# Scientific

## Trading linearity for ellipticity: a nonsmooth approach to Einstein's theory of gravity and Lorentzian splitting theorems

While Einstein’s theory of gravity is formulated in a smooth setting, the celebrated singularity theorems of Hawking and Penrose describe many physical situations in which this smoothness must eventually breakdown. In positive-definite signature, there is a highly successful theory of metric and metric-measure geometry which includes Riemannian manifolds as a special case, but permits the extraction of nonsmooth limits under dimension and curvature bounds analogous to the energy conditions in relativity: here sectional curvature is reformulated through triangle comparison, while and Ricci curvature is reformulated using entropic convexity along geodesics of probability measures.

This lecture explores recent progress in the development of an analogous theory in Lorentzian signature, whose ultimate goal is to provide a nonsmooth theory of gravity. In work in progress, we aim to establish a low regularity splitting theorem by sacrificing linearity of the d’Alembertian to recover ellipticity. We exploit a negative homogeneity $p-$ d’Alembert operator for this purpose. The same technique yields a simplified proof of Eschenberg (1988) Galloway (1989) and Newman’s (1990) confirmation of Yau’s (1982) conjecture, bringing all three Lorentzian splitting results into a framework closer to the Cheeger-Gromoll splitting theorem from Riemannian geometry.

## Condensation phenomena in random trees - Lecture 3

Consider a population that undergoes asexual and homogeneous reproduction over time, originating from a single individual and eventually ceasing to exist after producing a total of n individuals. What is the order of magnitude of the maximum number of children of an individual in this population when n tends to infinity? This question is equivalent to studying the largest degree of a large Bienaymé-Galton-Watson random tree. We identify a regime where a condensation phenomenon occurs, in which the second greatest degree is negligible compared to the greatest degree. The use of the "one-big jump principle" of certain random walks is a key tool for studying this phenomenon. Finally, we discuss applications of these results to other combinatorial models.

## Random walks and branching random walks: old and new perspectives - Lecture 16

This course will focus on two well-studied models of modern probability: simple symmetric and branching random walks in ℤd. The focus will be on the study of their trace in the regime that this is a small subset of the ambient space.

We will start by reviewing some useful classical (and not) facts about simple random walks. We will introduce the notion of capacity and give many alternative forms for it. Then we will relate it to the covering problem of a domain by a simple random walk. We will review Lawler’s work on non-intersection probabilities and focus on the critical dimension $d=4$. With these tools at hand we will study the tails of the intersection of two infinite random walk ranges in dimensions d≥5.

A branching random walk (or tree indexed random walk) in ℤd is a non-Markovian process whose time index is a random tree. The random tree is either a critical Galton Watson tree or a critical Galton Watson tree conditioned to survive. Each edge of the tree is assigned an independent simple random walk in ℤd increment and the location of every vertex is given by summing all the increments along the geodesic from the root to that vertex. When $d\geq 5$, the branching random walk is transient and we will mainly focus on this regime. We will introduce the notion of branching capacity and show how it appears naturally as a suitably rescaled limit of hitting probabilities of sets. We will then use it to study covering problems analogously to the random walk case.

## Random matrix theory of high-dimensional optimization - Lecture 16

Optimization theory seeks to show the performance of algorithms to find the (or a) minimizer x∈ℝd of an objective function. The dimension of the parameter space d has long been known to be a source of difficulty in designing good algorithms and in analyzing the objective function landscape. With the rise of machine learning in recent years, this has been proven that this is a manageable problem, but why? One explanation is that this high dimensionality is simultaneously mollified by three essential types of randomness: the data are random, the optimization algorithms are stochastic gradient methods, and the model parameters are randomly initialized (and much of this randomness remains). The resulting loss surfaces defy low-dimensional intuitions, especially in nonconvex settings.

Random matrix theory and spin glass theory provides a toolkit for theanalysis of these landscapes when the dimension $d$ becomes large. In this course, we will show

how random matrices can be used to describe high-dimensional inference

nonconvex landscape properties

high-dimensional limits of stochastic gradient methods.

## Subshifts with very low word complexity

he word complexity function p(n) of a subshift X measures the number of n-letter words appearing in sequences in X, and X is said to have linear complexity if p(n)/n is bounded. It’s been known since work of Ferenczi that linear word complexity highly constrains the dynamical behaviour of a subshift.

## A computational phase transition for pressure approximation

In this talk, we will review some new techniques and limitations for achieving efficient approximation algorithms for entropy and pressure in the context of Gibbs measures defined over countable groups. Our starting point will be a deterministic formula for the Kolmogorov-Sinai entropy of measure-preserving actions of order-able amenable groups. Next, we will review techniques based on random orderings, mixing properties of Markov random fields, and percolation theory to generalize previous work. As a by-product of these results, we will obtain conditions for the uniqueness of the equilibrium state and the locality of pressure, among other implications that are not strictly algorithmic.

## Subset Problems in Combinatorial Number Theory Coding Problems in Symbolic Dynamics

TBA

## Divisibility of integer Laurent polynomials and dynamical systems

Let f, p, and q be Laurent polynomials in one or several variables with integer coefficients, and suppose that f divides p + q. In joint work with Klaus Schmidt, we establish sufficient conditions to guarantee that f individually divides p and q. These conditions involve a bound on coefficients, a separation between the supports of p and q, and, surprisingly, a requirement on f called atorality about how the complex variety of f intersects the multiplicative unit torus. The proof uses an algebraic dynamical system related to f and the fundamental dynamical notion of homoclinic point. Without the atorality assumption this method fails, the validity of our result in this case remains an open problem. We have recently learned that if the general case could be proved (even a very special version of it), there would be important consequences in determining whether certain upper triangular groups have trivial Poisson boundary, but already the proven case does have implications for this.

## Shifts of finite type defined by one forbidden block and a Theorem of Lind

We focus on shift of finite type (SFTs) obtained by forbidding one word from an ambient SFT. Given a word in a one-dimensional full shift, Guibas and Odlyzko, and Lind showed that its auto-correlation, zeta function and entropy (of the corresponding SFT) all determine the same information. We first prove that when two words both have trivial auto-correlation, the corresponding SFTs not only have the same zeta function, but indeed are conjugate to each other, and this conjugacy is given by a chain of swap conjugacies. We extend this result to the case when the ambient SFT is the golden mean shift, with an additional assumption on the extender set of the words. Then, we introduce SFTs obtained by forbidding one finite pattern from a higher dimensional ambient SFT. We prove that, when two patterns have the same auto-correlation, then there is a bijection between the language of their corresponding SFTs. The proof is a simple combination of a replacement idea and the inclusion-exclusion principle. This is joint work with Nishant Chandgotia, Brian Marcus and Jacob Richey.

## Shift equivalence implies flow equivalence for reducible shifts of finite type

Explain the implication stated in the title.