Gregory R. Chambers
Since September 2014, I have been an L. E. Dickson Instructor at the University of Chicago. I completed my PhD at the University of Toronto in June 2014 under the supervision of Alexander Nabutovsky and
Regina Rotman.
My research is in metric geometry and geometric analysis. I am particularly interested in quantitative topology, min-max theory, isoperimetric inequalities, stability estimates for geometric inequalities, and symmetrizations and their applications.
My CV can be downloaded here.
Contact Information
Mailing Address:
Department of Mathematics
University of Chicago
5734 S University Ave
Chicago, IL
60637
Phone: (773) 702-9177
Email: chambers@math.uchicago.edu
Office: Eckhart 325
Teaching
I am currently teaching MAT205 Section 45 (Accelerated). All course information is available on Chalk.
Publications
Submitted
- Area of convex disks, with C. Croke, Y. Liokumovich, and H. Wen
arXiv:1701.06594
This paper considers metric balls \(B(p,R)\) in two dimensional Riemannian manifolds when \(R\) is less than half the convexity radius. We prove that \(Area(B(p,R))\geq\frac{8}{\pi}R^2\). This inequality has long been conjectured for \(R\) less than half the injectivity radius. This result also yields the upper bound \(\mu_2(B(p,R))\leq 2(\frac{\pi}{2 R})^2\) on the first nonzero Neumann eigenvalue \(\mu_2\) of the Laplacian in terms only of the radius. This has also been conjectured for \(R\) up to half the injectivity radius.
- Quantitative nullhomotopy and rational homotopy type, with F. Manin, and S. Weinberger
arXiv:1611.03513
In [GrOrang], Gromov asks the following question: given a nullhomotopic map \(f:S^m \rightarrow S^n\) of Lipschitz constant \(L\), how does the Lipschitz constant of an optimal nullhomotopy of \(f\) depend on \(L\), \(m\), and \(n\)? We establish that for fixed \(m\) and \(n\), the answer is at worst quadratic in \(L\). More precisely, we construct a nullhomotopy whose thickness (Lipschitz constant in the space variable) is \(C(m,n)(L+1)\) and whose width (Lipschitz constant in the time variable) is \(C(m,n)(L+1)^2\).
More generally, we prove a similar result for maps \(f : X \rightarrow Y\) for any compact Riemannian manifold \(X\) and \(Y\) a compact simply connected Riemannian manifold in a class which includes complex projective spaces, Grassmannians, and all other simply connected homogeneous spaces. Moreover, for all simply connected \(Y\), asymptotic restrictions on the size of nullhomotopies are determined by rational homotopy type.
- Quantitative null-cobordism, with D. Dotterrer, F. Manin, and S. Weinberger
arXiv:1610.04888
For a given null-cobordant Riemannian \(n\)-manifold, how does the minimal
geometric complexity of a null-cobordism depend on the geometric complexity of
the manifold? In [GrQHT], Gromov conjectured that this dependence should
be linear. We show that it is at most a polynomial whose degree depends on
\(n\).
This construction relies on another of independent interest. Take \(X\) and \(Y\)
to be sufficiently nice compact metric spaces, such as Riemannian manifolds or
simplicial complexes. Suppose \(Y\) is simply connected and rationally homotopy
equivalent to a product of Eilenberg-MacLane spaces: for example, any simply
connected Lie group. Then two homotopic \(L\)-Lipschitz maps \(f,g:X \to Y\) are
homotopic via a \(CL\)-Lipschitz homotopy. We present a counterexample to show
that this is not true for larger classes of spaces \(Y\).
- Existence of minimal hypersurfaces in complete manifolds of finite volume, with Y. Liokumovich
arXiv:1609.04058
We prove that every complete non-compact manifold of finite volume contains a (possibly non-compact) minimal hypersurface of finite volume.
- A note on the affine-invariant plank problem
arXiv:1604.00456
Suppose that \(C\) is a bounded convex subset of \(\mathbb{R}^n\), and that \(P_1,\dots,P_k\) are planks which cover \(C\) in respective directions \(v_1,\dots,v_k\) and with widths \(w_1,\dots,w_k\). In 1951, Bang conjectured that $$\sum_{i=1}^k \frac{w_i}{w_{v_i}(C)} \geq 1,$$
generalizing a previous conjecture of Tarski. Here, \(w_{v_i}(C)\) is the width of \(C\) in the direction \(v_i\). In this note we give a short proof of this conjecture under the assumption that, for every \(m\) with \(1 \leq m \leq k\), \(C \setminus \bigcup_{i=1}^m P_i\) is a convex set.
Published
- Monotone homotopies and contracting discs on Riemannian surfaces, with R. Rotman
Journal of Topology and Analysis, to appear, arXiv:1311:2995
We prove a "gluing" theorem for monotone homotopies; a monotone
homotopy is a homotopy through simple contractible closed curves which
themselves are pairwise disjoint. We show that two monotone homotopies
which have appropriate overlap can be replaced by a single monotone homotopy.
The ideas used to prove this theorem are used in (10) to prove an
analogous result for cycles, which forms a critical step in their proof of the
existence of minimal surfaces in complete non-compact manifolds of finite
volume.
We also show that, if monotone homotopies exist, then fixed point contractions
through short curves exist. In particular, suppose that \(\gamma\) is a simple
closed curve of a Riemannian surface, and that there exists a monotone contraction
which covers a disc which \(\gamma\) bounds consisting of curves of length
\( \leq L\). If \(\epsilon > 0\) and \(q \in \gamma\), then there exists a homotopy that contracts \(\gamma\) to
\(q\) over loops that are based at \(q\) and have length bounded by \(3L + 2d + \epsilon\),
where \(d\) is the diameter of the surface. If the surface is a disc, and if \(\gamma\) is the
boundary of this disc, then this bound can be improved to \(L + 2d + \epsilon\).
- Optimal sweepouts of a Riemannian 2-sphere, with Y. Liokumovich
Journal of the European Mathematics Society, to appear, arXiv:1411:6349
Given a sweepout of a Riemannian 2-sphere which is composed of curves of length less than \(L,\) we construct a second sweepout composed of curves of length less than \(L\) which are either constant curves or simple curves.
This result, and the methods used to prove it, have several consequences; we answer a question of M. Freedman concerning the existence of min-max embedded geodesics, we partially answer a question due to N. Hingston and H.-B. Rademacher, and we also extend the results of (2) concerning converting homotopies to isotopies in an effective way.
- Proof of the Log-Convex Density Conjecture
Journal of the European Mathematics Society, to appear, arXiv:1311.4012
We completely characterize isoperimetric regions in \(\mathbb{R}^n\) with density \(e^h\), where \(h\) is convex, smooth, and radially symmetric. In particular, balls around the origin constitute isoperimetric regions of any given volume, proving the Log-Convex Density Conjecture due to Kenneth Brakke.
- Ergodic properties of folding maps on spheres, with A. Burchard and A. Dranovski
Discrete and Continuous Dynamical Systems - Series A 37(3):1183-1200 (2017), DOI 10.3934/dcds.2017049, arXiv:1509.02454
We consider the trajectories of points on the \((d-1)\)-dimensional sphere under certain folding maps associated with reflections. The main result gives a condition for a collection of such maps to produce dense trajectories. At least \(d+1\) directions are required to satisfy the conditions.
- Isoperimetric regions in \(\mathbb{R}^n\) with density \(r^p\), with W. Boyer, B. Brown, A. Loving, and S. Tammen
Analysis and Geometry in Metric Spaces 4(1):236-265 (2016), DOI 10.1515/agms-2016-0009, arXiv:1504:01720
We show that the unique isoperimetric hypersurfaces in \(\mathbb{R}^n\) with density \(r^p\) for \(n \geq 3\) and \(p>0\) are spheres that pass through the origin.
- Splitting a contraction of a simple curve traversed \(m\) times, with Y. Liokumovich
Journal of Topology and Analysis (2016), DOI 10.1142/S1793525317500157, arXiv:1510.03445
Suppose that \(M\) is a 2-dimensional oriented Riemannian manifold, and let \(\gamma\) be a simple closed curve on \(M\). Let \(\gamma^m\) denote the curve formed by tracing \(\gamma\) \(m\) times. We prove that if \(\gamma^m\) is contractible through curves of length less than \(L\), then \(\gamma\) is contractible through curves of length less than \(L\). In the last section we state several open questions about controlling length and the number of self-intersections in homotopies of curves on Riemannian surfaces.
- Geometric stability of the Coulomb energy, with A. Burchard
Calculus of Variations and PDE 54(3):3241-3250 (2015), DOI 10.1007/s00526-015-0900-8, arXiv:1407.1918
The Coulomb energy of a charge that is uniformly distributed on some set is maximized (among sets of given volume) by balls. It is shown here that near-maximizers are close to balls.
- Perimeter under multiple Steiner symmetrizations, with A. Burchard
Journal of Geometric Analysis (2015) 25:871, DOI 10.1007/s12220-013-9448-z, arXiv:1209.4521
Steiner symmetrization along \(n\) linearly independent directions transforms every compact subset of \(\mathbb{R}^n\) into a set of finite perimeter.
- Converting homotopies to isotopies and dividing homotopies in half in an effective way, with Y. Liokumovich
Geometric and Functional Analysis (2014) 24:1080, DOI 10.1007/s00039-014-0283-6, arXiv:1311.0779
We prove two theorems about homotopies of curves on 2-dimensional Riemannian manifolds. We show that, for any \(\epsilon > 0\), if two simple closed curves are homotopic through curves of bounded length \(L\), then they are also isotopic through curves of length bounded by \(L + \epsilon\). If the manifold is orientable, then for any \(\epsilon > 0\) we show that, if we can contract a curve gamma traversed twice through curves of length bounded by \(L\), then we can also contract gamma through curves bounded in length by \(L + \epsilon\).
Our method involves cutting curves at their self-intersection points and reconnecting them in a prescribed way. We consider the space of all curves obtained in this way from the original homotopy, and use a novel approach to show that this space contains a path which yields the desired homotopy.