Talk 1: Spectral Turan problems for intersecting even cycles
Dheer Desai
University of Wyoming

Turán numbers are a cornerstone of extremal graph theory. For graphs with chromatic number at least three, the asymptotics of the Turán numbers are completely known and follow from celebrated results of Erdős, Stone and Simonovits. However, these remain unknown for several basic bipartite graphs. Nikiforov introduced a spectral analogue to these, called spectral Turán problems. Instead of maximizing the number of edges, our objective is to maximize the spectral radius of the adjacency matrices of graphs not containing some subgraphs. Such a study promises to give upper bounds for the associated Turán problems. While the asymptotics of spectral Turán numbers are known for graphs with chromatic number at least three, several families of bipartite graphs remain open in this scenario too.

In this talk we will discuss an overview comparing extremal graphs for both kinds of problems and then focus more on some bipartite families. We share some recent progress on the spectral Turán numbers for intersecting even cycles (graphs formed by intersecting multiple even cycles at a unique vertex).  In some cases, these will strengthen the previous upper bounds for the associated Turán numbers, that follow from works of Alon, Krivelevich and Sudakov. This extends past ideas developed for the spectral even cycle problem and a spectral Erdős-Sós theorem.

Talk 2: Fibbinary Zippers, Frozen Wedding Cakes, Spiraling Prototiles,
Fourteeners, and Half-Dominos: The Art of Generalized Hilbert
Curve Motifs

Douglas McKenna
Mathemaesthetics Inc, Boulder, Colorado

Space filling curve motifs form visual patterns whose aesthetic freedoms are
tempered by unexpected and elegant combinatorial constraints. Accessible to a
general audience, this talk concerns those order-$n$ motifs, each a
spatially recursive arrangement of $n^2$ pairwise-adjacent, oriented square tiles,
that use Hilbert-style threading. The focus will be on those motifs that under
iterated edge-replacement—exponentiation in a monoid—build Hamiltonian
(i.e., self-avoiding) tile or dual paths on toroidal or planar grid-graphs.

Exponentiated order-$n$ motifs that fill a square while their edge-based
tile paths remain finitely self-avoiding have boundaries governed by one
of $F_{(n-3)/2}$ Fibbinary zipper modes, where $n\ge 1$ can only be odd
and $F_i$ is the $i$th Fibonacci number.  The zipper mode with the most 0
bits in its Zeckendorf representation freezes over half of the $n \times n$ square
tiles into “wedding cake” patterns, essentially emanating constraint at toroidal
lattice points.

For motifs built from pairwise edge-adjacent squares, my interactive eBook-app
“Hilbert Curves” is a dynamically illustrated compendium of their prototiles
and center-connected dual paths.  It presents enumerative evidence for full-turn
spiral prototiles solving a Hamiltonian path constraint; an infinite sequence of
motifs whose curves’ fractal tile borders subsume the curves’ interior area to
converge to fourteen square-filling curves; and new “half-domino” curves,
whose almost-everywhere linear, self-similar, and infinitely detailed boundaries
are often reminiscent of rug, pottery, basket, or other self-negative, geometric
craft designs.

I’ll show a few pertinent art pieces, including “A Unit Domino” which won
first prize in 2D media in the 2020 Joint Mathematics Meetings art show in Denver.