Overview

Catherine Searle is a professor of mathematics in the Department of Mathematics, Statistics, and Physics. After obtaining her PhD in mathematics at the University of Maryland at College Park in 1992 under the direction of Professor Karsten Grove, she started her research and teaching career as an assistant professor at the CINVESTAV-IPN in Mexico City, Mexico. She then worked at the Mathematics Institute of the UNAM in Cuernavaca, Morelos from 1996–2011. She was a visiting professor at Oregon State University from 2012–2014, and then joined the faculty at Wichita State University in 2014, where she was promoted to full professor in 2019. 

Catherine Searle works in Differential Geometry with an emphasis on Comparison Geometry. Her research has been focussed on positively and non-negatively curved Riemannian manifolds, which admit “large” isometric group actions, where “large” can be defined in a number of ways. The existence of an isometric group action G on a metric space X leads to information about the space itself and can be used both as a tool to identify the space and as a means to improve the metric on that space. More recently she has been studying isometric group actions in these two contexts, namely, as a tool to identify both Riemannian manifolds and Alexandrov spaces with a lower curvature bound and as a tool to improve the metric on a Riemannian manifold with a G-invariant metric. 

Information

Academic Interests and Expertise
  • Differential Geometry
  • Alexandrov Geometry
  • Transformation Groups
Areas of Research Interest

Catherine Searle studies manifolds and singular spaces with curvature lower bounds and symmetries from two different points of view:  

(1) using continuous and discrete symmetries to better understand the topology of positively, non-negatively, and almost non-negatively curved Riemannian manifolds and Alexandrov spaces and

(2) finding new examples of Riemannian manifolds  of positive Ricci curvature, almost non-negative sectional curvature, and positive intermediate Ricci curvature, using  symmetries and topology as tools to do so. 

Areas of Teaching Interest
  • Geometry
  • Topology
  • Algebra
Publications

Articles 
1.Positively curved manifolds with discrete abelian symmetries, with L. Kennard and E. Khalili Samani, available at arXiv:2110.13345 (2021). 
2. Positive (p, n)-intermediate scalar curvature and cobordism, with M. Burkemper and M. Walsh, available at arXiv:2110.12069 (2021). 
3. Odd dimensional GKM manifolds of non-negative curvature, with C. Escher and O. Goertsches, to appear in Int. Math. Res. Not., available at
arXiv:1912.02466 (2019) 
4. Almost non-negatively curved 4-manifolds with circle symmetry, with J. Harvey, Proceedings of the AMS, 148, no. 11 (2020), 4933–4950 https:/
doi.org/10.1090/proc/15093. 
5. Almost torus manifolds of non-negative curvature, with Z. Dong, and C. Escher, available at arXiv:1811.01493v1 (2018). 
6. Positively curved Alexandrov spaces with circle symmetry in dimension 4, with J. Harvey, to appear in Documenta Math., available at
arXiv:math.DG/1805.09362v1 (2018). 
7. Alexandrov Spaces with Integral Current Structure, with M. Jaramillo, R. Perales, P. Rajan, A. Siffert, accepted for publication in Communications
in Analysis and Geometry, available at arXiv:1703.08195 (2017). 
8. Non-negatively curved 6-manifolds with almost maximal symmetry rank, with C. Escher, Journal of Geometric Analysis, doi.org/10.1007
s12220-018-0026-2 (2018). 
9. Torus actions, maximality and non-negative curvature, with C. Escher, J. Reine Angew. Math., doi.org/10.1515/crelle-2021-0035 (2021). 
10. Orientation and symmetries of Alexandrov spaces with applications in positive curvature, with J. Harvey, Journal of Geometric Analysis, 27 (2) ,
1636–1666 (2017). 
11. Regularization via Cheeger Deformations, with P. Solorzano, F. Wilhelm, Annals of Global Analysis and Geometry, doi:10.1007
s10455-015-9471-3, pp. 1–9 (2015). 
12. How to lift positive Ricci curvature, with F. Wilhelm, Geometry and Topology, 19 (3), pp. 1409-1475 (2015). 
13. An introduction to isometric group actions with applications to spaces with curvature bounded below, Geometry of Manifolds of Non-negative
Sectional Curvature, Lecture Notes in Mathematics 2110, DOI 10.1007/978-3-319-06373-7_3, Springer International (2014). 
14. Non-negatively curved 5-manifolds of almost maximal symmetry rank, with F. Galaz-Garcia, Geometry & Topology 18 pp. 1397–1435 (2014). 
15. Initial Structure of Cetyltrimethylammonium Bromide Micelles in Aqueous Solution from Molecular Dynamics Simulations, with G. Fernandez
Cata, H. Comas Rojas, A. Perez Gramatges, C. Zicovich-Wilson, L.J. Alvarez, Soft Matter, vol. 7, pp. 8508–8515 (2011).

16. Cohomogeneity one Alexandrov spaces, with F. Galaz-Garcia, Transformation Groups, Vol. 16, No. 1, pp. 91–107 (2011). 
17. Low dimensional manifolds with non-negative curvature and maximal symmetry rank, with F. Galaz-Garcia, Proceedings of the American
Mathematical Society, Volume 139, Number 7, pp. 2559–2564 (2011). 
18. Diameters of 3-sphere quotients, with W. Dunbar, S. Greenwald, J. McGowan, Differential Geometry and its Applications, vol 27, no. 2, pp. 307
319 (2009). 
19. How Tightly Can You Fold a Sphere?, with J. McGowan, Differential Geometry and its Applications, v. 22, no. 1, pp. 81–104 (2005). 
20. The Hopf Conjecture for Manifolds with Low Cohomogeneity or High Symmetry Rank, with T. Puttmann, Proceedings of the AMS, vol. 130, no.
1, pp. 163–166 (2002). 
21. Global G-Manifold Resolutions and Reductions, with K. Grove, Annals of Global Analysis and Geometry, vol. 18, pp 437–446 (2000). 
22. Differential Topological Restrictions by Curvature and Symmetry, with K. Grove, Journal of Differential Geometry, vol 47, pp. 530–559 (1997),
Correction, JDG, vol. 49, p. 205 (1998). 
23. On the Topology of Nonnegatively Curved Simply Connected 4-Manifolds with Continuous Symmetry, with D.G. Yang, Duke Mathematical
Journal, vol. 74, no. 2, pp. 547–556 (1994). 
24. Positively Curved Manifolds with Maximal Symmetry Rank, with K. Grove, Journal for Pure and Applied Algebra, vol. 91, pp. 137–142 (1994). 
25. Positively Curved Manifolds with Maximal Symmetry Rank, with K. Grove, Aportaciones Matematicas, serie: Comunicaciones 12, ISBN
968-36-3280-7, pp. 153–156 (1993). 
26. Cohomogeneity and Positive Curvature in Low Dimensions, Mathematische Zeitschrift, vol. 214, no.3, pp. 491–498 (1993), Corrigendum, Math.
Z., vol 226, pp. 165–167 (1997). 
27. Cohomogeneity One Manifolds of Positive Curvature, Aportaciones Matematicas, serie: Notas de Investigacion no. 8, ISBN 968-36-2793-5, pp.
109–110 (1992). 
28. Low-Dimensional Chaotic Attractors for an Unstable, Inhomogeneously-Broadened Single Mode Laser, with A.M. Albano, T.H. Chyba, S. Yong,
R.S. Gioggia, N.B. Abraham, Journal Opt. Sci. of America B, vol.125, pp. 47–55 (1985). 
29. Measurement of Impurities in a Neutral Beam by Laser-Induced Fluorescence, with C.F. Burrell, A.S. Schlachter, R.V. Pyle, Journal Vac. Sci.
Technol. A2 (2), pp. 708–709 (1984). 
30. Laser Induced Flourescence as Probe of Fast Impurity Atoms in a Neutral Beam, with C.F. Burrell, A.S. Schlachter, R.V. Pyle, Bulletin American
Physical Society, Series II, v. 28, no.8, p. 1119 (1983). 
Expository Articles 
1. An Introduction to Spherical Orbit Spaces, with J. McGowan, IJMMS, Vol. 32, no. 8, pp. 453–469 (2002). 
2. Algunos Ejemplos de Espacios Orbitales Esfericos de Cohomogeneidad 2, with J. McGowan, Divulgaciones Matematicas, Vol. 9, no. 1, pp. 1–23
(2001)

 

Grants
  •  PI for continuing NSF grant Curvature and Symmetry, DMS #2204324 for $261,767.00, 2022-2025
  •  PI for NSF grant Curvature and Symmetry, DMS #1906404 for $248,098.00, 2019-2022
  • PI for NSF grant Lower Curvature Bounds, Topology and Symmetries, DMS #1611780 for $150,000.00, 2016-2019
  • PI for  Simons Collaboration Grant for Mathematicians #355508,  Lower curvature bounds and symmetries, 2015--2018