A cornerstone result in geometry is the Szemerédi–Trotter
theorem, which gives a sharp bound on the maximum number of
incidences between m points and n lines in the real plane. A
natural generalization of this is to consider
point-hyperplane...
Triangulated surfaces are Riemann surfaces formed by gluing
together equilateral triangles. They are also the Riemann surfaces
defined over the algebraic numbers. Brooks, Makover, Mirzakhani and
many others proved results about the geometric...
In this talk, I will discuss lower bounds for a certain
set-multilinear restriction of algebraic branching programs. The
significance of the lower bound and the model is underscored by the
recent work of Bhargav, Dwivedi, and Saxena (2023), which...
Cayley graphs provide interesting bridges between graph theory,
additive combinatorics and group theory. Fixing an ambient finite
group, random Cayley graphs are constructed by choosing a
generating set at random. These graphs reflect interesting...
To any unital, associative ring R one may associate a family of
invariants known as its algebraic K-groups. Although they are
essentially constructed out of simple linear algebra data over the
ring, they see an extraordinary range of information...
I will discuss an adaptation of Gromov's ideal-valued measures
to symplectic topology. It leads to a unified viewpoint at three
"big fiber theorems": the Centerpoint Theorem in combinatorial
geometry, the Maximal Fiber Inequality in topology, and...
Subsets A of an abelian group with a small doubling |A+A|/|A|
have been extensively studied, and results of Freiman, Ruzsa and
Green give fundamental structural descriptions of such sets. These
have important applications across combinatorics and...
We consider an energy model for N ordered elastic membranes
subject to forcing and boundary conditions. The heights of the
membranes are described by real functions u_1, u_2,...,u_N, which
minimize an energy functional involving the Dirichlet...
We study multiply warped product geometries
MN:=Bn×Fn1×···×FnA
g = g_B + \sum_{a=1}^A v_a^2 g_{F^{n_a}} and show that for an
open set of initial data within multiply warped product geometries
the Ricci flow starting at any of those develops...
An old question of Poincaré concerns creating periodic orbits
via perturbations of a flow/diffeomorphism. While pseudoholomorphic
methods have successfully addressed this question in dimensions
2-3, the higher-dimensional case remains less...
