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)9786786
Fax: (416) 9784107
Email:
vtk@math.toronto.edu
"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
Teaching:
 Fall 2020: MAT1312 Characteristic classes,
M1112, Th 122. First class: Thursday, September
10, 122.
 The course will cover chapters 119 of the
book Characteristic Classes by
Milnor and Stasheff.
 Prerequisites: Topology I and Topology II.
 Please email me for a Zoom link if
you are not registered for the course but would like to
attend the lectures.
Research
Interests:
I'm interested in global
Riemannian geometry. More specifically,
I'm interested in convergence with lower
sectional and Ricci curvature bounds.
Here are the
notes from my lectures at SMS
2011
Summer
School
on
“Metric
Measure
Spaces:
Geometric
and
Analytic
Aspects”.
I'm currently working on a book on
spaces with curvature bounded below and
above together with Stephganie
Alexander and Anton Petrunin.
Several complete chapters from
this book can
be found here.
My book Invitation to "Alexandrov
geometry: CAT(0) spaces" has
been published by Springer.
Workshop on
Curvature and Global Shape.
Publications/Preprints
(which
can also be downloaded here)

Structure
of submetries, with A.
Lytchak, 2020.
 On
gluing Alexandrov spaces with
lower Ricci curvature bounds,
with C. Ketterer and K.T.
Sturm, 2020.
 Mixed
curvature almost flat
manifolds, 2019,
accepted for publication in
Geometry & Topology
 On
the structure of RCD spaces
with upper curvature bounds,
with
M. Kell and C. Ketterer, 2019
 On the
topology and the boundary of
Ndimensional RCD(K,N) spaces,
with A. Mondino, 2019, accepted
for publication in Geometry &
Topology
 Weakly
noncollapsed 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.
(Crelles Journal), 0.0 (2019), doi:10.1515/crelle20190021.
 Metricmeasure
boundary and geodesic
flow on Alexandrov
spaces, with A.
Lytchak and A. Petrunin, 2017,
accepted for publication in JEMS.
 Invitation
to Alexandrov geometry,
(with
S. Alexander
and A.
Petrunin),
Springer
Briefs, 2019.
 On
noncollapsed almost Ricciflat
4manifolds, (with John
Lott), Amer. J. Math. 141
(2019), no. 3, 737755.
 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,
345374.
 Manifolds
without conjugate points and
their fundamental groups,
(with S. Ivanov),
J. Differential Geom.
96
(2014),
no. 2, 223240.
 On
fibrations with formal elliptic
fibers, (with M. Amann),
Adv. Math.
231
(2012),
no. 34, 20482068.
 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
10311033
 Perelman's
Stability Theorem,
Surveys in
Differential
geometry,
Metric and
Comparison
geometry, vol XI
,
International
press,
(2007),
103136
 Nilpotency,
almost
nonnegative
curvature and the gradient push,
(with A.
Petrunin and W.
Tuschmann), Annals of
Mathematics, Vol. 171
(2010), No. 1, 343373
 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. 365383.
 Pinching
estimates
for negatively curved manifolds
with nilpotent fundamental
groups,(with I.
Belegradek),
Geom. Funct.
Anal. 15,
(2005), no. 5,
929938.
 Curvature
bounds via Ricci smoothing, Illinois Journal of
Mathematics, 49,
(2005) no. 1, pp.259263,
 (with I.
Belegradek) Classification of
negatively pinched manifolds
with amenable fundamental
groups , Acta Math. 196
(2006), no. 2, 229260
 Biquotients
with
singly
generated
rational
cohomology, (with W.
Ziller), Geometriae Dedicata,
104 (1): 149160,
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,
121137.
 (with
I.Belegradek) Obstructions to
nonnegative curvature and
rational homotopy theory,
Journal of AMS, 16
(2003), 259284.
 (with
I.Belegradek) Finiteness
theorems for nonnegatively
curved vector bundles, Duke
Mathematical Journal, 108
(2001), no. 1, pp. 109134.
 (with
I.Belegradek)
Topological
obstructions
to nonnegative curvature,
Math. Ann. 320 (2001),
no. 1, 167190.
 (with L.
Guijarro) Restrictions on the
geometry at infinity of
nonnegatively curved
manifolds, Duke Math. J.
78 (1995), no. 2,
257276.
Here are some geometry links:
This
home page is
always under
construction