Vitali Kapovitch's Home Page

Department of Mathematics,
University of Toronto,
Toronto, Ontario, Canada M5S 2E4

Office: 45 St George st (PGB), room 201c
Telephone: (416)-978-6786
Fax: (416) 978-4107

"God put me on Earth to accomplish a certain number of things. Right now I am so far behind I will never die."

--Bill Watterson

Research Interests: I'm interested in global Riemannian geometry. More specifically, I'm interested in convergence with lower sectional and Ricci curvature bounds.

I am coorganizing the Thematic Program on Nonsmooth Riemannian and Lorentzian Geometry at the Fields Institute for Research in Mathematical Sciences in Toronto.

During the program I am coorganizing the workshop Workshop on Geometry of Spaces with Upper and Lower Curvature Bounds which will take place September 12 - 16, 2022.

My book with Anton Petrunin and Stephanie Alexander "Invitation to Alexandrov geometry: CAT(0) spaces" has been published by Springer.

Winter 2023: MAT1318: Comparison Geometry

Past courses

Fall 2015: MAT1300 Topology I
Winter 2015: MAT246 Concepts in Abstract Mathematics
Fall 2014: MAT1300 Topology I
Winter 2014: MAT246 Concepts in Abstract Mathematics
Winter 2014: MAT363 Differential geometry I
Fall 2013: MAT1313 Comparison Geometry.

Jost Eschenburg's Notes on Comparison Geometry

Wolfgang Meyer's notes on comparison geometry.

Winter 2013: MAT246 Concepts in Abstract Mathematics
Winter 2013: MAT1120 Lie Groups and Lie Algebras
Winter 2012: MAT247 Algebra II
Winter 2012: MAT246
Winter 2011: MAT246
Fall 2011: MAT464/1342 Differential geometry
Fall 2010-Winter 2011: MAT257

Fall 2009-Winter 2010: MAT257

Publications/Preprints (which can also be downloaded here)

On the intrinsic and extrinsic boundary for metric measure spaces with lower curvature bounds, with Xingyu Zhu, 2022

Alexandrov geometry: foundations, vol 1, with S. Alexander and A. Petrunin, 2022.

Remarks on manifolds with two sided curvature bounds, with A. Lytchak, 2021, Anal. Geom. Metr. Spaces 9 (2021), no. 1, 53--64

The Structure of Submetries, with A. Lytchak, Geom. Topol. 26 (2022), no. 6, 2649--2711

On gluing Alexandrov spaces with lower Ricci curvature bounds, with C. Ketterer and K.-T. Sturm, accepted for publication in Communications in Analysis and Geometry

Mixed curvature almost flat manifolds, Geom. Topol. 25 (2021), no. 4, 2017--2059

On the structure of RCD spaces with upper curvature bounds, with M. Kell and C. Ketterer, Math. Z. 301 (2022), no. 4, 3469--3502

On the topology and the boundary of N-dimensional RCD(K,N) spaces, with A. Mondino, Geometry & Topology 25, (2021), 445--495

Weakly non-collapsed RCD spaces with upper curvature bounds, with C. Ketterer, Analysis and Geometry in Metric Spaces, 7 (2019), no. 1, 197--211

On Torsion in the Center Conjecture, (with A. Petrunin and W. Tuschmann), Electron. Res. Announc. Math. Sci. 25 (2018), 27--35.

CD meets CAT, with C. Ketterer, J. Reine Angew. Math. 766 (2020), 1 -- 44

Metric-measure boundary and geodesic flow on Alexandrov spaces, with A. Lytchak and A. Petrunin, J. Eur. Math. Soc. (JEMS) 23 (2021), no. 1, 29--62

Invitation to Alexandrov geometry, (with S. Alexander and A. Petrunin), Springer Briefs, 2019.

On noncollapsed almost Ricci-flat 4-manifolds, (with John Lott), Amer. J. Math. 141 (2019), no. 3, 737--755.

On dimensions of tangent cones in limit spaces with lower Ricci curvature bounds, (with Nan Li), J. Reine Angew. Math. (Crelles Journal), 742 (2018), 263--280.

Space of nonnegatively curved metrics and pseudoisotopies, (with Igor Belegradek and Tom Farrell), J. Differential Geom. 105 (2017), no. 3, 345--374.

Manifolds without conjugate points and their fundamental groups, (with S. Ivanov),
J. Differential Geom. 96 (2014), no. 2, 223--240.

On fibrations with formal elliptic fibers, (with M. Amann), Adv. Math. 231 (2012), no. 3-4, 2048--2068.

Structure of fundamental groups of manifolds with Ricci curvature bounded below, (with B. Wilking), 2011

Alexandrov meets Kirzbraun, (with S. Alexander and A. Petrunin) , Proceedings of the 17th Gökova Gemometry and Topology Conference, 2010

An optimal lower curvature bound for convex hypersurfaces in Riemannian manifolds, (with S. Alexander, and A. Petrunin) , Illinois Journal of Mathematics, vol. 52, (2009), No 3, pp 1031--1033

Perelman's Stability Theorem, Surveys in Differential geometry, Metric and Comparison geometry, vol XI , International press, (2007), 103--136

Nilpotency, almost nonnegative curvature and the gradient push, (with A. Petrunin and W. Tuschmann), Annals of Mathematics, Vol. 171 (2010), No. 1, 343--373

Nonnegative pinching, moduli spaces and bundles with infinitely many souls,(with A. Petrunin and W. Tuschmann), Journal of Diff. Geometry, 71, (2005), no. 3, pp. 365--383.

Pinching estimates for negatively curved manifolds with nilpotent fundamental groups,(with I. Belegradek), Geom. Funct. Anal. 15, (2005), no. 5, 929--938.

Curvature bounds via Ricci smoothing, Illinois Journal of Mathematics, 49, (2005) no. 1, pp.259--263,

(with I. Belegradek) Classification of negatively pinched manifolds with amenable fundamental groups , Acta Math. 196 (2006), no. 2, 229--260

Biquotients with singly generated rational cohomology, (with W. Ziller), Geometriae Dedicata, 104 (1): 149--160, 2004

Restrictions on collapsing with a lower sectional curvature bound, Mathematische Zeitschrift, 249, (2005),no.3, pp. 519-- 539

A note on rational homotopy of biquotients, preprint

Regularity of limits of noncollapsing sequences of manifolds, Geom. Funct. Anal. 12 (2002), no. 1, 121--137.

(with I.Belegradek) Obstructions to nonnegative curvature and rational homotopy theory, Journal of AMS, 16 (2003), 259--284.

(with I.Belegradek) Finiteness theorems for nonnegatively curved vector bundles, Duke Mathematical Journal, 108 (2001), no. 1, pp. 109--134.

(with I.Belegradek) Topological obstructions to nonnegative curvature, Math. Ann. 320 (2001), no. 1, 167--190.

(with L. Guijarro) Restrictions on the geometry at infinity of nonnegatively curved manifolds, Duke Math. J. 78 (1995), no. 2, 257--276.

Here are some geometry links:

This home page is always under construction