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.

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)

Sharp bounds and rigidity for volumes of boundaries of Alexandrov spaces, with Qin Deng, 2023

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