On the basis of final reports submitted by all eligible participants in last summer's REGS program, four Fellows were selected to give 15-minute presentations on their projects:

• Jennifer Wise
  Title: Edge-Antipodal Colorings of Hypercubes
  Abstract: Edge-antipodal 2-edge-colorings of hypercubes are looked at. We will see an explicit proof that every edge-antipodal 2-edge-coloring of a hypercube with at most 5 dimensions contains a monochromatic geodesic between some pair of antipodal vertices, something previously only known through computer search. (Work done with Hannah Kolb and Oliver Pechenik.)

• Dan Schultz
  Title: Cubic Theta Functions
  Abstract: The cubic generalizations of the Jacobi theta functions are introduced, and the theory of these functions is developed analogously to the classical theory of elliptic functions.

• June Huh
  Title: Coloring graphs, counting solutions, and attaching cells
  Abstract:The chromatic polynomial of a graph counts the number of colorings. We give an affirmative answer to the conjecture of Read and Rota that the absolute values of the coefficients of the chromatic polynomial form a log-concave sequence. The proof is a combination of combinatorics, intersection theory, convex geometry, and Morse theory.

• Joseph Vandehey
  Title: Information, probability, and randomness in even and odd continued fractions
  Abstract: Suppose someone walked up to you and handed you a random irrational number. As you read through the digits of the irrational number, what is the probability that you know its value within epsilon error right in the middle of reading a particular string of digits? This question is very easy to answer for decimal respresentations, and surprisingly easy for continued fractions as well. We show some work on answering this question for more generalized continued fractions.