Yevgeny Liokumovich

Department of Mathematics, University of Toronto

Email
ylio [at] math.toronto.edu
Office
BA6189 (St. George) · DH3052 (UTM)
Yevgeny Liokumovich

Research interests

AI Safety, geometric analysis, metric geometry, low-dimensional topology.

AI safety for mathematicians

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I am accepting Master's and PhD students who want to work on mathematical aspects of AI safety, control and alignment.

In the fall of 2026 I am teaching a graduate course on Mathematics for AI Safety at the Fields Institute.

Papers

Most of my papers can be found on the arXiv.

AI safety and interpretability

  1. “Sky sphere representation in language models” with A. Berdnikov · preprint, 2026 · arXiv:2607.27092
  2. “Quantifying stability of non-power-seeking in artificial agents” with E. R. Gunter and V. Krakovna · preprint, 2024 · arXiv:2401.03529

3-manifold topology

  1. “The Smale conjecture and min-max theory” with D. Ketover · Invent. Math. 241 (2025), 1–34 · arXiv:2310.05756
  2. “On the existence of minimal Heegaard surfaces” with D. Ketover and A. Song · preprint, 2019 · arXiv:1911.07161

Scalar curvature

  1. “Length of a closed geodesic in 3-manifolds of positive scalar curvature” with D. Maximo and R. Rotman · Geom. Funct. Anal. 36 (2026), no. 3, 919–946 · arXiv:2504.05459
  2. “On the waist and width inequality in complete 3-manifolds with positive scalar curvature” with Z. Wang · Camb. J. Math. 14 (2026), no. 2, 349–375 · arXiv:2308.04044
  3. “Classifying sufficiently connected PSC manifolds in 4 and 5 dimensions” with O. Chodosh and C. Li · Geom. Topol. 27 (2023), 1635–1655 · arXiv:2105.07306
  4. “Waist inequality for 3-manifolds with positive scalar curvature” with D. Maximo · Perspectives in Scalar Curvature, Vol. 2, World Scientific (2023), 799–831 · arXiv:2012.12478

Geodesics and minimal surfaces

  1. “Parametric inequalities and Weyl law for the volume spectrum” with L. Guth · Geom. Topol. 29 (2025), 863–902 · arXiv:2202.11805
  2. “Generic density of geodesic nets” with B. Staffa · Selecta Math. 30 (2024), no. 1, art. 14 · arXiv:2107.12340
  3. “On the existence of closed C1,1 curves of constant curvature” with D. Ketover · Calc. Var. PDE 62 (2023), art. 246 · arXiv:1810.09308
  4. “Geodesic nets on non-compact Riemannian manifolds” with G. R. Chambers, A. Nabutovsky and R. Rotman · J. Reine Angew. Math. (2023), 287–303 · arXiv:2205.09242
  5. “Singular behavior and generic regularity of min-max minimal hypersurfaces” with O. Chodosh and L. Spolaor · Ars Inveniendi Analytica (2022), paper no. 2, 27 pp. · arXiv:2007.11560
  6. “Existence of minimal hypersurfaces in complete manifolds of finite volume” with G. R. Chambers · Invent. Math. 219 (2020), 179–217 · arXiv:1609.04058
  7. “Weyl law for the volume spectrum” with F. C. Marques and A. Neves · Ann. of Math. 187 (2018), 933–961 · arXiv:1607.08721
  8. “Sweeping out 3-manifolds of positive Ricci curvature by short 1-cycles via estimates of min-max surfaces” with X. Zhou · Int. Math. Res. Not. IMRN 2018, no. 4, 1129–1152 · arXiv:1510.02896
  9. “Lengths of three simple periodic geodesics on a Riemannian 2-sphere” with A. Nabutovsky and R. Rotman · Math. Ann. 367 (2017), 831–855 · arXiv:1410.8456
  10. “Width, Ricci curvature and minimal hypersurfaces” with P. Glynn-Adey · J. Differential Geom. 105 (2017), no. 1, 33–54 · arXiv:1408.3656

Metric geometry

  1. “Filling metric spaces” with B. Lishak, A. Nabutovsky and R. Rotman · Duke Math. J. 171 (2022), no. 3, 595–632 · arXiv:1905.06522
  2. “Optimal sweepouts of a Riemannian 2-sphere” with G. R. Chambers · J. Eur. Math. Soc. 21 (2019), no. 5, 1361–1377 · arXiv:1411.6349
  3. “Area of convex disks” with G. R. Chambers, C. Croke and H. Wen · Proc. Amer. Math. Soc. (2018) · arXiv:1701.06594
  4. “Determinantal variety and normal embedding” with K. U. Katz, M. G. Katz and D. Kerner · J. Topol. Anal. 10 (2018), no. 1, 27–34 · arXiv:1602.01227
  5. “Splitting a contraction of a simple curve traversed m times” with G. R. Chambers · J. Topol. Anal. 9 (2017), no. 3, 409–418 · arXiv:1510.03445
  6. “Families of short cycles on Riemannian surfaces” Duke Math. J. 165 (2016), no. 7, 1363–1379 · arXiv:1410.8436
  7. “Contracting the boundary of a Riemannian 2-disc” with A. Nabutovsky and R. Rotman · Geom. Funct. Anal. 25 (2015), 1543–1574 · arXiv:1205.5474
  8. “Converting homotopies to isotopies and dividing homotopies in half in an effective way” with G. R. Chambers · Geom. Funct. Anal. 24 (2014), 1080–1100 · arXiv:1311.0779
  9. “Slicing a 2-sphere” J. Topol. Anal. 6 (2014), no. 4, 573–590 · arXiv:1401.0060
  10. “Surfaces of small diameter with large width” J. Topol. Anal. 6 (2014), no. 3, 383–396 · arXiv:1307.2306
  11. “Spheres of small diameter with long sweep-outs” Proc. Amer. Math. Soc. 141 (2013), 309–312 · arXiv:1105.6159

Teaching

Fall 2026
MAT1510 — Math for AI Safety
MAT232 — Multivariable Calculus (UTM)
Fall 2024
MAT337 — Introduction to Real Analysis
Fall 2024
MAT232 — Multivariable Calculus
Spring 2024
MAT1341HS — Topics in Differential Geometry: Manifolds with Positive Scalar Curvature
Fall 2022
MAT337 — Introduction to Real Analysis
MAT232 — Multivariable Calculus
Fall 2021
MAT1300HF — Differential Topology
Fall 2020
MAT337H1F — Introduction to Real Analysis
Spring 2020
MAT1309HS — Geometric Inequalities
Fall 2019
MAT337H1F — Introduction to Real Analysis

Students and postdocs

I am accepting students to work on mathematical problems related to reducing catastrophic risk from artificial intelligence.

Current

Fields–PrincInt Mathematics for AI Safety postdocs

Past students

Past postdocs

Seminars

I co-organize the Mathematical AI Seminar and the Fields Geometric Analysis Colloquium, and I serve on the steering committee of the Fields Institute Centre for Mathematical AI.

Personal

My wife Merey Ismailova is the artistic director of the Ismailova Theatre of Dance.