הרצאת המכון ללימודים מתקדמים: מסע מתמטי על אופניים
מרצה אורח: פרופסור סרגיי טבצ'ניקוב המחלקה למתמטיקה' אוניברסיטת המדינה של פנסילבניה
HOW CAN THE MOTION OF A BICYCLE GIVE RISE TO UNEXPECTED MATHEMATICAL STRUCTURES?
As part of his visit to the Institute for Advanced Studies at Tel Aviv University as an IAS Distinguished Scholar, Professor Sergei Tabachnikov (Pennsylvania State University) will offer a short lecture course titled "Flavors of Bicycle Mathematics".
The course explores a simple yet remarkably rich mathematical model inspired by bicycle motion. Despite its elementary formulation, the model leads to surprising connections with geometry, dynamical systems, and completely integrable systems, and raises a number of intriguing open problems. The lectures will combine geometric intuition with rigorous mathematical ideas.
COURSE DETAILS
Where: Schreiber Building, Tel Aviv University
Monday, 13 April 2026 | 15:00–16:00 | Room 008
Thursday, 16 April 2026 | 13:00–14:00 | Room 309
Light refreshments will be served at the opening session.
The workshop will be conducted in English.
This course is closed and intended only for enrolled students.
MORE ABOUT THE SPEAKER
Sergei Tabachnikov is a Professor of Mathematics at Pennsylvania State University and a Fellow of the American Mathematical Society. His research spans geometry, topology, and dynamical systems, including mathematical billiards, projective geometry, integrable systems, and cluster algebras. He is also one of the founders of the field known as bicycle mathematics.
Professor Tabachnikov has mentored numerous young researchers and has held leadership roles in several international programs and institutes, including ICERM, MASS, and the Heidelberg Laureate Forum. He currently serves as editor-in-chief of the Arnold Mathematical Journal and The Mathematical Intelligencer, and sits on the editorial boards of several other mathematics journals.
FULL ABSTRACT
FLAVORS OF BICYCLE MATHEMATICS
This course will focus on a naive model of bicycle motion: a bicycle is a segment of fixed length that can move so that the velocity of the rear end is always aligned with the segment. Surprisingly, this simple model is quite rich and has connections with several areas of research, including completely integrable systems, and many questions remain open. Here is the list of problems that I hope to discuss:
(i) The trajectory of the front wheel and the initial position of the bicycle uniquely determine its motion and its terminal position; the monodromy map that sends the initial position to the terminal one arises. This mapping is a Möbius transformation, a remarkable fact with various geometrical and dynamical consequences.
(ii) The rear wheel track and a choice of the direction of motion uniquely determine the front wheel track; changing the direction to the opposite yields another front track. These two front tracks are related by the bicycle (Bäcklund, Darboux) correspondence, which defines a discrete-time dynamical system on the space of curves. This system is completely integrable and is closely related to another well-studied, completely integrable dynamical system, the filament (a.k.a. binormal, smoke ring, local induction) equation.
(iii) Given the rear and front tracks of a bicycle, can one tell which way the bicycle went? Usually, one can, but sometimes one cannot. The description of these ambiguous tire tracks is an open problem, intimately related to Ulam's problem in flotation theory (in dimension two): is the round ball the only body that floats in equilibrium in all positions? This problem is also related to the motion of a charge in a magnetic field of a special kind. It turns out that the known solutions are solitons of the planar version of the filament equation.
(iv) Can one discretize the previous problem, i.e., replace curves by polygons? I shall present some partial results in this direction.
(v) Bicycle geodesics are bicycle paths whose front track length is critical among all bicycle paths connecting two given placements of the line segment. In the plane, these geodesic front tracks are elastica, and in space they are Kirchhoff rods.
(vi) Is it possible to ride a bicycle so that the rear wheel track coincides with the front wheel track (other than going straight, of course)? Such "unicycle" tracks tend to behave very chaotically, but so far these are mostly only experimental observations.
The Institute of Advanced Studies
https://ias.tau.ac.il
