Where: Math 1308

Where: Math 1308

Speaker: Mike Kreisel (UMD) -

Where: Math 1308

Speaker: Scott Schmieding (University of Maryland - College Park) -

Where: Math 1308

Speaker: Scott Schmieding (UMD) -

Where: Math 1308

Speaker: Maxim Arnold (University of Illinois at Urbana-Champaign) -

Abstract: Rapidly exploring Random Trees (RRT) have become

increasingly popular as a way to explore high-dimensional spaces for

problems in robotics, motion planning, virtual prototyping,

computational biology, and other fields. It was established that the

vertex distribution converges in probability to the sampling

distribution. It was also noted that there is a "Voronoi bias" in the

tree growth because the probability that a vertex is selected is

proportional to the volume of its Voronoi region. I shall explain the

evolution of the shape of convex hull of RRT when the size of the

search space is increased.

Where: Math 1308

Speaker: Brendan Berg (UMD)

Where: MTH 1308

Speaker: Scott Schmieding (University of Maryland)

Where: MTH 1308

Speaker: Hillel Furstenberg (Hebrew University) -

Where: Math 1308

Speaker: Mike Kreisel (UMD) -

Where: Math 3206

Speaker: Rune Johansen (University of Copenhagen)

Where: Math 1308

Speaker: Marco Lenci, Universita' di Bologna, Italy

Abstract: Finding a satisfactory definition of mixing for dynamical systems

preserving an infinite measure (in short, infinite mixing) is an

important open problem. Virtually all the definitions that have been

attempted so far use ’local observables’, that is, functions that

essentially only “see” finite portions of the phase space. We

introduce the concept of ’global observable’, a function that gauges a

certain quantity throughout the phase space. This concept is based on

the notion of infinite-volume average, which plays the role of the

expected value of a global observable. Endowed with these notions,

which are to be specified on a case-by-case basis, we give a number of

definitions of infinite mixing. These fall in two categories:

global-global mixing, which expresses the “decorrelation” of two

global observables, and global-local mixing, where a global and a

local observable are considered instead. Time permitting, we will see

how these definitions respond on some examples of

infinite-measure-preserving dynamical systems.

Where: Math 1308

Where: MATH 1308

Speaker: Scott Schmieding (UMD)

Where: MATH 1308

Speaker: Scott Schmieding (UMD)

Where: MATH 1308

Speaker: Mike Kreisel (UMD)

Where: Math 1308

Speaker: Mike Kreisel

Where: Math 1308

Speaker: Brendan Berg (UMD)

Where: MATH 1308

Speaker: Olena Karpel (Institute for Low Temperature Physics, Kharkov, Ukraine)

Abstract: Two measures \mu and \nu on a topological space X are called homeomorphic if there exists a homeomorphism f of X such that \mu(A) = \nu(f(A)) for every Borel subset A. We are interested in the problem of classification of Borel probability and infinite measures on a Cantor set with respect to a homeomorphism.

For a wide class of probability measures which E. Akin called good, a criterion of being homeomorphic is known. A full non-atomic measure \mu is good if whenever U, V are clopen sets with \mu(U) < \mu(V), there exists a clopen subset W of V such that \mu(W) = \mu(U). For the class of good probability measures, the set S(\mu) of values of measure \mu on all clopen subsets of X is a complete invariant.

We consider ergodic probability and infinite invariant measures for aperiodic substitution dynamical systems. S. Bezuglyi, J.Kwiatkowski, K.Medynets and B.Solomyak showed that these measures can be described as ergodic measures on non-simple stationary Bratteli diagrams invariant with respect to the cofinal (tail) equivalence relation. We also consider a wide class of infinite measures on a Cantor set. We find necessary and sufficient condition for good measures to be homeomorphic. It turns out, that for good infinite measures, the set S(\mu) is not a complete invariant, we find a new invariant which is complete.

Where: Math 1308

Speaker: Scott Schmieding (UMD)

Where: Math 1308

Speaker: Jacopo De Simoi (Università di Roma - Tor Vergata)