Research interests
AI Safety, geometric analysis, metric geometry, low-dimensional topology.
AI safety for mathematicians
- Fields–PrincInt postdoctoral fellowships in Mathematics for AI Safety
Two-year Fields Postdoctoral Fellowships, run by the Fields Institute
in partnership with Principles of Intelligence, supporting research at the
intersection of mathematics and AI safety. Research directions include
rigorous mathematical
foundations of deep learning for AI interpretability, formal verification
methods, modeling emergent risks from AI R&D automation. Part of the
Fields Institute Centre for Mathematical AI;
apply via MathJobs.
- AI Safety for Mathematicians
An introduction to the field written for working mathematicians,
maintained by Jacob Tsimerman.
- ILIAD Intensive
A four-week, full-time course on foundational AI alignment research
for mathematicians, theoretical physicists and theoretical computer scientists.
Tuition is free and comes with a travel and housing allowance; run in London and
Berkeley.
- Toronto AI Safety Initiative
Runs a fellowship and an intensive course training for students in
Toronto to become AI safety researchers.
- Trajectory Labs
A Toronto AI safety office providing workspace, talks and a peer
network for people working in the field.
Papers
Most of my papers can be found on the
arXiv.
AI safety and interpretability
- “Sky sphere representation in language models”
with A. Berdnikov · preprint, 2026 ·
arXiv:2607.27092
- “Quantifying stability of non-power-seeking in artificial agents”
with E. R. Gunter and V. Krakovna · preprint, 2024 ·
arXiv:2401.03529
3-manifold topology
- “The Smale conjecture and min-max theory”
with D. Ketover ·
Invent. Math. 241 (2025), 1–34 ·
arXiv:2310.05756
- “On the existence of minimal Heegaard surfaces”
with D. Ketover and A. Song · preprint, 2019 ·
arXiv:1911.07161
Scalar curvature
- “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
- “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
- “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
- “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
- “Parametric inequalities and Weyl law for the volume spectrum”
with L. Guth ·
Geom. Topol. 29 (2025), 863–902 ·
arXiv:2202.11805
- “Generic density of geodesic nets”
with B. Staffa ·
Selecta Math. 30 (2024), no. 1, art. 14 ·
arXiv:2107.12340
- “On the existence of closed C1,1 curves of constant curvature”
with D. Ketover ·
Calc. Var. PDE 62 (2023), art. 246 ·
arXiv:1810.09308
- “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
- “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
- “Existence of minimal hypersurfaces in complete manifolds of finite volume”
with G. R. Chambers ·
Invent. Math. 219 (2020), 179–217 ·
arXiv:1609.04058
- “Weyl law for the volume spectrum”
with F. C. Marques and A. Neves ·
Ann. of Math. 187 (2018), 933–961 ·
arXiv:1607.08721
- “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
- “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
- “Width, Ricci curvature and minimal hypersurfaces”
with P. Glynn-Adey ·
J. Differential Geom. 105 (2017), no. 1, 33–54 ·
arXiv:1408.3656
Metric geometry
- “Filling metric spaces”
with B. Lishak, A. Nabutovsky and R. Rotman ·
Duke Math. J. 171 (2022), no. 3, 595–632 ·
arXiv:1905.06522
- “Optimal sweepouts of a Riemannian 2-sphere”
with G. R. Chambers ·
J. Eur. Math. Soc. 21 (2019), no. 5, 1361–1377 ·
arXiv:1411.6349
- “Area of convex disks”
with G. R. Chambers, C. Croke and H. Wen ·
Proc. Amer. Math. Soc. (2018) ·
arXiv:1701.06594
- “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
- “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
- “Families of short cycles on Riemannian surfaces”
Duke Math. J. 165 (2016), no. 7, 1363–1379 ·
arXiv:1410.8436
- “Contracting the boundary of a Riemannian 2-disc”
with A. Nabutovsky and R. Rotman ·
Geom. Funct. Anal. 25 (2015), 1543–1574 ·
arXiv:1205.5474
- “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
- “Slicing a 2-sphere”
J. Topol. Anal. 6 (2014), no. 4, 573–590 ·
arXiv:1401.0060
- “Surfaces of small diameter with large width”
J. Topol. Anal. 6 (2014), no. 3, 383–396 ·
arXiv:1307.2306
- “Spheres of small diameter with long sweep-outs”
Proc. Amer. Math. Soc. 141 (2013), 309–312 ·
arXiv:1105.6159
Teaching
- Fall 2026
- 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.
