## The UCLA Geometry Group |

Geometry research at UCLA is represented by nine permanent faculty members and several postdocs and graduate students. It covers the areas of Riemannian geometry, complex manifolds, algebraic geometry, gauge theory, symplectic geometry, contact geometry, Teichmüller theory, and mirror symmetry.

There are significant interactions and overlap with the Topology Group, the Algebra Group, and the Analysis Group.

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Permanent faculty:
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- Mario Bonk (Ph.D. TU Braunschweig, 1988) does research in
fields at the interface of geometry and analysis, including classical complex analysis, the geometry of
negatively curved spaces, geometric group theory, dynamics of rational maps, and analysis on metric spaces. His
current work often relies on an extension of classical results in geometry and analysis to a non-smooth or
fractal setting. He was an invited speaker at the International Congress of Mathematicians (ICM) in 2006.

- David Gieseker (Ph.D. Harvard, 1970) works in
algebraic geometry. He has been very influential in the area of geometric invariant theory, where he
introduced a
novel concept of stability. Gieseker gave a Plenary Invited Address at an annual meeting of the
AMS in 1983. He was also an ICM invited speaker in 1978.

- Robert Greene (Ph.D Berkeley, 1969) is an expert
in complex and Riemannian geometry. He is known for his work on isometric embeddings of
Riemannian manifolds. Greene is a
former Putnam Fellow and
Sloan Research Fellow.

- Ko Honda (Ph.D.
Princeton, 1997) works in
low-dimensional topology and
symplectic geometry, with an emphasis on contact geometry. He introduced
topological tools such as the bypass, which enabled him (in collaboration with John Etnyre) to give the
first example of a 3-manifold without a tight contact structure. He also established the
equivalence between Heegaard Floer homology and embedded contact homology (in joint work with
Vincent Colin and Paolo Ghiggini). He is a recipient of a Sloan Fellowship, a CAREER Grant, and the
2009 Geometry Prize of the Mathematical Society of Japan. He was an invited speaker at ICM 2006.

- Gang Liu (Ph.D. Stony Brook, 1995) does research in
symplectic geometry, particularly in the areas of Gromov-Witten theory and Floer theory. He is known
for his work (with Gang Tian) on the Arnold Conjecture, and on a stabilized version of the
Weinstein Conjecture. G.Liu is a
recipient of the AMS Centennial Fellowship.

- Kefeng Liu (Ph.D. Harvard, 1993) uses diverse methods to
address questions in geometry and mathematical physics. He is known for his work in elliptic genus and mirror
symmetry, including the proof of general rigidity theorems using modular forms and being a co-author of the
proofs of the Mirror Principle for the quintic, and the Marino-Vafa Conjecture about Hodge integrals on moduli
spaces of Riemann surfaces and the Chern-Simons knot invariants. K.Liu is a winner of the Guggenheim Fellowship,
the Morningside Gold Medal, and was an ICM Invited Speaker in 2002.

- Ciprian Manolescu (Ph.D. Harvard, 2004) works in
gauge theory, low-dimensional topology, and
symplectic geometry. With several collaborators, he found combinatorial descriptions for Heegaard
Floer homology and the related mixed four-manifold invariants. He has also worked on Seiberg-Witten
theory, and used it to disprove the Triangulation Conjecture in high dimensions. Manolescu is a former
Clay Research Fellow, Royal Society University Research Fellow, and three-time Putnam Fellow. He was
awarded the Morgan Prize in 2001 and the EMS Prize in 2012. He is an ICM
invited speaker in 2018.

- Peter Petersen (Ph.D. Maryland, 1987) does
research in Riemannian
geometry. He has been working on constructing a Riemannian metric on the Gromoll-Meyer sphere with
positive sectional curvature, which is a long-standing open problem in the field. He also does research
on quasi-Einstein and gradient soliton metrics, topics which have received much attention with Perelman
classification of certain gradient solitons in three dimensions. Petersen is a winner of the Sloan
Research Fellowship and an NSF Young Investigator Award. Petersen's "Riemannian Geometry" has become a standard
graduate textbook.

- Burt Totaro (Ph.D. Berkeley, 1989) works in
algebraic geometry, often using ideas from homotopy theory. He has done research on birational
geometry and on torsion algebraic cycles.
He is a winner of the Whitehead Prize (awarded by the
London Mathematical Society) and of the Prix Franco-Britannique. He was an invited speaker at the
2002 ICM. In 2009 he was elected a Fellow of
the Royal Society.

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Postdoctoral fellows:
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- Alex
Austin (Ph.D. UIC, 2016) does research in geometric mapping
theory. He is particularly interested in quasiconformal mappings and their
infinitesimal generators. Work to date has focused on rigidity problems
for the Heisenberg groups.
- Omprokash Das
(Ph.D. Utah, 2015) works on algebraic geometry. He studies
the minimal model program and the corresponding singularities, both in
characteristic 0 and in positive characteristic.
- Sylvester
Eriksson-Bique (Ph.D. NYU, 2017) works on problems at the interface of
geometry and analysis, as well as geometric analysis. He is focused on
problems related to establishing and understanding analytic inequalities,
such as Poincare inequalities on metric spaces, and how these interact
with the geometry of the space and quasisymmetries on the space.
Additionally, he is interested in curvature constrained spaces such as
Alexandrov spaces and their embedding problems, as well as algorithmic
problems arising in computational geometry.
- Martin
Gallauer Alves
de Souza (Ph.D. Zurich, 2015) works on algebraic geometry
(especially the theory of motives) and abstract homotopy theory.
With Utsav Choudhury, he proved that two proposed definitions
of the motivic Galois group, one by Ayoub and one by Nori,
are in fact the same.
- Wenhao
Ou
(Ph.D. Grenoble, 2015) works on algebraic geometry, including
rationally
connected varieties, holomorphic symplectic varieties, and Fano
varieties.