The mechanics of geometr

Dr. Shira Tanny is fascinated by the dynamicsof particles on non-flat spaces. Currently a postdoctoral fellow at the School of Mathematics at the Institute for Advanced Study (IAS) in Princeton, New Jersey, she is scheduled to join the Weizmann Institute’s Department of Mathematics in August 2024.

Dr. Tanny studies symplectic geometry, a field of math that emerged from classical mechanics. Symplectic geometry is the study of movement of particles through spaces that are not flat, seeking to understand the relationship between the potential dynamics of the particle and the geometry and topology of the space in which it is moving.

“In classic mechanics, we have a mechanical system, we have forces that act on the system, we have energy that accompanies these forces, and then we have the physical motions—how things actually move,” she explains.

If a particle is moving in free space, she continues, researchers can consider the particle’s position and its momentum, and then write equations that determine the particle’s motion depending on the forces and the system.

Dr. Tanny’s research takes this one step further, studying particles’ motion in particular spaces, such as a torus (a donut-shaped curved surface) or a sphere.

“We want to understand the relationship between the dynamics that could happen and the geometry and topology of the space,” she says. “The whole field is about connections, many of which we don’t yet understand.”

Answering tricky questions

One problem Dr. Tanny has been studying as an IAS postdoc is a question posed by the French mathematician Henri Poincaré, who believed that periodic orbits—points in space that return to themselves after some time—are key to understanding dynamics.

An example would be launching a satellite into space that repeats the same motions and asking whether small changes to a given system could produce periodic orbits.

The answer, Dr. Tanny explains, depends on what is meant by “small changes of the system.” Using the satellite example, changes to the system could come in the form of adding a small amount of fuel or having the satellite change course gradually.

And when the changes to the system are sufficiently small in one sense or another—such as the satellite’s velocity changing slowly—then the question becomes “very tricky.”

“There are nuances to the notion of ‘slight changes’ to the system,” she says.

At the Weizmann Institute, one of the topics Dr. Tanny plans to explore is the mysterious relationship between symplectic geometry and convex geometry. In the eyes of a symplectic geometer, convex sets are the same as some non-convex sets because symplectic geometry looks at the objects’ dynamics, rather than their shapes.

As an undergraduate student at Tel Aviv University, Dr. Tanny found inspiration in two popular science books about math: The Man Who Loved Only Numbers: The Story of Paul Erdős and the Search for Mathematical Truth by Paul Hoffman and The Poincaré Conjecture: In Search of the Shape of the Universe by Donal O’Shea.

“It might be a little weird, but I like not understanding,” she explains of her fascination with these books. “I like the feeling that I’m getting into something new that I know nothing about, or understand nothing about, even if it’s in my field. It makes me really curious.”

“I like the fact that in science in general, and math, we get the chance to understand things for the first time.” She describes mathematical research as starting from a place of non-understanding. With work, she says, bit by bit the pieces of the puzzle fall into place.

“I think that was evident in those books, the mystery of problems you don’t understand,” she adds.

Returning to Israel

When asked what life has been like for an Israeli researcher in the United States following the October 7 Hamas attacks, which prompted a storm of antisemitism and anti-Israel protests on college campuses, Dr. Tanny describes the IAS as a “bubble.”

“Everybody has been completely supportive and kind, and all the conversations I’ve had [about the situation in Israel] have been intelligent and patient. The atmosphere here is really comforting and helpful,” she notes.

Dr. Tanny grew up in Elyakhin, a town in the center of Israel, and earned a combined BSc in physics and mathematics (summa cum laude) at Tel Aviv University and an MSc in mathematics at the Weizmann Institute, where she studied under Prof. Sergei Yakovenko in the Department of Mathematics.

She returned to Tel Aviv University for her PhD in mathematics under the supervision of Profs. Lev Buhovski and Leonid Polterovich. She has been a postdoctoral fellow at the IAS since 2021. Dr. Tanny is married with two children, who will be starting preschool and first grade when the family returns to Israel.

Education and select awards

BSc summa cum laude, Tel Aviv University (2012)

MSc, Weizmann Institute of Science (2014)

PhD, Tel Aviv University (2021)

Postdoc, Institute of Advanced Study, New Jersey (2021-2024)

Dean’s List, Tel Aviv University Faculty of Exact Sciences (2011-12); Outstanding Student Award, Departments of Mathematics and Physics, TAU (2012); Maria Proner- Pogonowska Scholarship for Women in Science, TAU (2010-2013); Rachel and Salim Benin Award (2015); Nehemia Levtzion Scholarship for Outstanding Doctoral Students from the Periphery (2018); Eric and Wendy Schmidt Postdoctoral Award for Women in Mathematical and Computing Sciences (2019); Rothschild Postdoctoral Fellowship (2021); Israeli Postdoctoral Scholar, Zuckerman STEM Leadership Program (2021)