Jeffrey F. BrockDirector of the Initiative on Data Science, Professor of Mathematics
Jeffrey Brock's research focuses on low-dimensional geometry and topology, particularly on spaces with hyperbolic geometry (the most prevalent kind of non-Euclidean geometry). His joint work with R. Canary and Y. Minsky resulted in a solution to the `ending lamination conjecture' of W. Thurston, giving a kind of classification theorem for hyperbolic 3-dimensional manifolds that are topologically finite in terms of certain pieces of `mathematical DNA' called laminations. He received his undergraduate degree in mathematics at Yale University and his Ph.D. in mathematics from U.C. Berkeley, where he studied under Curtis McMullen. After holding postdoctoral positions at Stanford University and the University of Chicago, he came to Brown as an associate professor. He was awarded the Donald D. Harrington Faculty Fellowship to visit the University of Texas, and has had continuous National Science Foundation support since receiving his Ph.D. In 2008 he was awarded a John S. Guggenheim Foundation Fellowship. He and his wife Sarah live in Barrington, RI, with their two boys Elliot and Sam and their daughter Nora.
Asymptotics of Weil-Petersson geodesics I: ending laminations, recurrence and flows. (With Howard Masur and Yair Minsky). Geom. & Funct. Anal., 19 (2010), pp. 1229-1257.
Coarse and synthetic Weil-Petersson geometry: quasi-flats, geodesics, and relative hyperbolicity. (With Howard Masur). Geometry and Topology, 12 (2008), pp. 2453-2495.
Curvature and rank of Teichmüller space (with B. Farb). Amer. J. Math. 128 (2006), pp. 1-22.
Algebraic limits of geometrically finite manifolds are tame (with J. Souto). Geom. and Funct. Anal. 16 (2006)
On the density of geometrically finite Kleinian groups (with K. Bromberg). Acta Mathematica 192 (2004), pp. 33-93.
The classification of Kleinian surface groups II: the Ending Lamination Conjecture (with R. Canary and Y. Minsky, 2004). Submitted for publication.
The Weil-Petersson metric and volumes of 3-dimensional hyperbolic convex cores. J. Amer. Math. Soc. 16 (2003), pp. 495-535
Weil-Petersson translation distance and volumes of mapping tori. Comm. Anal. Geom. 11 (2003), pp. 987-999.
Tameness on the boundary and Ahlfors' measure conjecture (with K. Bromberg, R. Evans, and J. Souto). Publ. Math. I.H.É.S. 98 (2003), pp. 145-166
Pants decompositions and the Weil-Petersson metric. In Complex Manifolds and Hyperbolic Geometry, American Mathematical Society, Providence, RI, 2002, pp. 27-40.
Boundaries of Teichmüller spaces and end-invariants for hyperbolic 3-manifolds. Duke Math. J., 106 (2001), pp. 527-552.
Iteration of mapping classes and limits of hyperbolic 3-manifolds. Invent. Math. 143 (2001), pp. 523-570.
Continuity of Thurston's length function. Geom. & Funct. Anal. 10 (2000), pp. 741-797.
The standard double soap bubble in R2 uniquely minimizes perimeter (with F. Morgan et. al.). Pacific Journal of Mathematics, 159 (1993), no. 1, pp. 47-59.
A recent trend in geometry and topology is to develop models for geometric spaces. Such models sacrifice a certain degree of precision in the interest of capturing more large-scale structure. In a recent result of Brock with his collaborators, such models were used to classify all 'hyperbolic' three-dimensional spaces of infinite volume. This result solved the long-standing conjecture of W. Thurston that a certain piece of 'mathematical DNA' for a space determines its structure.
Brock's research has sought to develop a complete picture of the geometry and topology of hyperbolic 3-manifolds, namely, 3-dimensional spaces that admit expanding geometry as opposed to our more familiar Euclidean or flat geometry. In the 1970s, William Thurston gave a conjectural classification of such manifolds in terms of their topology and certain pieces of asymptotic data called ending laminations. Brock's recent collaboration with R. Canary (Michigan) and Y. Minsky (Yale) solved this conjecture, the so-called ending lamination conjecture , and led to the classification of all hyperbolic 3-manifolds with infinite volume and finite topological type. These results were described in the recent article Taming the Hyperbolic Jungle by Pruning its Unruly Edges in Science (2004) , by Dana Mackenzie.
Brock's more recent research has sought to employ the techniques used in the solution to the ending lamination conjecture to develop a more complete understanding of the geometry and structure of finite-volume hyperbolic 3-manifolds, which account for most compact 3-manifolds. It appears that the effective geometric estimates obtained in the infinite volume case lead to a new way to tie together geometric features of finite volume hyperbolic 3-manifolds with more classical topological invariants that have been the lifeblood of the field for many years.
PI, NSF Grant, DMS-1608759, 9/2016-9/2019. Volume and Combinatorics, in Hyperbolic Geometry. Amount: $239,997.
PI, NSF Grant, DMS-1207572, 6/2012-6/2016. Combinatorics, Models, and Bounds in Hyperbolic Geometry. Amount: $364,703.
Co-PI, NSF Mathematical Sciences Research Institute Grant, DMS-0931908. The Institute for Computational and Experimental Research in Mathematics. Amount: $15,495,376.
PI, NSF Grant DMS-0906229, 6/2009-6/2012 Teichmüller Theory, Kleinian Groups, and the Complex of Curves Amount $210,754.
Co-PI, NSF FRG (Focused Research Group) DMS-0553694 with co-PIs K. Bromberg (Utah), R. Canary (Michigan), and Y. Minsky (Yale), 6/2006-6/2009. Focused Research Group - Collaborative Research: Geometry and Deformation Theory of Hyperbolic 3-Manifolds . Amount: $255,420.
PI, NSF Grant DMS-0354288, 6/2005-6/2008 Effective Rigidity, Combinatorial Models and Parameter Spaces for Low-Dimensional Hyperbolic Manifolds . Amount: $294,624.
PI, NSF Grants DMS-0204454 (Chicago) and DMS-0354288 (Brown), 6/2002-6/2005 The Classification Problem for Hyperbolic 3-Manifolds . Amount: $101,601.
Postdoctoral Associate, NSF Research Grant DMS-0072133, (PI James Milgram, Co-PIs Steven Kerckhoff, Ralph Cohen, Stanford University), 2000-2002. Amount: $667,998.
PI, NSF Mathematical Sciences Postdoctoral Research Fellowship, Stanford University, 1997-2000. Algebraic and Geometric Limits of Hyperbolic 3-Manifolds and their Classification . Amount: $75,000.