In this talk, we begin by recalling Arnold’s geometric
formulation of hydrodynamics and then extend this framework to a
broader class of Hamiltonian systems, incorporating various PDEs
arising in mathematical physics. This motivates the study of...
We consider the symplectic area functional, constrained to loops
of vanishing Hamiltonian mean value: It has the same critical
points as the Rabinowitz action functional, and can be used to
define a similar Floer homology. In contrast to RFH, it...
Consider a collection of forms of odd degree with rational
coefficients. Birch proved in 1957 that if the number of variables
is sufficiently large, then the forms must have a nontrivial
rational zero. The bounds resulting from Birch's proof...
I will discuss questions pertaining to geometric unlikely
intersections and transcendence in the setting of torii in positive
characteristic. This is based on work in progress joint with Anup
Dixit, Philip Engel, and Ruofan Jiang.
Every o-minimal structure determines a collection of "tame" or
"definable" subsets of bbRn. We can then ask about the fragment of
complex geometry present in the structure: Which holomorphic
functions are definable, and which spaces are cut out by...
We present solutions to two problems on indefinite integral
ternary quadratic forms. The first, highlighted by Margulis in
1990, concerns the distribution of the ternary Markoff spectrum
associated with minima of forms. The second, initiated by...
In the theory of error-correcting codes, list decoding allows a
decoder to output a list of candidates when attempting to remove
noise from a corrupted input. The constructions and algorithms for
such list decodable codes has had numerous...
This talk, which is based on two joint works, one with Pedro
Salomão and Richard Siefring and another with Michael Hutchings and
Vinicius Ramos, revolves around the role that restrictions on the
knot types of periodic Reeb orbits imposed by the...
I will prove Gromov's conjecture that every 3-manifold of
positive scalar curvature contains a short closed geodesic. The
proof uses Min-Max theory of minimal surfaces and a combinatorial
version of mean curvature flow. Time permitting, I will...
