Thursday, September 16Yu. Baryshnikov, Bell Labs
2:30pmHard disks in a box: topology and complexity
ABSTRACTConfiguration spaces of hard disks (or balls) in a box support archetypal
dynamical systems of statistical physics. Yet, we still know precious little
about the structure of these space (say are they connected, for low densities?).
This talk will be dealing with their topology.
Joint work with M. Kahle, IAS.
Thursday, September 23Adrian Ocneanu, Penn State
2:30pmSymmetry, regular solids and quantum field theory
ABSTRACTWe discuss the structures of 3-dimensional space motions and symmetry and their relation to 4-dimensional regular solids as illustrated by the Octacube sculpture.
We present applications of these and of higher symmetry structures in models of quantum field theory.
Thursday, September 30David Futer, Temple University
2:30pmThe rental harmony theorem
ABSTRACTThe rental harmony theorem (proved by Francis Su a decade ago) is quite possibly the most practically useful theorem that I have come across. Suppose that n housemates are renting a house that has n unequal rooms. The theorem says that under mild hypotheses, there is a way to partition the total rent into rents for the individual rooms, such that every housemate will prefer a different room. Furthermore, it provides an algorithm to find this harmonious partition of the rent. I will explain the theorem and its proof.
Thursday, October 7Richard Kenyon, Brown University
2:30pmTilings with rational polygons
ABSTRACTIn 1903, Dehn showed that an aXb rectangle can be tiled by squares if and only if a/b is rational. We generalize this as follows. A convex polygon P can be tiled by rational polygons if and only if P is rational.
Here by rational polygon we mean a polygon whose side length ratios are rational. The proof uses the notions of the signature of a quadratic form, and of rational linear maps from R to R, both of which we will introduce
as necessary.
Thursday, October 14Marl Sapir, Vanderbilt University
2:30pmAmenability of the Thompson group $F$ and some elementary properties of $Z^n$
ABSTRACTI will explain the notion of amenability of groups introduced by von Neumann in 1929., and define the R. Thompson group $F$ using diagram groups. I will formulate some elementary but still not proved properties of $Z^k$ (the standard cubical lattice in $R^k$) which, if true, would shed some light on the still open question about amenability of $F$. This is a joint work with A. Dranishnikov.
Thursday, October 21Svetllana Jitomirskaya, University of California Irvine
11:15amBehind the Hofstadter's butterfly: the competition between order and chaos
ABSTRACTHofstadter's butterfly was maybe the first beautiful fractal produced in a numerical experiment (in 1976, before the word fractal came into existence). We will discuss some interesting features of the models behind it, called quasiperiodic operators, an area of strong interplay of ergodic theory, dynamical systems, probability, functional and harmonic analysis. Quasiperiodic operators feature a fascinating competition between randomness (ergodicity) and order (periodicity), that is often resolved on a deep arithmetic level. The interest in those models was enhanced by strong connections with some major discoveries in physics, such as integer quantum Hall effect, experimental quasicrystals, quantum chaos theory, as well as graphene. Quasiperiodic operators provide central or important models for all four. We will not talk much about physics but will touch upon a related question of how the difference between rationals and irrationals can play a role in nature.
Thursday, November 4Kenneth Gross, University of Vermont
2:30pmSymmetry: From Triangles to Quantum Physics
ABSTRACTSymmetry is a fundamental concept in nature and to the human mind. It occurs everywhere, from patterns of wallpaper, to the paintings of the great masters, to the structure of poetry and music, and throughout science. The purpose of this presentation is to explain the mathematical formulation of the concepts of symmetry and invariance, and to provide an indication of their breadth of applications. The presentation assumes familiarity with complex numbers, matrices, and calculus.
Thursday, November 11Ron Perline, Drexel University
2:30pmSome reflections on mirrors
ABSTRACTSome simple geometry can make surprisingly important contributions to aspects of mirror design. I will talk about three mirror designs and their simple associated mathematics: rectifying mirrors, equiareal mirrors, and sideways mirrors. Basic differential equations theory, vector calculus and some geometry are the only pre-requisites.
Thursday, November 18Richard Schwartz, Brown University
2:30pmOuter Billiards, Polytope Exchange Maps, and Renormalization
ABSTRACTOuter billiards is a simple dynamical system, defined relative to a convex shape in the plane. In my talk I will explain some connections between polygonal outer billiards, higher dimensional polytope exchange maps, and renormalization. I will exploit these connections to get a fairly complete understanding
of outer billiards defined relative to the Penrose kite.
Thursday, December 2Robert Finn, Stanford Univrsity
2:30pmCapillary surfaces
ABSTRACTIt would be hard to get through a day without encountering surfaces separating
distinct adjacent fluids. In a grotesque accident of history, all such things have come to be
labeled capillary surfaces (capillus = hair in Latin). You encounter them whenever you drink a
glass of water or wash your face, or sing in the rain or enjoy a candlelit dinner or see a
spiderweb. They appear in all sizes, from microscopic to planetary and beyond. Experience has
let us adapt to everyday occurrences, but the needs of quantitative theory lead to challenges that
defy traditional methods. The formal equations have beautiful geometric form, but almost no
exact solutions are known, and surprising things happen that have yet to be clarified. One of
them is a striking relation to billiards theory. Some of the simplest problems were solved in
approximate form during the 19th century. These “solutions” are still routinely used by engineers
as models for general behavior, however recent discoveries (in some of which undergraduate
students participated) have shown that such procedures can lead to major errors. I will discuss
the state of the theory, with emphasis on current unsolved problems.
Thursday, October 28A. Petrunin, Penn State
2:30pmAlexandrov's theorem on development of polyhedron