NSF Funding
Below are my NSF Funded Research Grants at Lehman College with the official descriptions and the resulting publications by the people I have funded with these grants. The resulting published work is listed under each grant. I have also obtained funding with collaborators through various mathematics research institutes (including Various Workshops and a Thematic Program in 2022 at the Fields Institute). I am currently part of a team applying for an NSF FRG grant this year.
NSF Grant DMS - 1612049 (2016-2020)
Geometric Compactness Theorems with Applications to General Relativity
In General Relativity, spacetime and spacelike slices of spacetime are manifolds (objects that locally resembles Euclidean spaces) satisfying certain geometric conditions determined by the Einstein Equation and other physically natural constraints. The manifolds arising in General Relativity are curved by gravity and they can contain black holes or thin deep gravity wells, making it technically difficult to estimate how close the manifold is to a simplified model, like Euclidean space. New compactness theorems with new notions of convergence are developed in this project providing fundamental new geometric tools that can be applied to address these challenges. The principal investigator has already been invited to present preliminary work in this direction at various mathematics and physics institutions around the world. As she has in the past, the PI will include young mathematicians of diverse backgrounds in this research project.
The PI will seek intrinsic flat limits of noncollapsing sequences of Riemannian manifolds with uniform lower bounds on scalar curvature. For example, the PI will consider sequences of asymptotically flat Riemannian manifolds with nonnegative scalar curvature whose ADM mass is approaching zero, or regions in such spaces with a uniform upper bound on Hawking mass. Compactness theorems for such sequences would be useful to prove the Almost Rigidity of the Schoen-Yau Positive Mass Theorem or the Bartnik Conjecture. Similar methods will also be applied towards proving Gromov's Almost Rigidity of Flat Tori Conjecture. To avoid cancellation and bubbling, the PI proposes to forbid the existence of arbitrarily small closed minimal surfaces in these and other conjectures stated within the proposal. Various Compactness Theorems for Intrinsic Flat convergence have been proven in different settings by Prof. Wenger, Dr. Portegies, Prof. LeFloch, Dr. Perales, Dr. Matveev, and the PI. Prior applications of intrinsic flat convergence to General Relativity have been completed in various papers by Prof. Lee, Prof. Huang, Prof. LeFloch, Prof. Stavrov, Prof. Jauregui and the PI.
Results funded in part by this grant:
A Sakovich and C Sormani, “Almost Rigidity of the Positive Mass Theorem for Asymptotically Hyperbolic Manifolds with Spherical Symmetry" General Relativity and Gravitation, Sept 2017 49:125. (arxiv) (reprint) (citations)
B Allen, L Hernandez, D Parise, A Payne, Wang. “Warped Tori of Almost Nonnegative Scalar Curvature” Geometriae Dedicata (reprint) (preprint)
B Allen and C Sormani “Contrasting various notions of convergence in geometric analysis” Pacific Journal of Mathematics Vol 303 No 1 (2019) (arxiv) (reprint)
J Basilio, D Kazaras, and C Sormani ``An intrinsic flat limit of Riemannian manifolds with no geodesics''. Geometriae Dedicata Vol 204, 265-284 (2020) (arxiv) (reprint)
E Bryden, “Stability of the Positive Mass Theorem for Axisymmetric Manifolds” Pacific Journal of Mathematics Vol 305 (2020) No 1 89-152 (reprint) (arxiv)
B Allen and E Bryden, “Sobolev bounds and convergence of Riemannian manifolds” with an appendix by B Allen and C Sormani (submitted) (arxiv)
A Cabrera Pacheco, C Ketterer, and R Perales ``Stability of graphical tori with almost nonnegative scalar curvature'' CVPDE Vol 59, No 134 (2020) (reprint) (arxiv)
C Sormani, D Kazaras, and students: D Afrifa, V Antonetti, M Dinowitz, H Drillick, M Farahzad, S George, A Lydeatte Hepburn, L Huynh, E Minichiello, J Mujo Pillati, S Rendla, A Yamin,``Smocked Metric Spaces and their Tangent Cones'' (arxiv) (DMS1612049)(SCGP)(CUNYGC) to appear in MJMS
E Bryden, M Khuri, and C Sormani “Stability of the Spacetime Positive Mass Theorem in Spherical Symmetry” Journal of Geometric Analysis Vol 31 Issue 4 (2020) 4191-4239 (reprint) (arxiv preprint) (NSF DMS1612049)(SCGP)
J Basilio and C Sormani ``Sequences of three dimensional manifolds with positive scalar curvature" (NSF DMS 1612049) (arxiv)
B Allen, R Perales and C Sormani ``Volume Above Distance Below'" to appear in JDG (arxiv) (DMS1612049)(IAS)
B Allen ``Almost Nonnegative Scalar Curvature on Manifolds Conformal to flat tori" (arxiv) (DMS1612049)
M Dominguez-Vazquez, D Gonzalez-Alvaro, and L Mouille “Infinite families of manifolds of positive kth-intermediate Ricci curvature with k small” (arxiv:2012.11640)
I Adelstein and F Vargas Pallete "The length of the shortest closed geodesic on positively curved two spheres" (NSF DMS 1612049) (arxiv)
C Sormani "Conjectures on Convergence and Scalar Curvature" Chapter for Perspectives in Scalar Curvature edited by Gromov and Lawson, Wold Scientific (2022).
Additional Papers not in the NSF Final Report because they were not yet completed:
L Mouille “Torus Actions on Manifolds with Positive Intermediate Curvature (arxiv) Journal of the London Math Society (reprint)
C Sormani, and students: V Antonetti, M Dinowitz, H Drillick, M Farahzad, L T Huynh, J Mujo Pillati, A Yamin,``SWIF Convergence of Smocked Metric Spaces" on the arxiv
Domínguez-Vázquez, M., González-Álvaro, D. & Mouillé, L. Infinite families of manifolds of positive kth-intermediate Ricci curvature with ksmall. Math. Ann. (2022). (reprint)
Christina Sormani, Wenchuan Tian, and Changliang Wang, ``An extreme limit with nonnegative scalar curvature" Nonlinear Analysis 239 (2024), 24 pp. (NSF DMS1006059) (PSC CUNY) (reprint) (arxiv)
A Sakovich and C Sormani "The null distance encodes causality," Journal of Mathematical Physics Vol.64, Issue 1 (Jan 2023) (NSF DMS1612049) (PSCCUNY) (arxiv) (reprint)
M Graf and C Sormani, “Lorentzian area and volume estimates for integral mean curvature bounds,” Developments in Lorentzian Geometry, Springer Proc. Math. Stat., 389 Springer, Cham, 2022, 105–128. (NSF DMS1612049) (arxiv)
B Allen and C Sormani “Relating Notions of Convergence in Geometric Analysis" Nonlinear Analysis Vol 200 Nov 2020 (NSF DMS 1612049) (arxiv)
Chen-Yun Lin and Christina Sormani, ``From Varadhan's limit to eigenmaps: a guide to the geometric analysis behind manifold learning.'' Mat. Contemp. 57 (2023), 32–115. (arxiv)
C Sormani "Conjectures on Convergence and Scalar Curvature" Chapter for Perspectives in Scalar Curvature edited by Gromov and Lawson, World Scientific (2023) (arxiv) (order here) (NSF DMS 1612049)
Still in progress:
Joint work with Sakovich, Lakzian, and Tian
Workshops funded by this grant:
2020 VWRS: Virtual Workshop on Ricci and Scalar Curvature (postdoc stipends)
Filling Volumes, Geodesics, and Intrinsic Flat Convergence at Yale (travel)
CUNY Metric Geometry Workshop January 14, 2019 (travel)
NSF Grant DMS - 1309360 (2013-2018)
Applications of the Convergence of Riemannian Manifolds to General Relativity
The PI will apply Intrinsic Flat convergence between Riemannian manifolds to better understand how close space-like manifolds studied in Mathematical General Relativity approximate the standard well known models. The Intrinsic Flat distance, first introduced by the PI with Stefan Wenger using methods of Ambrosio-Kirchheim, is particularly well-suited to some questions arising in General Relativity because increasingly thin gravity wells disappear under this convergence. In joint work with Dan Lee, the PI has shown that spherically symmetric Riemannian manifolds with increasingly small ADM mass converge to Euclidean space in the pointed intrinsic flat sense, and here proposes to generalize this result. In addition, the PI proposes to develop two new notions of convergence: the first will allow mathematicians to study Lorentzian manifolds directly, and the second will prevent regions from disappearing due to orientation and cancellation. Both notions are specifically adapted to questions arising in General Relativity.
Einstein's Theory of General Relativity describes how space is curved by gravity. Even within our own solar system, when computing the trajectories of spacecraft heading to Mars, engineers must take into account the curvature caused by the mass of the planets and the sun. Each planet forms a gravity well. If the mass of a planet is small, one would like to know in what sense the space around it is almost flat. In fact, the space around a planet of arbitrarily small mass could be very highly curved (and have a very deep but thin gravity well). In joint work with Dr. Stefan Wenger, the PI has developed a new means of measuring the closeness between curved spaces and, in joint work with Dr. Dan Lee, she has estimated how close the space around a single perfectly spherical planet is to Euclidean space. In this project, she will develop tools allowing one to better understand the space around groups of planets which are not perfect spheres: like the ones in our own solar system.
Results funded in part by this grant:
C Sormani and I Stavrov``Geometrostatic manifolds of small ADM Mass" to appear in Communications on Pure and Applied Mathematics (arxiv preprint) 42 pp
C Sormani “Spacetime Intrinsic Flat Convergence” Oberwolfach Report 2018 Mathematical General Relativity (arxiv)
J Nunez-Zimbron and R Perales “A generalized tetrahedral property for spaces with conical singularities” (arxiv)
C Sormani,"Intrinsic Flat Arzela-Ascoli Theorems" Communications in Analysis and Geometry Vol. 27, No 1, 2019 (citations) (arxiv preprint) 44 pages,
M Jaramillo, R Perales, P Rajan, C Searle, A Siffert ”Alexandrov Spaces with Integral Current Structure” (arxiv) to appear in Communications in Analysis and Geometry
C Sormani, ``Scalar Curvature and Intrinsic Flat Convergence" (arxiv preprint) (reprint) Chapter 9 in Measure Theory in Non-Smooth Spaces, edited by Nicola Gigli, De Gruyter Press, (2017) pp 288-338.
P LeFloch and C Sormani ``The Nonlinear stability of rotationally symmetric spaces with low regularity" Journal of Functional Analysis, 268 (2015) no. 7 2005-2065. (arxiv)
C-Y Lin and C Sormani, "Bartnik's Mass and Hamilton's Modified Ricci Flow" Annales Henri Poincare October 2016, Volume 17, Issue 10, pp 2783–2800 (arxiv preprint) (reprint) (citations)
C Sormani and C Vega, "Null distance on a spacetime" Classical and Quantum Gravity, 33 (2016) no 8 29 pages (arxiv preprint) (reprint)
L-H Huang, D A Lee and C Sormani, "Intrinsic flat stability of the Positive Mass Theorem for graphical hypersurfaces in Euclidean space" Journal fur die Riene und Angewandte Mathematik, Crelle's Journal, Vol. 727 (2017), 269-299. (arxiv preprint) (reprint) (citations)
J Park, W Tian, and C Wang ``A Compactness Theorem for sequences of rotationally symmetric Riemannian manifolds with positive scalar curvature" arxiv to appear in PAMQ
NSF Grant DMS - 1006059 (2010-2014)
Convergence of Riemannian Manifolds
Over the past three decades mathematicians have gained deep new insight into Riemannian manifolds by applying the methods of Gromov-Hausdorff, Lipschitz and metric measure convergence. Such techniques have been particularly useful for studying manifolds with bounds on sectional or Ricci curvature, but a new weaker notion of convergence is needed to understand manifolds without such strong conditions. Recently the PI and Dr. Wenger have applied work of Drs. Ambrosio and Kirchheim to introduce a new distance between manifolds: the intrinsic flat distance. While the convergence is weaker than previous forms of convergence, the limit spaces, called Integral Current Spaces, are countably H^m rectifiable. Applying work of Cheeger-Colding, Gromov and Perelman, the PI and Dr. Wenger have shown that the Gromov-Hausdorff and intrinsic flat limits of manifolds with nonnegative Ricci curvature agree. However, in general the limit spaces are different and sequences which do not converge in the Gromov-Hausdorff sense may still converge in the intrinsic flat sense. The PI will study the properties of these limit spaces under a variety of conditions on the sequence of manifolds and prove stability theorems under these weaker conditions. In particular the PI proposes to improve her results on the stability of the Friedmann model.
The spacelike universe is described in Friedmann cosmology as an isotropic three dimensional Riemannian manifold that expands in time starting from the initial Big Bang. In reality the universe is not isotropic because it is bent by gravity in a nonuniform way. Weak gravitational lensing (due to dust) and strong gravitational lensing (due to massive objects) has been observed by the Hubble to distort regions of space. The universe is thus, at best, close to the Friedmann model in some sense. In prior work, under strong assumptions, the PI has shown that a Riemannian manifold which is almost isotropic (in a way which allows for weak gravitational lensing and localized strong gravitational lensing) is close to the Friedmann model in the Gromov-Hausdorff sense. This is proven by studying the Gromov-Hausdorff limits of increasingly isotropic manifolds. Now the PI proposes to prove that under weaker assumptions, the universe is close to the Friedmann model in the intrinsic flat sense by studying the intrinsic flat limits of Riemannian manifolds. Using the intrinsic flat distance will not only allow for weak and strong gravitational lensing but also allow for the possible existence of wormholes.
Results funded in part by this grant:
J Basilio, J Dodziuk, and C Sormani, ``Sewing Riemannian Manifolds with Positive Scalar Curvature" to appear in Journal of Geometric Analysis (arxiv preprint) (reprint) 46 pages (DMS 1006059)
J Basilio, D Kazaras, and C Sormani ``An intrinsic flat limit of Riemannian manifolds with no geodesics''. (arxiv)
J Portegies and C Sormani, "Properties of the Intrinsic Flat Distance" (arxiv preprint) (8 citations)(reprint) solicited for the Volume in Honor of Yuri Burago, editted by D Burago and S Buyalo, Algebra i Analiz Issue 3, Volume 29 (2017), pages 70-143). (DMS 1006059)
Z Sinaei and C Sormani, "Intrinsic Flat Convergence of Covering Spaces" Geometriae Dedicata, Vol 184 (2016) pages 83–114 (arxiv preprint) (reprint)
and more that appeared after the final report. See also 2013 Research Description.
NSF Grant DMS - 0102279 (2001-2006)
The Topology of Open Manifolds with Nonnegative Ricci Curvature
Abstract:
Dr. Sormani proposes to study the topology of complete manifolds with nonnegative Ricci curvature and their limit spaces. In particular she plans to investigate various approaches to Milnor's conjecture that the fundamental group of an open manifold with nonnegative Ricci curvature is finitely generated. She also plans to study the higher dimensional homology of these spaces. Techniques which will be employed involve Gromov-Hausdorff limits, the almost rigidity theory of Cheeger-Colding, and Busemann functions. In particular, the properness of Busemann functions on these manifolds will be investigated. It should be noted that there are direct applications of this project to the theory of topological censorship in general relativity. The condition of nonnegative Ricci curvature on space-time is called the null energy condition and it arises in the Einstein equation.
Roughly speaking, Dr. Sormani proposes to study the existence and prevalence of holes in a space which has no boundary, extends to infinity and has a condition imposed upon the way in which it can bend. The universe we live in is such a space. Simpler examples are cylinders (i.e. tubes) and paraboloids (i.e. bowls). The cylinder has a hole but the paraboloid does not. The spaces studied in this project are of arbitrary dimension and so the holes come in various dimensions as well. The universe is one such higher dimensional space and its holes, which may or may not exist, are often called wormholes. By furthering our understanding of this geometric problem, it is hoped that we will further our understanding of the universe.
Results Funded in part by this grant:
C. Sormani, "Convergence and the Length Spectrum" Advances in Mathematics, Volume 213, Issue 1, 1 August 2007, Pages 405-439. ( arxiv preprint Feb 2006) (reprint)
C. Sormani and Guofang Wei, "The Covering Spectrum of a Compact Length Space" Journal of Differential Geometry, Vol 67 (2004) 35-77. (Arxiv posting Nov 2003) (reprint) [An Erratum in JDG 74 (2006) 523-3: Example 10.3 is not correct. All else OK.]
C. Sormani, "Friedmann Cosmology and Almost Isotropy" Geometric and Functional Analysis, Vol. 14 (2004) 853-912. (Arxiv posting Feb 2003, Mar 2004) (reprint)
C. Sormani and Guofang Wei, "Universal Covers for Hausdorff Limits of Noncompact Spaces" Transactions of the American Mathematical Society 356 (2004), no. 3 pp. 1233-1270. (Arxiv preprint July 2002) (reprint)
C. Sormani and Guofang Wei,"Hausdorff Convergence and Universal Covers" Transactions of the American Mathematical Society 353 (2001), no. 9, 3585--3602. pdf copy (preprint Aug 2000) (reprint)
Zhongmin Shen and Christina Sormani. "The Codimension One Homology of a Complete Manifold with Nonnegative Ricci Curvature" American Journal of Mathematics 123 (2001), no. 3, 515--524. (reprint) (preprint Sep 1999, Nov 2000)
C Sormani, "On Loops Representing Elements of the Fundamental Group of a Complete Manifold with Nonnegative Ricci Curvature" Indiana Journal of Mathematics, 50 (2001) no. 4, 1867-1883. (reprint) (preprint April 1999)
and more that appeared after the final report
NSF Doctoral Fellowship (1991-1996)
C Sormani, "The Rigidity and Almost Rigidity of Manifolds with Lower Bounds on Ricci Curvature and Minimal Volume Growth" Communications in Analysis and Geometry. Vol 8, No. 1, 159-212, January 2000. (preprint May 1996, March 1999) (reprint)
C Sormani, "Busemann Functions on Manifolds with Lower Ricci Curvature Bounds and Minimal Volume Growth" Journal of Differential Geometry 48, (1998) 557-585. (reprint) (preprint May 1996, Nov 1997) (see also my doctoral dissertation)