# Scientific

## Brauer classes in moduli problems and arithmetic: Lecture 3

We cover Brauer classes, how they arise as obstructions on moduli spaces of sheaves, and how they can be used to obstruct rational points, highlighting recent links between the two.

## Theory of rational curves and its arithmetic applications: Lecture 2

We discuss deformation theory of rational curves and Mori’s famous Bend and Break techniques as well as their applications to Geometric Manin’s Conjecture. The lecture series contain introductory components as well as problem sessions and they aim for graduate students and postdocs.

## Brauer classes in moduli problems and arithmetic: Lecture 2

The Pacific Rim Mathematical Association Congress meets in December 2022. A number of summer schools will take place prior to the main event at the end of the year. This summer school is part of the PRIMA Special Session on Arithmetic geometry: theory and computation. In this summer school, we cover two topics:(1) Brauer classes in moduli problems and arithmetic and (2) theory of rational curves and its arithmetic applications.

## Theory of rational curves and its arithmetic applications: Lecture 1

We discuss deformation theory of rational curves and Mori’s famous Bend and Break techniques as well as their applications to Geometric Manin’s Conjecture. The lecture series contain introductory components as well as problem sessions and they aim for graduate students and postdocs.

## Brauer classes in moduli problems and arithmetic: Lecture 1

We cover Brauer classes, how they arise as obstructions on moduli spaces of sheaves, and how they can be used to obstruct rational points, highlighting recent links between the two.

## Environmental Escape from the Prisoner's Dilemma

During reproduction, viruses manufacture products that diffuse within the host cell. Because a virus does not have exclusive access to its own gene products, coinfection of multiple viruses allows for strategies of cooperation and defection— cooperators produce large amounts of gene product while defectors produce less product but specialize in appropriating a larger share of the common pool. Experimental data shows that, under conditions where coinfection is common, bacteriophage $\Phi$6 becomes trapped in a Prisoner’s dilemma, with defectors spreading to fixation, causing lowered population fitness. However, these experiments did not allow for fluctuation in the density of the external viral population. Here, I’ll discuss a model formulated to see if environmental feedback can free $\Phi$6 from the Prisoner’s dilemma. I’ll also discuss the concept of the Effective Game, which incorporates the frequency and density of different viral types in the environment.

## Random walks on Gromov hyperbolic spaes and Teichmüller spaces. Pacific Dynamics Seminar

n this talk, I will discuss random walks on Gromov hyperbolic spaces. Due

to the hyperbolicity of the spaces, random walks exhibit behaviors that

differ from the classic (Euclidean) ones. These behaviors include the

escape to infinity, central limit theorems when centered at the escape

rate, and geodesic tracking. I will explain how one can sharpen these

behaviors based on the recent observations by Gouëzel and Baik-Choi-Kim. If

time allows, I will also explain how one can implement this theory on

(non-hyperbolic) Teichmüller spaces.

## Dynamic Self Organization and Microscale Fluid Properties of Nucleoplasm

The principal function of the nucleus is to facilitate storage, retrieval, and maintenance of the genetic information encoded into DNA and RNA sequences. A unique feature of nucleoplasm—the fluid of the nucleus—is that it contains chromatin (DNA) and RNA.

In contrast to other important biological polymer hydrogels, such as mucus and extracellular matrix, the nucleic acid polymers have a sequence. Recent experiments have shown that during the growth phase of the cell cycle, chromatin condenses in a sequence specific manner into regions within the nucleoplasm, possibly so that functionally related genes are grouped together spatially even though they might be far apart in terms of sequence distance.

At the same time, we are becoming increasingly aware of the role of liquid-liquid phase separation (LLPS) in cellular processes in the nucleus and the cytoplasm. Complex molecular interactions over a wide range of timescales can cause large biopolymers (RNA, protein, etc) to phase separate from the surrounding nucleoplasm into distinct biocondensates (spherical droplets in the simplest cases).

I will discuss recent work modelling the role of nuclear biocondensates in neurodegenerative disease and several ongoing projects related to

modelling and microscopy image analysis.

## Epidemic arrivals and Antibiotic Calenders

Here I give two tiny talks on some of my research from the past couple years. In the first half of the talk I re-examine some popular heuristics for epidemic "time of spread" through the world airline network, and use hitting times and branching processes to explore the mathematical underpinnings of these observations. In the second half of the talk, we zoom in to exploring how antibiotics spreads through a single hospital, the various models and their conflicting recommendations. Mostly just some straightforward dynamical systems, with the opportunity for some cute asymptotic arguments on the side.

## Dynamical inference for biological processes through the lens of optimal transport

Understanding how cells change their identity and behaviour over time in living systems is a key question in many fields of biology. Measurement of cell states is inherently destructive, and so the relationship of the current state of a cell to some future state, or ‘fate’, cannot be observed experimentally. Trajectory inference refers to the general problem of trying to estimate various aspects of the state-fate relationship. We discuss optimal transport as a useful analytical tool for trajectory inference, and we develop a mathematical framework for recovering trajectories in both non-equilibrium as well as equilibrium systems.