# Scientific

## Rigidity of Geodesic Planes in Hyberbolic Manifolds

TBA

## Factors of Gibbs measures on subshifts (1 of 2)

Classical results of Dobrushin and Lanford-Ruelle establish, in rough terms, that for a local energy function on a subshift without too much long-range order, the translation-invariant Gibbs measures are precisely the equilibrium measures. There are multiple definitions of a Gibbs measure in the literature, which do not always coincide. We will discuss two of these definitions, one introduced by Capocaccia and the other used by Dobrushin-Lanford-Ruelle, and outline a proof (available at [arxiv.org/abs/2003.05532]) that they are equivalent.

We will also discuss forthcoming work, in which we show that Gibbsianness is preserved by pushforward through a certain kind of almost invertible factor map. As an application in one dimension, we show that for a sufficiently regular potential, any equilibrium measure on an irreducible sofic shift is Gibbs. As far as we know, this is the first reasonably general result of the Lanford-Ruelle type for a class of subshifts without the topological Markov property.

Joint work with Luísa Borsato, with extensive advice from Brian Marcus and Tom Meyerovitch.

This lecture was given in two parts. The video on this page was distributed as a pre-recorded session ahead of a second live lecture.

## Quantum Unique Ergodicity

TBA

## Caroline Series' The modular surface and continued fractions

Caroline Series' The modular surface and continued fractions

## Effective Equidistribution of Horospherical Flows in Infinite Volume

The horospherical flow on finite-volume hyperbolic surfaces is well-understood. In particular, effective equidistribution of non-closed horospherical orbits is known. New difficulties arise when studying the infinite-volume setting. We will discuss the setting in finite- and infinite-volume manifolds, and the measures that play a crucial role in the latter. Joint work with Jacqueline Warren.

## Modeling strict age-targeted mitigation strategies for COVID-19

In this talk, we will use compartmental models to examine the power of age-targeted mitigation strategies for COVID-19. We will present evidence that, in the context of strategies which end with herd immunity, age-heterogeneous strategies are better for reducing direct mortalities across a wide parameter regime. And using a model which integrates empirical data on age-contact patterns in the United States and recent estimates of COVID-19 mortality and hospitalization rates, we will present evidence that age-targeted approaches have the potential to greatly reduce mortalities and ICU utilization for COVID-19, among strategies which ultimately end the epidemic by reaching herd immunity. This is joint work with Maria Chikina.

## Boshernitzan's criterion for unique ergodicity

TBA

## Multiple fission cycles in Chlamydomonas

In this talk I will present a "dynamical paradigm" for modeling networks of interacting genes and proteins that regulate every aspect of cell physiology. The paradigm is based on dynamical systems theory of nonlinear ODEs, especially one- and two-parameter bifurcation diagrams. I will show how we have used this paradigm to unravel the mechanisms controlling "multiple fission" cycles in the photosynthetic green alga Chlamydomonas. While most eukaryotic cells maintain a characteristic size by executing binary division after each mass doubling, Chlamydomonas cells can grow more than eight-fold during daytime before undergoing rapid cycles of DNA replication, mitosis and cell division at night, which produce up to 16 daughter cells. We propose that this unusual strategy of growth and division (which is clearly advantageous for a photosynthetic organism) can be governed by a size-dependent bistable switch that turns on and off a limit cycle oscillator that drives cells through rapid cycles of DNA synthesis and mitosis. We show that this simple ‘sizer-oscillator’ arrangement reproduces the experimentally observed features of multiple-fission cycles and the response of Chlamydomonas cells to different light-dark regimes. Our model makes unexpected predictions about the size-dependence of the time of onset of cell-cycle oscillations after cells are transferred from light to dark conditions, and these predictions are confirmed by single-cell experiments.

Collaborators: Stefan Heldt and Bela Novak (Oxford Univ) on the modeling; Fred Cross (Rockefeller Univ) on the experiments.

## Explicit results about primes in Chebotarev's density theorem

Let $L/K$ be a Galois extension of number fields with Galois group $G$, and let $C⊂G$ be a conjugacy class. Attached to each unramified prime ideal p in OK is the Artin symbol $\sigma p$, a conjugacy class in $G$. In 1922 Chebotarev established what is referred to his density theorem (CDT). It asserts that the number $\pi C(x)$ of such primes with $\sigma p=C$ and norm $Np≤x$ is asymptotically $\left|C\right|\left|G\right|\mathrm{Li} (x)$ as $x\rightarrow\infty$ where $\mathrm{Li} (x)$ is the usual logarithmic integral. As such, CDT is a generalisation of both the prime number theorem and Dirichlet's theorem on primes in arithmetic progressions. In light of Linnik's result on the least prime in an arithmetic progression, one may ask for a bound for the least prime ideal whose Artin symbol equals C. In 1977 Lagarias and Odlyzko proved explicit versions of CDT and in 1979 Lagarias, Montgomery and Odlyzko gave bounds for the least prime ideal in the CDT. Since 2012 several unconditional explicit results of these theorems have appeared with contributions by Zaman, Zaman and Thorner, Ahn and Kwon, and Winckler. I will present several recent results we have proven with Das, Ng, and Wong.

## Regular Representations of Groups

A natural way to understand groups visually is by examining objects on which the group has a natural permutation action. In fact, this is often the way we first show groups to undergraduate students: introducing the cyclic and dihedral groups as the groups of symmetries of polygons, logos, or designs. For example, the dihedral group $D_8$ of order 8 is the group of symmetries of a square. However, there are some challenges with this particular example of visualisation, as many people struggle to understand how reflections and rotations interact as symmetries of a square.

Every group G admits a natural permutation action on the set of elements of $G$ (in fact, two): acting by right- (or left-) multiplication. (The action by right-multiplication is given by $\left{t_g : g \in G\right}, where $t_g(h) = hg$ for every $h \in G$.) This action is called the "right- (or left-) regular representation" of $G$. It is straightforward to observe that this action is regular (that is, for any two elements of the underlying set, there is precisely one group element that maps one to the other). If it is possible to find an object that can be labelled with the elements of $G$ in such a way that the symmetries of the object are precisely the right-regular representation of $G$, then we call this object a "regular representation" of $G$.

A Cayley (di)graph $Cay(G,S)$ on the group $G$ (with connection set $S$, a subset of $G$) is defined to have the set $G$ as its vertices, with an arc from $g$ to $sg$ for every $s$ in $S$. It is straightforward to see that the right-regular representation of $G$ is a subset of the automorphism group of this (di)graph. However, it is often not at all obvious whether or not $Cay(G,S)$ admits additional automorphisms. For example, $Cay(Z_4, {1,3})$ is a square, and therefore has $D_8$ rather than $Z_4$ as its full automorphism group, so is not a regular representation of $Z_4$. Nonetheless, since a regular representation that is a (di)graph must always be a Cayley (di)graph, we study these to determine when regular representations of groups are possible.

I will present results about which groups admit graphs, digraphs, and oriented graphs as regular representations, and how common it is for an arbitrary Cayley digraph to be a regular representation.