# Giulio Tiozzo and Tristan C. Collins are awarded the 2021 André Aisenstadt Prize

Giulio Tiozzo (University of Toronto) and Tristan C. Collins (Massachusetts Institute of Technology) are winners of the 2021 André Aisenstadt prize. Many brilliant mathematicians were in the running for the André Aisenstadt prize this year. Once again, the CRM International Scientific Committee was particularly impressed with the achievements of two nominees and thus decided that both be awarded the prize. This rare situation is an expression of the high caliber of their work.

#### Giulio Tiozzo (University of Toronto)

#### Entropy along the Mandelbrot set

*Summary*: The notion of topological entropy, arising from information theory, is a fundamental tool to understand the complexity of a dynamical system. When the dynamical system varies in a family, the natural question arises of how the entropy changes with the parameter.

In the last decade, W. Thurston introduced these ideas in the context of complex dynamics by defining the “core entropy” of a quadratic polynomials as the entropy of a certain forward-invariant set of the Julia set (the Hubbard tree). As we shall see, the core entropy is a purely topological/combinatorial quantity which nonetheless captures the richness of the fractal structure of the Mandelbrot set. In particular, we will relate the variation of such a function to the geometry of the Mandelbrot set. We will also prove that the core entropy on the space of polynomials of a given degree varies continuously, answering a question of Thurston. Finally, we will provide a new interpretation of core entropy in terms of measured laminations, and discuss its finer regularity properties such as its Holder exponent.

*Biography*: Giulio Tiozzo is an Assistant Professor of Mathematics at the University of Toronto. He obtained his PhD from Harvard in 2013, under the supervision of C.T. McMullen. Prior to joining University of Toronto, he was a Gibbs Assistant Professor at Yale University, and in 2018 he was awarded the Alfred P. Sloan Fellowship. His field of research is dynamical systems and ergodic theory, with applications to complex analysis, probability, and geometric group theory.

#### Tristan C. Collins (MIT)

#### Exploring string vacua through geometric transitions

*Summary*: A fundamental problem in string theory is the multitude of distinct geometries which give rise to consistent solutions of the vacuum equations of motion. One possible resolution of this “vacuum degeneracy” problem is the “fantasy” that the moduli space of string vacua is connected through the process of “geometric transitions”. I will discuss some geometric problems associated to this fantasy and their applications.

*Biography*: Tristan Collins is an Assistant Professor in the Mathematics Department at MIT. Formerly he was a Benjamin Peirce Assistant Professor at Harvard University. He completed his Ph.D. under the supervision of D.H. Phong at Columbia University in New York City. Before studying at Columbia, he studied mathematics at the University of British Columbia in Vancouver, B.C., where he worked with Malabika Pramanik. He does research in geometry and analysis. He is particularly interested in geometric flows, and applications of algebraic geometry to analysis, and vice versa. His research is partially supported by NSF grant DMS-1810924, NSF CAREER grant DMS-1944952, and an Alfred P. Sloan Fellowship.