Applied Mathematics Seminar

Speaker: 
Nicholas Marshall
Event time: 
Monday, September 11, 2017 - 3:45pm to 5:00pm
Location: 
LOM 206 See map
12 Hillhouse Avenue
New Haven , CT 06511
Event description: 

“Domains capturing many lattice points: from triangles to convex domains”

 

We discuss two problems about lattice points. (1) Motivated by problems in

mathematical physics, Antunes & Freitas proved in 2012 that among all 

axis-parallel ellipses in the plane centered at the origin and having fixed 

area, the one containing the most lattice points asymptotically converges

to a circle as area becomes large. We give a far-ranging generalization to convex bodies with

nonvanishing Gauss curvature. The methods are based on Fourier analysis.

(2) Secondly, in joint work with S. Steinerberger, we answer a question

posed by Laugesen & Liu about right-angled triangles in the plane capturing

many lattice points. Instead of a single optimal shape, there is a countably 

infinite limit set of triangles each of which capture a maximal number of 

positive lattice points for arbitrarily large areas. Moreover, this limit set is 

fractal with Minkowski dimension at most 3/4. The proof involves elements

from combinatorics and dynamical systems.

Event Type: 
Applied Mathematics