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Geometry

The modern discipline of geometry is affecting virtually every branch of mathematics, and is in a period of great progress. Many old problems are being solved using techniques from the area (notably Perelman's resolution of the Poincaré conjecture) and new directions are being pioneered. An important theme in this area has been the development and use of sophisticated techniques from the theory of PDEs to study natural equations that arise in geometry. The Atiyah-Singer index theorem from the 1960s connects the theory of linear PDEs to topology and geometry. The development of tools for nonlinear PDEs in geometry has been slower but has led to many of the most dramatic developments in mathematics including Donaldson's breakthroughs in the theory of four-manifolds using the Yang-Mills equations of high-energy physics.

The study of lower dimensional manifolds (dimensions four or less) has particular significance to theoretical physics and has many applications. Floer homology is a mathematically rigorous way of constructing parts of a quantum field theory. Another important and growing area is the mathematics of general relativity. The Lorentz version of the Einstein equations is now at the cutting edge of our hyperbolic PDE technology. One branch of geometric analysis involves the recovery of a function from its integrals over various domains. A well-known application of this idea is Computed Tomography Scanning (CT scans).

The research interest of this group covers geometric analysis as well as symplectic topology and its role in mirror symmetry, low dimensional topology and gauge theory, Riemannian geometry and minimal surfaces and mathematical physics.

Faculty

Tobias Holck Colding Differential Geometry, Partial Differential Equations

Tristan Collins Geometric Analysis, PDEs

Victor Guillemin Differential Geometry

Larry Guth Metric geometry, harmonic analysis, extremal combinatorics

William Minicozzi Geometric Analysis, PDEs

Tomasz Mrowka Gauge Theory, Differential Geometry

Paul Seidel Mirror Symmetry

Instructors & Postdocs

Daniel Alvarez-Gavela Symplectic geometry, h-principles, algebraic K-theory

Anthony Conway Low dimensional topology

Irving Dai

Pei-Ken Hung

Yang Li Differential Geometry

Maggie Miller Low-dimensional topology

Tristan Ozuch-Meersseman Geometric analysis

Yair Shenfeld Probability, Convex Geometry

Abigail Ward Symplectic Geometry, Homological Mirror Symmetry

Junho Whang

Yiming Zhao PDE, geometric analysis

Graduate Students*

Julius Baldauf Geometric analysis

Alexey Balitskiy

Deeparaj Bhat Gauge theory, Low-dimensional topology

Zihong Chen

Luis Kumanduri Metric Geometry, Quantitative Topology

Jae Hee Lee

Tang-Kai Lee

Elia Portnoy

Sahana Vasudevan

Donghao Wang Gauge Theory

*Only a partial list of graduate students