# Scientific

## Random walks and branching random walks: old and new perspectives - Lecture 3

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 2

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.

## Random walks and branching random walks: old and new perspectives - Lecture 2

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 1

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.

## Random walks and branching random walks: old and new perspectives - Lecture 1

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.

## Introduction to agent-based evolutionary game theory

Evolutionary game theory is a discipline devoted to studying populations of individuals that are subject to evolutionary pressures, and whose success generally depends on the composition of the population. In biological contexts, individuals could be molecules, simple organisms or animals, and evolutionary pressures often take the form of natural selection and mutations. In socioeconomic contexts, individuals could be humans, firms or other institutions, and evolutionary pressures often derive from competition for scarce resources and experimentation.

In this talk I will give a very basic introduction to agent-based evolutionary game theory, a bottom-up approach to modelling and analyzing these systems. The defining feature of this modelling approach is that the individual units of the system and their interactions are explicitly and individually represented in the model. The models thus defined can be usefully formalized as stochastic processes, whose dynamics can be explored using computer simulation and approximated using various mathematical theories.

## Reproductive value, prevalence, and perturbation theory of Perron vectors

In a linear population model that has a unique “largest” eigenvalue and is suitably irreducible, the corresponding left and right (Perron) eigenvectors determine the long-term relative prevalence and reproductive value of different types of individuals, as described by the Perron-Frobenius theorem and generalizations. It is therefore of interest to study how the Perron vectors depend on the generator of the model. Even when the generator is a finite-dimensional matrix, there are several approaches to the corresponding perturbation theory. We explore an approach that hinges on stochasticization (re-weighting the space of types to make the generator stochastic) and interprets formulas in terms of the corresponding Markov chain. The resulting expressions have a simple form that can also be obtained by differentiating the renewal-theoretic formula for the Perron vectors. The theory appears well-suited to the study of infection spread that persists in a population at a relatively low prevalence over an extended period of time, via a fast-slow decomposition with the fast/slow variables corresponding to infected/non-infected compartments, respectively. This is joint work with MSc student Tareque Hossain.

## Phase dynamics of cyclic reptilian tooth replacement

For over a century, scientists have studied striking spatiotemporal patterns during the continual tooth replacement of reptiles. Aside from the compelling aesthetics of this phenomenon, it is thought that understanding the underlying mechanisms may provide the insight required to trigger adult tooth replacement in humans. Theoretical frameworks have long been proposed to understand the rules behind the observed spatiotemporal order, but have only been analyzed mathematically more recently. Starting from Edmund's observations in crocodiles and proposed theory of replacement waves, we show how a simple model consisting of a row of non-interacting phase oscillators predicts several experimental observations. Next, inspired by the hypothesis put forth by Osborn, we consider a variation of the phase model with ODEs that account for mutual inhibition between tooth sites, and use continuation methods to thoroughly search parameter space for experimentally validated solutions. We then extend the model to a PDE that explicitly accounts for the diffusion of inhibitory signals between teeth, yielding some novel solution types. Using continuation methods once again, we delineate parameter regimes with solutions that closely resemble experimental observations in leopard geckos.

## Almost periodicity and large oscillations of prime counting functions

If we assume the relevant Riemann hypotheses, after a suitable rescaling many functions counting certain primes become almost periodic. There are different notion of almost periodicity in use; here we consider the notion induced by the norm $||f|| = \sup_{x∈\mathbb{R}} \int_x^{x+1} |f(t)|^2\,dt$. We show that if a function $f$ can be approximated by linear combinations of periodic functions with respect to this norm, then the level sets $\left\{x: f(x) \geq t\right\}$ are almost periodic for all real $t$ with at most countably many exceptions. We also compare this notion to other notions of almost periodicity in use.

*Please note, the wrong video feed was captured for this lecture so the writing on the blackboard is not legible.*

## A race problem arising from elliptic curves

Given an elliptic curve $E/\mathbb{Q}$, we can consider its trace of Frobenius, denoted as $a_p(E)$, where $p$ is a prime. We will discuss the race problem arising from these ap values and the general strategy in attacking these problems.