Georgia Institute of Technology College of Sciences

In this talk, for an ergodic probability preserving system $(X,\mathcal{B},m,T)$, we will discuss the existence of a function $f:X\to \mathbb{Z}^d$, whose corresponding cocycle satisfies the $d$-dimensional local central limit theorem.
As an application, we resolve a question of Huang, Shao and Ye, and Franzikinakis and Host regarding non-convergence in $L^2$ of polynomial multiple averages of non-commuting zero entropy transformations. If time allows, we will also discuss the first examples of failure of multiple recurrence for zero entropy transformations along polynomial iterates. This is joint work with Zemer Kosloff (arXiv:2409.05087). 

I will discuss some results from our ongoing work with Ian D. Morris which aims at a systematic study of projections of self-affine fractals.
After explaining the extension of classical results of Falconer to the projections of self-affine fractals, I will discuss:

• the existence of equilibrium states having non-exact dimensional linear projections (equilibrium states themselves are exact dimensional by Feng);
• the existence of self-affine fractals in dimensions at least 4, whose set of exceptional projections in the sense of Marstand Projection Theorem contains higher degree algebraic varieties in Grassmannians (such constructions are not possible even in Borel category in dimension 3 by the solution of a conjecture of Fassler-Orponen by Gan et.al., neither in any dimension if the linear parts of affinities acts strongly irreducibly on all exterior powers, by Barany, Hochman, Rapaport);
• the existence of self-affine fractals whose sumsets have lower than expected dimension without satisfying an arithmetic resonance (impossible in dimension 1 by Hochman, Shmerkin, Peres and in dimension 2 by Pyorala).

Travelling pulses and waves are a rich subset of feasible patterns in reaction-diffusion systems. Many have investigated their existence, stability, and other properties, but what happens if the deterministic dynamics is affected by random occurrences? How does the interplay between diffusion and noise influence the velocity, curvature, and stability of multidimensional patterns?

We consider reaction-diffusion systems with multiplicative noise on a spatial domain of dimension two or higher. The noise process is white in time, coloured in space, and invariant under translations; based on applications. Inspired by previous works on the real line, we establish the multidimensional stability of planar waves on a cylindrical domain on time scales that are exponentially long with respect to the noise strength. In the deterministic setting, multidimensional stability of planar waves on the whole space has been obtained, and we show to what extend we can do this in the stochastic case.

The metastability result above is achieved by means of a stochastic phase tracking mechanism that can be maintained over such long-time scales. The corresponding mild formulation of our problem features stochastic integrals with respect to anticipating integrands, which hence cannot be understood within the well-established setting of Itô-integrals. To circumvent this problem, we exploit and extend recently developed theory concerning forward integrals.

The geometry of non-arithmetic hyperbolic manifolds is mysterious in spite of how plentiful they are. McMullen and Reid independently conjectured that such manifolds have only finitely many totally geodesic hyperplanes and their conjecture was recently settled by Bader-Fisher-Miller-Stover in dimensions larger than 3. Their works rely on superrigidity theorems and are not constructive. In this talk, we strengthen their result by proving a quantitative finiteness theorem for non-arithmetic hyperbolic manifolds that arise from a gluing construction of Gromov and Piatetski-Shapiro. Perhaps surprisingly, the proof relies on an effective density theorem for certain periodic orbits. The effective density theorem uses a number of ideas including Margulis functions, a restricted projection theorem, and an effective equidistribution result for measures that are nearly full dimensional. This is joint work with K. W. Ohm.