Christopher Leininger
Professor, Rev. John A. Zahm, C.S.C., Collegiate Professor in the Department of Mathematics
Education
Ph.D., Mathematics, The University of Texas, Austin, TX, May 2002
B.S. in Mathematics, Ball State University, Muncie, IN, May 1997
Research Groups
Algebra and Algebraic Geometry, Topology
Research Area
Low dimensional geometry and topology, and geometric group theory
Bio
Research Statement:
My research involves the interplay between geometry, topology, and group theory, particularly in low dimensions. One facet of my work centers around coarse negative curvature features of surface bundles and their connection to geometric properties of the monodromy homomorphism into the mapping class group. Studying the geometry of the mapping class group has naturally led me to investigate a variety of geometric structures on surfaces. Some of the geometric structures I have studied connect to other areas of mathematics, including the dynamics of polygonal billiards.
My work on surface bundles is often motivated by the theory of hyperbolic surface bundles over the circle, and another prong of my research involves the study of geometric 3-manifolds, more generally. Free-by-cyclic groups share many properties with surface bundles over the circle, and my research also pushes these connections in new directions. Many features of the classical work on surface bundles over the circle have fascinating analogues in the setting of infinite type surfaces. I am particularly interested in associated geometric structures on infinite type surfaces, special subgroups of their mapping class groups, and relationships with hyperbolic geometry in dimension 3, particularly via foliations.
Selected Publications:
- A.E. Kent and C.J. Leininger, Atoroidal surface bundles, to appear, Annals of Math.
- M. Duchin, V. Erlandsson, C.J. Leininger, C. Sadanand, You can hear the shape of a billiard table: Symbolic dynamics and rigidity for flat surfaces, Comment. Math. Helv., 96 (2021), no. 3.
- M. Bestvina, K. Bromberg, A.E. Kent, and C.J. Leininger, Undistorted purely pseudo-Anosov groups, J. Reine Angew. Math. (Crelle's Journal) 760 (2020)
- M. Duchin, C.J. Leininger and K. Rafi, Length spectra and degeneration of flat metrics, Invent. Math., 182 (2010) No. 2
- C.J. Leininger and S. Schleimer, Connectivity of the space of ending laminations, Duke Math. J. 150 (2009) no. 3.
Email: cleining@nd.edu
Fax: 574-631-6579
Office: 110 Hayes-Healy Center