Monday, April 22, 2013
Laplacian Growth and the Mystery of the Abelian Sandpile: a Visual
We compare several growth models on the two dimensional lattice. In
some models, like internal DLA and rotor-router aggregation, the scaling limits
are universal; in particular, starting from a point source yields a disk. In
the abelian sandpile, particles are added at the origin and whenever a site has
four particles or more, the top four particles topple, with one going to each
neighbor. Despite similarities to other models, for the sandpile, the
intriguing pattern that arises is not circular and depends on the particular
lattice. A scaling limit exists for the sandpile, as was recently shown by
Pegden and Smart, but it is not universal and is still mysterious. This research
has been greatly influenced by pictures of the relevant sets, which I will show
in the talk. They suggest a connection to conformal mapping which has not been
established yet. (Talk based on joint works with Lionel Levine.)
Following Monday's lecture, there will be a reception in honor of
Professor Peres at the Koenig Alumni Center, 1202 University
Avenue (the SE corner of Broadway and University).
Tuesday, April 23, 2013
Hunter, Cauchy Rabbit, and Optimal Kakeya Sets
A planar set that contains a unit segment in every direction is called a
Kakeya set. These sets have been studied intensively in geometric measure
theory and harmonic analysis since the work of Besicovich (1928); we find a new
connection to game theory and probability. A hunter and a rabbit move on an
n-vertex cycle without seeing each other until they meet. At each step, the
hunter moves to a neighboring vertex or stays in place, while the rabbit is
free to jump to any node. Thus they are engaged in a zero sum game, where the
payoff is the capture time. We show that every rabbit strategy yields a
Kakeya set; the optimal rabbit strategy is based on a discretized Cauchy random
walk, and it yields a Kakeya set K consisting of 4n triangles, that has minimal
area among such Kakeya sets (the area of K is of order 1/log(n)). Passing to
the scaling limit yields a simple construction of a random Kakeya set with zero
area from two Brownian motions. (Joint work with Y.
Sousi and P.
Yuval Peres is a Principal Researcher in the Theory group at Microsoft Research.
Before joining MSR in 2006, he was a Professor at UC Berkeley.
He has also taught at Yale and at the Hebrew University.
Yuval has published
more than 200 papers with 100 co-authors and has mentored 19 PhD theses. His
research encompasses many areas of probability theory, including random walks,
Brownian motion, percolation, point processes and random graphs as well as
connections with Ergodic Theory, PDEs, Combinatorics, Fractals, and Theoretical
Computer Science. He has recently co-authored books on Markov chains and mixing
times, on zeros of Gaussian analytic functions, and on Brownian motion. Yuval
is a fellow of the AMS and a recipient of the Rollo Davidson Prize. In 2001 he
received the Loeve Prize, awarded once every two years to a leading
probabilist. Yuval was an invited speaker at the International Congress of
Mathematics (2002) and in the European Congress of Mathematics (2008). His
favorite quote is from his son Alon, who was overheard at age 6 asking a
friend: "Leo, do you have a religion? You know, a religion, like Christian, or
Jewish, or Mathematics....?"
DeLong Lecture Series
This Lecture Series is funded by an endowment given by Professor Ira M.
DeLong, who came to the University of Colorado in 1888 at the age of 33.
Professor DeLong essentially became the mathematics department by teaching
not only the college subjects but also the preparatory mathematics courses.
Professor DeLong was a prominent citizen of the community of Boulder as
well as president of the Mercantile Bank and Trust Company, organizer of the
Colorado Education Association, and president of the charter convention that
gave Boulder the city manager form of government in 1917. After his death
in 1942 it was decided that the bequest he made to the mathematics
department would accumulate interest until income became available to fund
DeLong prizes for undergraduates and DeLong Lectureships to bring outstanding
mathematicians to campus each year. The first DeLong Lectures were delivered
in the 1962-63 academic year.