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.

Permanent faculty:

• 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 was 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.
  
  

Postdoctoral fellows:

• 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.
  
  
• Sherry Gong (Ph.D. MIT, 2018) does research in gauge theory, specifically on the Yang-Mills and Seiberg-Witten equations and their applications to knot theory.
  
  
• Mikhail Hlushchanka (Ph.D. Jacobs University Bremen, 2017) does research in dynamical systems, conformal geometry, and self-similar groups. Work to date has focused on combinatorial aspects, Thurstons theory, and rigidity questions in holomorphic dynamics. Additionally, he is interested in properties and applications of iterated monodromy groups.
  
  
• Wenhao Ou (Ph.D. Grenoble, 2015) works on algebraic geometry, including rationally connected varieties, holomorphic symplectic varieties, and Fano varieties
  
  
• Matthias Wink (D.Phil. Oxford, 2018) works in differential geometry and geometric analysis. His research focuses in particular on geometric flows and their solitons and recent work includes constructions of gradient Ricci solitons.