**Alexei Kovalev**

*a.g.kovalev *at* dpmms.cam.ac.uk*

Department of Pure Mathematics
and Mathematical Statistics, University
of Cambridge

## Information for potential PhD students

I may take a PhD student for 2018.

**Research**

My general research area is differential geometry and global analysis;
occasionally it includes topics in algebraic geometry or topology.
I am interested in `special' differential-geometric structures and
their moduli spaces. These structures are often expressed as solutions
of partial differential equations, typically non-linear and elliptic, on
manifolds or vector bundles. Examples of `special geometries' that I
studied include Ricci-flat 7- and 8-dimensional manifolds of holonomy
G_{2} and Spin(7), their calibrated minimal submanifolds, and
Calabi–Yau and hyper-Kähler manifolds. Projects that I offer may include
applications of Analysis (the PDE methods). Familiarity with Algebraic
Geometry and Topology is an advantage. Some of the results might be of
interest also to Theoretical Physicists, especially String Theorists.

2017–2018
Part
III courses. If you are a Part III student considering doing a PhD with
me, then I recommend taking (many of) these courses:

*Differential Geometry* and *Algebraic Topology* — lectured in
Michaelmas Term,

*Complex manifolds* and *Elliptic Partial Differential Equations*
— lectured in Lent Term.

I strongly recommend that you do a
**Part III essay**
on a geometry topic, preferably related to the above.

PhD student(s) finished to date

2008 Johannes Nordström,
"Deformations
and gluing of asymptotically cylindrical manifolds with exceptional
holonomy"

2015 Matthias Ohst,
"Deformations of Cayley submanifolds"

2017 Kimberley Moore (supervised jointly with Jason Lotay, Unversity College London),
"Deformation theory of Cayley submanifolds"

2017 Timothy Talbot,
"Asymptotically cylindrical Calabi–Yau and special Lagrangian geometry"