The main focus of my recent research is magnitude of metric spaces and
related aspects of metric geometry. Tom Leinster maintains a bibliography of
works on this subject.
Here is a list of blog posts about
magnitude, diversity, and closely related topics at the n-Category
Café.
The bulk of my past research has been in probability, especially
random matrix theory and aspects related to convex geometry.
Magnitude and Holmes–Thompson intrinsic volumes of convex
bodies. Canadian Mathematical Bulletin 66 (2023) no. 3, pp. 854–867.
arXiv
/ published version
On the magnitude and intrinsic volumes of a convex body in
Euclidean space. Mathematika 66 (2020) no. 2, 343–355.
arXiv
/ published version
The magnitude of a metric space: from category theory to geometric
measure theory (with T. Leinster). Measure Theory in Non-Smooth Spaces, 156–193,
DeGruyter Open, 2017.
arXiv
/ published version
Maximizing diversity in biology and beyond (with T. Leinster). Entropy 18 (2016) no. 3, article 88.
arXiv
/ published version Note: Due to the journal's editorial
policy, the numbering of theorems, examples, etc. in this paper was
changed between the arXiv preprint and the published
version.
Magnitude, diversity, capacities, and dimensions of metric spaces. Potential Anal. 42 (2015) no. 2, 549–572.
arXiv
/ published version
Positive definite metric spaces. Positivity 17 (2013) no. 3, 733–757.
arXiv
/
published
version Note: The last paragraph of the published
version of misstates the consequences for magnitude dimension of
Theorems 4.4 and 4.5. The discussion around those results has been
revised in the arXiv version to clarify this matter.
Random matrix theory
Fluctuations of the spectrum in rotationally invariant random matrix ensembles (with E. Meckes). Random Matrices Theory Appl. 10 (2021) no. 3, Article No. 2150025.
arXiv
/
published version
Random matrices with prescribed eigenvalues and expectation values for random quantum states (with E. Meckes). Trans. Amer. Math. Soc. 373 (2020) no. 7, 5141–5170.
arXiv
/
published version
A sharp rate of convergence for the empirical spectral measure of a random unitary matrix (with E. Meckes). Zapiski Nauchnykh Seminarov POMI 457 (2017), 276–285 /
Journal of Mathematical Sciences.
arXiv
/
published version
Rates of convergence for empirical spectral measures: a soft approach (with E. Meckes). Convexity and Concentration, 157–181, IMA Volumes
in Mathematics and its Applications 161, Springer, 2017.
arXiv
/
published
version
A rate of convergence for the circular law for the complex
Ginibre ensemble (with E. Meckes). Ann. Fac. Sci. Toulouse Math. Series 6, 24 (2015) no. 1, 93–117.
arXiv
/ published version
Spectral measures of powers of random matrices (with
E. Meckes). Electron. Commun. Probab. 18 (2013) no. 78, 1–13.
arXiv
/
published version
Concentration and convergence rates for spectral measures of random matrices (with
E. Meckes). Probab. Theory Related Fields 156 (2013), 145–164.
arXiv
/
published
version
The spectra of random abelian G-circulant matrices. ALEA Lat. Am. J. Probab. Math. Stat. 9 (2012) no. 2, 435–450.
arXiv
/
published
version
Concentration for noncommutative polynomials in random matrices (with
S. Szarek). Proc. Amer. Math. Soc. 140 (2012), 1803–1813.
arXiv
/
published version
Another observation about operator compressions (with
E. Meckes). Proc. Amer. Math. Soc. 139 (2011), 1433–1439.
arXiv /
published version
Some results on random circulant matrices. High Dimensional Probability V: The Luminy Volume, 213–223,
IMS Collections 5, Institute of Mathematical Statistics, Beachwood, OH,
2009.
arXiv /
published version
On the spectral norm of a random Toeplitz matrix. Electron. Commun. Probab. 12 (2007), 315–325.
arXiv
/
published version
Concentration of norms and eigenvalues of random matrices. J. Funct. Anal. 211 (2004) no. 2, 508–524.
arXiv
/
published
version
Probability in convexity and convexity in probability
On the equivalence of modes of convergence for log-concave
measures (with E. Meckes). Geometric Aspects of Functional Analysis, 385–394,
Lecture Notes in Math. 2116, Springer, 2014.
arXiv
/ published version
Gaussian marginals of convex bodies with symmetries. Beiträge Algebra Geom. 50 (2009) no. 1, 101–118.
arXiv /
published
version
The central limit problem for random vectors with symmetries (with
E. Meckes). J. Theoret. Probab. 20 (2007), 697–720.
arXiv /
published
version Note: The arXiv preprint contains a section on background
on Stein's method which does not appear in the published version. As
a result, some theorem numbers are different in the two versions.
Some remarks on transportation cost and related inequalities. Geometric Aspects of Functional Analysis, 237–244,
Lecture Notes in Math. 1910, Springer, 2007.
arXiv /
published version
Sylvester's problem for symmetric convex bodies and related problems. Monatsh. Math. 145 (2005) no. 4, 307–319.
arXiv
/
published version Note: The published version contains several
references to the literature which are missing in the arXiv preprint, and
has improved proofs of Propositions 13 and 16.
Volumes of symmetric random polytopes. Arch. Math. 82 (2004) no. 1, 85–96.
arXiv
/
published version
Other
Quenched central limit theorem in a corner growth setting (with H.C. Gromoll
and L. Petrov). Electron. Commun. Probab. 23 (2018), paper no. 101, 1–12.
arXiv
/
published version