Geometry, Dynamics, and Topology Day 2013
Old Main, (West) 2nd floor
Friday, April 19 (OM 2221)
|3:00-3:20||Vadim Zharnitsky||Three-period Orbits in Billiards on the Surfaces of Constant Curvature|
|3:30-3:50||A Ping-pong Problem (Estimating (k,n)-chunks of the Alternating Series|
|4:00-4:20||Alex Tumanov||Scarcity of Periodic Orbits in Billiards|
|4:30-4:50||Charles Delman||A Separation Principle for Geodesics on Euclidean Cone Manifolds with Pockets|
|5:00-5:20||Maxim Dm. Arnold||Pinball Dynamics|
Saturday, April 20 (OM 2231)
|9:00-9:50||Valentin Afraimovich||Dimension-like Characteristics of Motions in Dynamical Systems|
|10:10-11:00||Brief History of the Boltzmann-Sinai Hypothesis|
|11:20-12:10||Appoximation of Locally Compact Groups by Finite Quasigroups and Related Geometric Problems|
|12:30-1:00||Open and Closed 20-13 Problems on Billiard Dynamics and Geometry|
|Lunch Break (1:00-2:00)|
|2:00-2:50||Greg Huber||The De Vicci Cube, and Other Mysteries from the Fourth Dimension (and Beyond)|
|Coffee Break (3:00-3:30)|
|3:30-4:20||Douglas Hofstadter||A Very Unexpected Energy Spectrum: A Tale of Solid-state Physics and Recursive Geometry|
|Closing Remarks (4:30)|
Title: Three-period orbits in billiards on the surfaces of constant curvature
Abstract: Wojtkovski's approach, based on the Jacobi fields, is applied to study sets of 3-period orbits in billiards on the hyperbolic plane and on the two-dimensional sphere. It is found that the set of 3-period orbits in billiards on the hyperbolic plane, as in the planar case, has zero measure. For the sphere, a new proof of Baryshnikov's theorem is obtained which states that 3-period orbits can form a set of positive measure if and only if a certain natural condition on the orbit length is satisfied. This was a joint work with V. Blumen, K.Y. Kim, and J. Nance.
Abstract: The study of periodic orbits has always been central in Hamiltonian Dynamics. The classical billiard introduced by Birkhoff is a popular example of a Hamiltonian dynamical system. Additional interest in periodic orbits comes from spectral theory of the Laplacian: Irvii showed that the so-called Weyl's asymptotics of the eigenvalues of the Laplacian for a planar domain holds if the set of all periodic orbits of the corresponding billiard has measure zero. The latter is a difficult open question, which will be our main theme. We consider another similar Hamiltonian system, the outer billiard introduced by B. Neumann and promoted by Moser. Until recently, the affirmative answer to the above question for both classical and outer billiards has been known only for orbits of period up to 3. For period 4 orbits, a similar conclusion has recently been drawn by Glyutsyuk and Kudryashov (for classical billiard) and Zharnitsky and the speaker (for outer billiard). We give a simple explanation of the last result and extend it to some period 5 orbits.
Title: A Separation Principle for Geodesics on Euclidean Cone Manifolds with Pockets
Abstract: In 2002, Gregory Galperin and I published a separation principle and consequent bound on the number of orbit types for Euclidean billiards with pockets. I revisit this work purely from the perspective of geometric topology, which lies at the heart of the proof. This renewed perspective suggests both simplifications of the arguments and an approach to generalizing the result.
Speaker: Maxim Dm. Arnold, University of Illinois at Urbana-Champaign
Title: Pinball dynamics
Abstract: The theory of small perturbations of completely integrable Hamiltonian systems has a long history. Summarily, one could say that the KAM theorem states that if a perturbation is sufficiently smooth, then positive measure of invariant tori survives, and thus, in particular for planar area-preserving transformations, there are no trajectories escaping to infinity. In some physical settings, one deals with non-smooth perturbations where the KAM-technique does not apply. However, there are no general instruments to work with such systems. We have discovered a natural family of such systems which covers many known examples. One of the representative cases we call Pinball transformation. I shall describe this system and show how to construct an escaping trajectory in it.
Title: A Very Unexpected Energy Spectrum: A Tale of Solid-state Physics and Recursive Geometry
Abstract: In 1974, as a graduate student in physics desperately seeking a Ph.D. topic, I was most surprised to hear of a very fundamental problem in solid-state physics -- the mysterious energy spectrum of electrons in a crystal in a magnetic field -- in which apparently the most crucial thing was whether the size of the magnetic field was a rational or an irrational real number. This idea sounded utterly crazy to me, since how can a physical phenomenon possibly depend on whether some number is rational or irrational? In fact, how does the supposed distinction even make sense, since the numerical value of any measured quantity in physics is arbitrary, being a function of the system of units someone randomly selected?
Well, it turned out that in this very special type of physical situation, the magnetic field was naturally measured in units of the "flux quantum" (a precise quantum-mechanical chunk), which made its value totally independent of any human-defined system of measurement. Well, that got rid of one of my two objections, but the other one was even stronger. How could nature possibly "know" whether the value of some physical quantity was rational or irrational, given that to know whether a number is rational or irrational number, you need to know its decimal expansion all the way out to infinity (and the first million, billion, or trillion digits don't even matter one whit)? How could any physical quantity be precisely defined to infinitely many decimals? And why on earth would nature "care" only about the trillionth decimal digit (or rather, only about the digits from
arbitrarily far on out)?
None of this made the slightest sense to me -- but that's precisely why the problem attracted me. As a former math graduate student who had been deeply in love with number theory and who had developed very strong intuitions about rational and irrational numbers, I found this paradox inside physics so crazy that I was hooked -- and so I decided to explore this problem that, at the time, theoretical solid-state physicists were completely stumped by. The startling geometrico-topological discovery that eventually emerged out of my Ph.D. work way back in the mid-1970s is the tale that I will tell in this talk.
Title: The De Vicci Cube, and other mysteries from the fourth dimension (and beyond)
Abstract: There is a secret cube that lives in four dimensions called the De Vicci Cube. "Surely you mean a 'hypercube' or a 'tesseract'?" you might be forgiven to ask. No, I mean an regular three-dimensional cube. But this is no ordinary cube. Leonardo Da Vinci didn't know of its existence, nor did the Freemasons, and nor does Dan Brown (for, if he did, there would surely be another bestseller and movie out). It was first foreshadowed by the Babylonians long ago. Then, in the seventeenth century, a certain Prince Rupert of the Rhine wagered on the outcome an optimization problem. He got his answer, and he won his bet, but he didn't solve the problem. The problem wasn't solved until Pieter Nieuwland examined it over 100 years later. The solution was
which, à la manière égyptienne,
= 1 + 1/17 + 1/545 + 1/561804 + ...
Nieuwland was getting close to the Cube, but soon after his discovery, he died under very mysterious circumstances. And still the De Vicci Cube remained a secret, lurking in the sea of undiscovered things. What is the De Vicci Cube? We know that De Vicci's Cube has linear dimension
= 1 + 1/135 + 1/36564 + …
This was calculated by at least six people (all dead now) before K. De Vicci discovered it. The Cube has acquired De Vicci's name because she supplied an actual proof. Erdös knew about the Cube, but soon after learning of it, he died too. There is much more to say. More than this, you must attend the talk, as I have written too much already.
Title: Brief History of the Boltzmann-Sinai Hypothesis
Abstract. The Boltzmann-Sinai Hypothesis dates back to 1963 as Sinai's modern formulation of Ludwig Boltzmann's statistical hypothesis in physics, actually as a conjecture: Every hard ball system on a flat torus is (completely hyperbolic and) ergodic (i. e. "chaotic", by using a nowadays fashionable, but a bit profane language) after fixing the values of the obviously invariant kinetic quantities. In the half century since its inception quite a few people have worked on this conjecture, made substantial steps in the proof, created useful concepts and technical tools, or proved the conjecture in some special cases, sometimes under natural assumptions. Quite recently I was able to complete this project by putting the last, missing piece of the puzzle to its place, getting the result in full generality. In the talk I plan to present the brief history of the proof by sketching the most important concepts and technical tools that the proof required.
Title: Open and Closed 20-13 Problems on Billiard Dynamics and Geometry
Abstract: In my short talk I will formulate several new and old problems on billiards and geometry, some of which are solved, some are being solved, and some are open. Among them problems on illumination of a polygon and of the plane (Euclidean and hyperbolic), problems on non-periodic but bounded billiard trajectories in a cut polygon; a jumping billiard particle in a gravitational field; and some purely geometry problems.
Speaker: Valentin Afraimovich, Universidad Autonoma de San Luis Potosi
Title: Dimension-like characteristics of motions in dynamical systems
Abstract: To characterize invariant sets in the phase space of a dynamical system, people sometimes use the Hausdorff or box dimensions. They reflect geometric (more precisely, metric) properties of these sets. But the temporal behavior of representative points on trajectories could be very different for systems having the same invariant set. To measure the motions, one needs to impose and apply other characteristics of dynamics. The Caratheodory-Pesin construction allows one to define such characteristics. In the lecture, the spectrum of dimensions for Poincare recurrences, as well as for escape times, will be defined, their main features will be listed, some examples will be presented where these characteristics can be easily calculated.best fit using geometric, topological and probabilistic approaches.