Georgia Institute of Technology College of Sciences

What is the maximum number of rainbow triangles in an edge-colored graph with m edges and r colors? Using entropic techniques, we prove an upper bound of C(r)*m^{3/2} rainbow triangles — the explicit constant C(r) we derive is best possible whenever there is an affine plane of order r-1. We also show that constructions attaining at least (C(r)−eps)*m^{3/2} rainbow triangles must exhibit an affine plane structure for sufficiently small epsilon, which further improves the upper bound if no such affine plane exists. Lastly, we consider the problem of counting properly edge-colored copies of K_d with d≥4. Surprisingly the optimal construction is the same for any d≥4, but differs from the affine plane construction that was optimal for d=3.

A unit fraction representation of a positive rational number writes it as a sum of finitely many unit fractions $\frac{1}{n}$ with distinct positive integers $n$. In 1202, Fibonacci showed that every positive rational number has such a representation. Since then, people have been interested in representations with special properties. In this spirit, Bloom asked: which sets $A \subseteq \mathbb{N}$ can represent every positive rational number, subject to some local conditions, using only denominators from $A$? Croot proved in 2003 that for any finite colouring of $\mathbb{N}$, $1$ can be written as sum of unit fractions with denominators in a single-color class. Bloom showed that any $A$ with positive upper logarithmic density works, and Liu and Sawhney improved this to sets with upper logarithmic density $(\log N)^{-1/5 + o(1)}$. Bloom also showed that the shifted primes $p-1$ have this property.

To characterize all sets $A$ with this special property, we need to understand which arithmetic properties make this possible. One way to do that is to study thinner sets that still have this special property. In this talk we show that the shifted Golomb primes have this property. The Golomb primes are a subset of primes defined by $q_1 = 3$ and, once $q_1, \dots, q_k$ are known, $q_{k+1}$ is the smallest prime $p$ with $p \not\equiv 1 \pmod{q_i}$ for all $i \le k$. They are a natural generalization of the primes, and their relative density in the primes is about $1/(loglogx)$, which goes to $0$.

Local invariant manifolds provide a natural way to separate center,
stable, and unstable dynamics near an invariant object and, at the
same time, to construct lower-dimensional models of the dynamics. In
this talk, I will present a unified numerical procedure for computing
high-order local invariant manifolds attached to fixed points and
reducible quasi-periodic invariant tori of Poincaré maps. The
fixed-point setting appears naturally as the zero-dimensional case of
the quasi-periodic formulation.

For a quasi-periodic skew-product, the normal variational dynamics
along the torus defines a linear cocycle over a rigid
rotation. Assuming reducibility, a Floquet transformation converts
this cocycle into a constant matrix, so that a spectral splitting can
be introduced as in the fixed-point case. The desired invariant
manifold is then represented as a graph over a selected spectral
subspace. Expanding this graph in Fourier–Taylor series leads, order
by order, to cohomological equations whose solvability is governed by
the interaction between the base rotation and the Floquet
spectrum. The same construction also provides the nonlinear dynamics
induced on the selected variables, yielding an explicit reduced model
together with a lifting to the full phase space.  

I will illustrate the method first on a (quasi-)periodically forced
pendulum, which provides a simple setting in which to visualize the
response torus, its center reduction, and the dynamics generated by
the reduced map. Time permitting, I will also discuss how the same
ideas arise in (quasi-)periodically forced models from celestial
mechanics.


Joint works with M. Barcelona, L. Domingo, and M. Jorba-Cusco.

Murmurations were first observed as surprising correlations between Frobenius traces and root numbers of elliptic curves. They were then rigorously established for the family of modular forms in the level aspect by Zubrillina, and have since been studied in many families of L-functions in the number field setting. As pointed out by Sarnak, murmurations can also be interpreted as a phase transition in the one-level density of the low-lying zeros of the family. The goal of this talk is to investigate the function field analogue of this phenomenon. We will compute the murmuration density for the family of quadratic characters, relate the mean of the murmuration density to the stable homology of the moduli space of hyperelliptic curves, and clarify the relationship this has to one-level density. As a corollary, we will also obtain non-vanishing results for the L-functions of these curves. This is joint work with Lalín, Lee, and Oliver.