025.
• Gave a talk in the Knots and representation theory seminar (online), December 2025.
• Coorganized the student low-dimensional topology seminar at UC Berkeley with Audrey Rosevear, Fall 2024 - Fall 2025.
• Participated in the workshop Low-dimensional topology and Floer theory at CRM, Montréal, Canada, August 2025.
• Gave a talk in the conference Knots, groups, and manifolds at CRM, Montréal, Canada, August 2025.
• Visited Tsinghua University, Beijing, China, August 2025.
• Gave a talk in the workshop New perspectives on skein modules at CIRM, Luminy, France, July 2025.
• Participated in the workshop and conference Categorification in Low Dimensional Topology, Ruhr University Bochum, Germany, July 2025.
• Gave a talk in the Max Planck Institute for Mathematics (MPIM) low-dimensional topology seminar, Bonn, Germany, July 2025.
• Participated in the workshop Multisectors at CIRM, Luminy, France, July 2025.
• Gave a talk in the Research Seminar on Quantum Topology and Categorification (QTCat) at the University of Hamburg, Germany, June 2025.
• Gave a talk in the geometric topology seminar at Humboldt University of Berlin, Germany, June 2025.
• Led exercise sessions and gave a talk in the Summer School and Conference on Modern Developments in Low-Dimensional Topology at ICTP, Triste, Italy, June 2025.
• Participated in the 2025 Georgia international topology conference, University of Georgia, May 2025.
• Gave a talk in the AMS 2025 Spring Western Sectional Meeting, California Polytechnic, May 2025.
• Participated in the Simons Collaboration on New Structures in Low-Dimensional Topology Annual Meeting, New York, March 2025.
• Gave a talk in the University of Oregon topology seminar, February 2025.
• Gave a talk (online) in the workshop Knots, Quivers & Beyond, February 2025.
• Gave a talk in the online Geometric Topology Grad and Postdoc Seminar (GT GAPS), February 2025.
• Gave a talk in the First International Online Knot Theory Congress, February 2025.
• Gave a talk at Sungkyunkwan University, South Korea, December 2024.
• Gave a talk in the UCSB topology seminar, December 2024.
• Gave a talk in the University of Georgia research topology seminar, October 2024.
• Gave a talk in the Knots and representation theory seminar (online), September 2024.
• Gave a talk in the Columbia University geometric topology seminar, September 2024.
• Gave a talk in the AMS 2024 Fall Central Sectional Meeting, University of Texas, San Antonio, September 2024.
• Gave lectures on lasagna s-invariants at a conference in quantum topology at Belalp, Switzerland, September 2024.
• Visited the University of Zürich, Switzerland, August 2024.
• Gave a talk in the Conference - New structures in low-dimensional topology at the Erdös Center, Rényi Institute, Budapest, Hungary, July 2024.
• Participated in the Summer school in Mathematics: Low dimensional topology at the Institut Fourier, Grenoble, France, June 2024.
• Gave a talk in the Knot Online Seminar (K-OS), June 2024.
• Gave a talk in the Research Seminar on Quantum Topology and Categorification (QTCat) at the University of Hamburg, Germany, June 2024.
• Participated in the conference Combinatorial and Gauge theoretical methods in low dimensional topology and geometry at the Centro di Ricerca Matematica Ennio De Giorgi, Pisa, Italy, June 2024.
• Participated in the 2024 Georgia topology summer school and conference, May 2024.
• Coorganized the student 3-manifold seminar at UC Berkeley with Ethan Dlugie, Fall 2023 to Spring 2024.
• Gave a talk in the University of South Florida geometry and topology seminar, April 2024.
• Gave a talk in the Simons Collaboration satellite conference: Floer homology, low-dimensional topology, and algebra, Princeton University, March 2024.
• Gave a talk in the UC Davis geometry and topology seminar, March 2024.
• Participated in the 2024 Geometry and Topology Workshop at UCLA, January 2024.
• Participated in the Workshop on exotic 4-manifolds at Stanford University, December 2023.
• Gave a talk in the Texas A&M University topology seminar, October 2023.
• Gave a talk in the Caltech geometry and topology seminar, September 2023.
• Gave a talk in the Stanford University topology seminar, September 2023.
• Gave a talk in the MIT geometry and topology seminar, September 2023.
• Participated in the conference Gauge Theory and Topology: in Celebration of Peter Kronheimer's 60th Birthday at the Clay Mathematics institute in Oxford, July 2022.
• Visited Instituto de Ciencias Matemáticas (ICMAT) in Madrid for the conference Geometric group theory and low-dimensional geometry and topology, June-July 2022.
• Participated in the Analytic and Geometric Aspects of Gauge Theory and Floer Homotopy Theory programs at MSRI/SLMath, Berkeley, Fall 2022.
• Coorganized a reading group on the Kobayashi-Hitchin correspondence with Yueqing Feng, Fall 2022.
• Organized a reading group on Donaldson & Kronheimer's The Geometry of Four-Manifolds, Summer 2021. I am also maintaining an errata list for the book.

Beyond Mathematics

• I am a marathon runner. I won first place in the Baystate Half Marathon (under 20 age group), 2018. I won third place in the Baystate Marathon (20-29 age group), 2019. I won third place in the Berkeley Half Marathon (20-29 age group), 2022. Here is a photo of me.
  My PRs are 1:19:02 (02/27/2022 Berkeley Half Marathon); 2:38:07 (12/04/2022 California International Marathon (CIM)).
• I'm a contributor to the FrontierMath benchmark, Humanity's Last Exam, and the GAUSS benchmark.
• If you're looking to pass the time and can read Chinese, here are some fun logic puzzles.