Rice Dynamics and Geometry Conference

All events will take place in Keck Hall, room 100, unless otherwise indicated.

Friday, April 10, 2026

• 3:00-4:00 pm Registration & Refreshments - Herman Brown Hall, 4th floor, room 438

• 4:00-5:00 pm Ben Dozier

• 5:00-5:15 pm Break

• 5:15-6:15 pm Franco Vargas Pallete

• 6:30-8:30 pm Free time/excursion

Saturday, April 11, 2026

• 8:00-9:40 am Breakfast - Hilton Houston Plaza/Medical Center (only for Hotel guests)

• 10:00-11:00 am Mikołaj Frączyk

• 11:00-11:15 am Break: coffee/tea

• 11:15-12:15 pm Yaiza Canzani

• 12:15-1:45 pm Lunch: box sandwiches

• 1:45-2:45 pm Elena Kim

• 2:45-3:00 pm Break: coffee/tea

• 3:00-4:00 pm Riccardo Caniato

• 4:15-4:45 pm Break: snacks

• 4:45-5:45 pm Problem session

• 6:15-8:45 pm Conference dinner - Ralph S. O’Connor Building for Engineering and Science, 5th floor ($20 banquet fee)

Sunday, April 12, 2026

• 8:00-9:40 am Breakfast - Hilton Houston Plaza/Medical Center (only for Hotel guests)

• 10:00-11:00 am Lewis Bowen

• 11:00-11:15 am Break: coffee/tea

• 11:15-12:15 pm Tina Torkaman

Titles and Abstracts

Benjamin Dozier (Cornell)

The boundary of a totally geodesic subvariety of moduli space

The moduli space of genus g Riemann surfaces equipped with the Teichmuller metric exhibits rich geometric, analytic, and dynamical properties.  A major challenge is to understand the totally geodesic subvarieties -- these share many properties with the moduli space itself.  For many decades, research focused on the one (complex) dimensional case, i.e. Teichmuller cuves. The discovery of interesting higher-dimensional examples in recent years has led to new questions.  In this talk, I will discuss joint work with Benirschke and Rached in which we study the boundary of a totally geodesic subvariety in the Deligne-Mumford compactification, showing that the boundary is itself totally geodesic.  

Franco Vargas Pallete (Arizona State University)

Counting commensurability classes of nearly Fuchsian surfaces in hyperbolic 3-manifolds

In this talk we will show that given any compact hyperbolic 3-manifold and $\epsilon>0$ the number of incommensurable (1+$\epsilon$) quasi-Fuchsian surfaces of genus $g$ grows like $g^{2g}$. Based on on-going joint work with Fernando Al Assal and Jia Wan.

Mikołaj Frączyk (Jagiellonian University)

Entropy support maps and the geometry of sparse factor of i.i.d. subsets

A subset of a group $\Gamma$ is called a factor of i.i.d. if it can be realized as the set of return times of a generic point to a positive measure set in the Bernoulli shift $[0,1]^\Gamma$. In certain problems, such as Gaboriau’s fixed price question, it is important to understand the geometry of sparse factor of i.i.d. subsets of $\Gamma$. I will talk about a new tool, called entropy support maps, that sheds some light on this problem. These maps turn a small subset $U$ of a hypercube $\{0,1\]^I$ into a subset $E(U)$ of $I\times \Omega$ (where $\Omega$ is an auxiliary probability space), which encodes the structural sensitivity of $U$ to bit flips. The assignment is equivariant under the full symmetric group of the the index set $I$ and satisfies several curious properties. Using entropy support maps, I was able to show that in any exact group, sparse factor of i.i.d. subsets are approximately hyperfine; i.e., they can be cheaply cut into finite pieces.

Yaiza Canzani (UNC)

Eigenfunction concentration and Weyl Laws via geodesic beams

A broad spectrum of physical phenomena, including the localization of quantum particles, is governed by the behavior of Laplace eigenvalues and eigenfunctions. This behavior is intrinsically connected to that of the geodesic flow, reflecting the deep correspondence between quantum and classical dynamics. To exploit this connection, in collaboration with J. Galkowski, we have developed a framework that hinges on decomposing eigenfunctions into a superposition of geodesic beams. In this talk, I will introduce these techniques and explain how to use them to derive refined bounds on the standard estimates for the eigenfunction’s pointwise behavior and the Weyl Law for the eigenvalue counting function. A significant consequence of this method is that a quantitatively improved Weyl Law holds on most manifolds.

Elena Kim (MIT)

The support of semiclassical measures in higher dimensions

A central question in quantum chaos is how classical chaotic dynamics influence quantum behavior. On compact Riemannian manifolds, pure quantum states correspond to Laplacian eigenfunctions. The quantum unique ergodicity (QUE) conjecture of Rudnick and Sarnak predicts that on hyperbolic manifolds, all high-energy eigenfunctions become uniformly distributed. The asymptotic behavior of eigenfunctions can be formulated in terms of semiclassical measures, which describe the microlocal distribution of eigenfunction mass. One approach towards the QUE conjecture applies microlocal analysis and uncertainty principles to characterize the support of semiclassical measures. I will discuss recent work that uses the breakthrough higher-dimensional fractal uncertainty principle of Cohen. Using this uncertainty principle, we prove the first result on the support of semiclassical measures on real hyperbolic n-manifolds.

Riccardo Caniato (Caltech)

Area rigidity for the regular representation of surface groups

Starting from the celebrated results of Eells and Sampson, a rich and flourishing literature has developed around equivariant harmonic maps from the universal cover of Riemann surfaces into negatively curved target spaces. In particular, such maps are known to be rigid, meaning that they are unique up to natural equivalence. This rigidity property fails when the target space has positive curvature, and comparatively little is known in this framework. In this talk, given a closed Riemann surface with strictly negative Euler characteristic, and a unitary representation of its fundamental group on a separable complex Hilbert space H which is weakly equivalent to the regular representation, we aim to discuss a lower bound on the Dirichlet energy of equivariant harmonic maps from the universal cover of the surface into the unit sphere S of H, and to give a complete classification of the cases in which the equality is achieved. As an interesting corollary, we obtain a lower bound on the area of equivariant minimal surfaces in S, and we determine all the representations for which there exists an equivariant, area-minimising minimal surface in S. The subject matter of this talk is a joint work with Antoine Song (Caltech) and Xingzhe Li (Cornell University).

Lewis Bowen (UT Austin)

Benjamini-Schramm convergence of high genus random translation surfaces 

Benjamini-Schramm convergence is a notion which captures the local geometry of a random point on a random space. It was originally introduced to study random rooted finite planar graphs (while sending the number of vertices to infinity), but it has since been generalized to a wide range of objects. A translation surface is a surface on which the local geometry is that of the Euclidean plane everywhere except for a discrete set of points called singularities. At each singularity, there is a multiple of 2pi extra cone angle; that is, the local geometry is identical to the k-fold branched cover of the complex plane corresponding to the map z - z^k. The set of translation surfaces of genus g and area g admits a natural, Lebesgue-class, finite measure called Masur-Smillie-Veech (MSV) measure. I will speak about joint work with Kasra Rafi and Hunter Vallejos in which we prove Benjamini-Schramm convergence of MSV-distributed random translation surfaces as genus tends to infinity. 

Tina Torkaman (University of Chicago)

Entropy and self-intersection number of geodesic currents on compact hyperbolic surfaces 

Let X be a compact hyperbolic surface and C a geodesic current which is a geodesic-flow invariant measure. Denote the measure theoretic entropy of C by h_X​(C). In this talk, assuming C is ergodic, we give an upper bound on h_X​(C) in terms of its self-intersection number i(C,C) and the systole of X. In particular, we show that small self-intersection number forces small entropy.