Presented By: Probability and Analysis Seminar - Department of Mathematics
Strongly log-concave measures and lattices
Brayden Letwin
Abstract: Let L be a lattice in R^n. Suppose that L has the property that the determinant of every
non-zero sublattice L' of L is larger than one. We prove that there exists a symmetric random vector X on R^n
with a smooth, strictly positive density f on R^n simultaneously enjoying the following two properties:
a) f is strongly log-concave.
b) X + L is uniform on R^n/L.
Moreover, the existence of such a density f on R^n characterizes this determinant condition imposed on L. This construction connects several problems
in lattice theory to established results in convex geometry. We discuss a range of applications. This is joint work with Victor Reis and Thomas Rothvoss.
non-zero sublattice L' of L is larger than one. We prove that there exists a symmetric random vector X on R^n
with a smooth, strictly positive density f on R^n simultaneously enjoying the following two properties:
a) f is strongly log-concave.
b) X + L is uniform on R^n/L.
Moreover, the existence of such a density f on R^n characterizes this determinant condition imposed on L. This construction connects several problems
in lattice theory to established results in convex geometry. We discuss a range of applications. This is joint work with Victor Reis and Thomas Rothvoss.
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