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Recent and Upcoming Talks

• Colloquium, Stanford U., January 7, 2020
• Geometric Analysis Seminar, U. Chicago, January 20, 2020
• Geometry Seminar, Carnegie Mellon U., February 20, 2020
• Colloquium, Carnegie Mellon U., February 21, 2020
• Joint Analysis Seminar, UCLA/Caltech, Apr 7, 2020
• Differential Geometry Seminar, TU Vienna, October 28, 2020
• Differential Geometry Seminar, Max Plank Institute, November 12, 2020
• Geometry and Topology Seminar, U. Luxembourg, December 14, 2020
• Differential Geometry Seminar, Rice University, January 27 12, 2021
• Calculus of Variations and PDE Conference, ETH, Zurich, June 21-25, 2021
• Frontiers in Mathematical Sciences, IPM, Iran, April 8, 2021
• Pangolin Seminar, Online, October 26, 2021
• Geometry Seminar, U. Georgia, April 22, 2022
• Workshop on Convexity, Georgia Tech, May 23-27, 2022
• Riemannian Geometry Conference, Florence, Italy, June 20-24, 2022
• Harmonic Analysis Methods in Geometric Tomography, ICERM, Brown U., Sep. 26-30, 2022
• Geometry and Topology Seminar, Indian Institute of Science, Bangalore, October 26, 2022
• Xavier Colloquium (Inaugural speaker), TCU, Fort-Woth, TX April 14, 2023
• Texas Geometry and Topology Conference, TCU, Fort-Woth, TX April 15, 2023
• International Conf. on Diff. Geometry, Jeju, South Korea, June 26-30, 2023
• Geometry Beyond Riemann, ESI Summer School, Vienna, September 18-22, 2023
• Geometria Aljamia, Williams Papermaking Museuem, Atlanta, November 8, 2023

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Slides of Previous Talks

• Topology of Riemannian submanifolds with prescribed boundary

  We prove that a smooth compact immersed submanifold of codimension 2 in Rⁿ, n>2, bounds at most finitely many topologically distinct compact nonnegatively curved hypersurfaces. This settles a question of Guan and Spruck related to a problem of Yau. Analogous results for complete fillings of arbitrary Riemannian submanifolds are obtained as well. On the other hand, we show that these finiteness theorems may not hold if the codimenion is too high, or the prescribed boundary is not sufficiently regular. Our proofs employ, among other methods, a relative version of Nash's isometric embedding theorem, and the theory of Alexandrov spaces with curvature bounded below, including the compactness and stability theorems of Gromov and Perelman. This includes joint works with Stephanie Alexander, Jeremy Wong, and Robert Greene.

• Relative isoperimetric inequality outside convex bodies

  We prove that the area of a hypersurface which traps a given volume outside of a convex body in Euclidean n-space must be greater than or equal to the area of a hemisphere trapping the given volume on one side of a hyperplane. The proof is based on a sharp estimate for total positive curvature of surfaces whose boundary meets a convex body orthogonally from the outside. This is joint work with Jaigyoung Choe and Manuel Ritore.

• Riemannian four vertex theorems

  We prove that every metric of constant curvature on a compact surface M with boundary bdM induces at least four vertices, i.e., local extrema of geodesic curvature, on a connected component of bdM, if, and only if, M is simply connected. Indeed, when M is not simply connected, we construct hyperbolic, parabolic, and elliptic metrics of constant curvature on M which induce only two critical points of geodesic curvature on each component of bdM. With few exceptions, these metrics are obtained by removing the singularities and a perturbation of flat structures on closed surfaces. Further, we uncover some connections between the topology of a complete Riemannian surface M and the minimum number of vertices, i.e., critical points of geodesic curvature, of closed curves in M. In particular we show that the space forms with finite fundamental group are the only surfaces in which every simple closed curve has more than two vertices. We also characterize the simply connected space forms as the only surfaces in which every closed curve bounding a compact immersed surface has more than two vertices.

• Deformations of unbounded convex bodies and hypersurfaces

  We study the topology of the space bd Kⁿ of complete convex hypersurfaces of Rⁿ which are homeomorphic to R^n-1. In particular, using Minkowski sums, we construct a deformation retraction of bd Kⁿ onto the Grassmannian space of hyperplanes. So every hypersurface in bd Kⁿ may be flattened in a canonical way. Further, the total curvature of each hypersurface evolves continuously and monotonically under this deformation. We also show that, modulo proper rotations, the subspaces of bd Kⁿ consisting of smooth, strictly convex, or positively curved hypersurfaces are each contractible, which settles a question of H. Rosenberg.

• Tangent cones and regularity of real hypersurfaces

  We characterize C¹ embedded hypersurfaces of Rⁿ as the only locally closed sets with continuously varying flat tangent cones whose measure-theoretic-multiplicity is at most m<3/2. It follows then that any (topological) hypersurface which has flat tangent cones and is supported everywhere by balls of uniform radius is C¹. In the real analytic case the same conclusion holds under the weakened hypothesis that each tangent cone be a hypersurface. In particular, any convex real analytic hypersurface X in Rⁿ is C¹. Furthermore, if X is real algebraic, strictly convex, and unbounded, then its projective closure is a C¹ hypersurface as well, which shows that X is the graph of a function defined over an entire hyperplane. This is joint work with Ralph Howard.

• Durer's problem on unfoldability of convex polyhedra

  A well-known problem in geometry, which may be traced back to the Renaissance artist Albrecht Durer, is concerned with cutting a convex polyhedral surface along some spanning tree of its edges so that it may be isometrically embedded into the plane. We show that this is always possible after an affine transformation of the surface. In particular, unfoldability of a convex polyhedron does not depend on its combinatorial structure, which settles a problem of Croft, Falconer, and Guy. On the other hand, we also construct a convex polyhedron which is not unfoldable when cut along a system of pseudo-edges, i.e., a network of geodesics which connect all vertices and have convex faces. This includes joint work with Nicholas Barvinok.

• Total curvature and isoperimetric inequality

  The classical isoperimetric inequality states that in Euclidean space spheres provide enclosures of least perimeter for any given volume. According to the Cartan-Hadamard conjecture, this inequality may be generalized to spaces of nonpositive curvature. In this talk we discuss an approach to proving this conjecture via a comparison formula for the total curvature of level sets of functions on nonpositively curved manifolds. In particular we show that the conjecture holds when the variation of the curvature of the ambient space is small. This is joint work with Joel Spruck.

• Geometric inequalities in spaces of nonpositive curvature

  We will discuss total mean curvatures, i.e., integrals of symmetric functions of the principle curvatures, of hypersurfaces in Riemannian manifolds. These quantities are fundamental in geometric variational problems as they appear in Steiner’s formula, Brunn-Minkowski theory, and Alexandrov-Fenchel inequalities. We will describe a number of new inequalities for these integrals in non positively curved spaces, which are obtained via Reilly's identities, Chern-type differential forms, and harmonic mean curvature flow. As applications we obtain several new isoperimetric inequalities, and Riemannian rigidity theorems. This is joint work with Joel Spruck.

• Convexity and rigidity of hypersurfaces in Cartan-Hadamard manifolds

  We show that in Cartan-Hadamard manifolds Mⁿ, n≥ 3, closed infinitesimally convex hypersurfaces Γ bound convex flat regions, if curvature of Mⁿ vanishes on tangent planes of Γ. This encompasses Chern-Lashof-sacksteder characterization of compact convex hypersurfaces in Euclidean space, and some results of Greene-Wu-Gromov on rigidity of Cartan-Hadamard manifolds. It follows that closed simply connected surfaces in M³ with minimal total absolute curvature bound Euclidean convex bodies, as stated by M. Gromov in 1985. The proofs employ the Gauss-Codazzi equations, a generalization of Schur comparison theorem to CAT(k) spaces, and other techniques from Alexandrov geometry outlined by A. Petrunin. We will also give a survey of related results and historical background on characterization of convex sets in differential geometry.

• Symmetries of tilings

  We give an introduction to symmetry groups of tilings and patterns with many historical references to architectural landmarks in Moorish Spain, and Persia. This talk was prepared for and delivered to a general audience in conjunction with an exhibit entitled Geometria Aljamia at the Williams Museum of Papermaking in Atlanta, which took place from September to December of 2023.