# Scientific

## Multiplicative functions in short intervals

In this talk, we are interested in a general class of multiplicative functions. For a function that belongs to this class, we will relate its “short average” to its “long average”. More precisely, we will compute the variance of such a function over short intervals by using Fourier analysis and by counting rational points on certain binary forms. The discussion is applicable to some interesting multiplicative functions such as

$$

\mu_k(n), \frac{\phi (n)}{n}, \frac{n}{\phi (n)}, \mu^2(n)\frac{\phi(n)}{n},

\sigma_\alpha (n), (-1)^{\#\left\{p: p^k | n \right\}}

$$

and many others and it provides various new results and improvements to the previous result

in the literature. This is a joint work with Mithun Kumar Das.

This event is part of the PIMS CRG Group on *L*-Functions in Analytic Number Theory. More details can be found on the webpage here: https://sites.google.com/view/crgl-functions/crg-weekly-seminar

## Joint value distribution of L-functions

It is believed that distinct primitive L-functions are “statistically independent”. The independence can be interpreted in many different ways. We are interested in the joint value distributions and their applications in moments and extreme values for distinct L-functions. We discuss some large deviation estimates in Selberg and Bombieri-Hejhal’s central limit theorem for values of several L-functions. On the critical line, values of distinct primitive L-functions behave independently in a strong sense. However, away from the critical line, values of distinct Dirichlet L-functions begin to exhibit some correlations.

This is based on joint works with Shota Inoue.

This event is part of the PIMS CRG Group on L-Functions in Analytic Number Theory. More details can be found on the webpage here: https://sites.google.com/view/crgl-functions/crg-weekly-seminar

## Rotary Molecular Motors Driven By Transmembrane Ionic Currents

There are two rotary motors in biology, ATP synthase and the bacterial flagellar motor. Both are driven by transmembrane ionic currents. We consider an idealized model of such a motor, essentially an electrostatic turbine. The model has a rotor and a stator, which are closely fitting cylinders. Attached to the rotor is a fixed density of negative charge, with helical symmetry. Positive ions move longitudinally by drift and diffusion on the stator. A key assumption is local electroneutrality of the combined charge distribution. With this setup we derive explicit formulae for the transmembrane current and the angular velocity of the rotor in terms of the transmembrane electrochemical potential difference of the positive ions and the mechanical torque on the motor. This relationship between "forces" and "fluxes" turns out to be linear, and given by a symmetric positive definite matrix, as anticipated by non-equilibrium thermodynamics, although we do not make any use of that formalism in deriving the result. The equal off-diagonal terms of this 2x2 matrix describe the electromechanical coupling of the motor. Although macroscopic, the model can be used as a foundation for stochastic simulation via the Einstein relation.

## Gaps in the sequence square root n mod 1

In this talk I will present some of aspects of the proof of a theorem of Elkies and McMullen (Duke Math Journal, 2004) on the asymptotic distribution of the gap sizes for the finite sequence ( √n (mod 1) : 1 ≤ n ≤ N ) as N goes to infinity. The proof relies, among other things, on tools from homogenous dynamics and relates the problem to one in the geometry of numbers.

## Fubini foiled: pathological foliations from symbolic codings

In this talk, I'll present a counterintuitive construction of A. Katok (exposited by Milnor) which at first glance seems to contradict Fubini's theorem. In particular, one can build a full-measure set E in the unit square and a foliation of the square by smooth curves such that any leaf of the foliation meets E in exactly one point. If time permits, I'll also mention work of Ruelle-Wilkinson and Shub-Wilkinson that shows these sorts of pathological examples are common in non-uniformly hyperbolic dynamics.

## Agent-based modelling and topological data analysis of zebrafish patterns

Patterns are widespread in nature and often form during early development due to the self-organization of cells or other independent agents. One example are zebrafish (Danio rerio): wild-type zebrafish have regular black and gold stripes, while mutants and other fish feature spotty and patchy patterns. Qualitatively, these patterns display impressive consistency and redundancy, yet variability inevitably exists on both microscopic and macroscopic scales. I will first discuss an agent-based model that suggests that both consistency and richness of patterning on zebrafish stems from the presence of redundancy in iridophore interactions. In the second part of my talk, I will focus on how we can quantify features and variability of patterns to facilitate predictive analyses. I will discuss an approach based on topological data analysis for quantifying both agent-level features and global pattern attributes on a large scale. The proposed methodology is able to quantify the differential impact of stochasticity in cell interactions on wild-type and mutant patterns and predicts stripe and spot statistics as a function of varying cellular communication. This is joint work with Alexandria Volkening and Melissa McGuirl.

## Combined modeling and experimental study of the interplay between tissue growth and shape regulation during Drosophila wing disc development

The regulation and maintenance of an organ’s shape is a major outstanding question in developmental biology. The Drosophila wing imaginal disc serves as a powerful system for elucidating design principles of the shape formation in epithelial morphogenesis. Yet, even simple epithelial systems such as the wing disc are extremely complex. A tissue’s shape emerges from the integration of many biochemical and biophysical interactions between proteins, subcellular components, and cell-cell and cell-ECM interactions. How cellular mechanical properties affect tissue size and patterning of cell identities on the apical surface of the wing disc pouch has been intensively investigated. However, less effort has focused on studying the mechanisms governing the shape of the wing disc in the cross-section. Both the significance and difficulty of such studies are due in part to the need to consider the composite nature of the material consisting of multiple cell layers and cell-ECM interactions as well as the elongated shape of columnar cells. Results obtained using iterative approach combining multiscale computational modelling and quantitative experimental approach will be used in this talk to discuss direct and indirect roles of subcellular mechanical forces, nuclear positioning, and extracellular matrix in shaping the major axis of the wing pouch during the larval stage in fruit flies, which serves as a prototypical system for investigating epithelial morphogenesis. The research findings demonstrate that subcellular mechanical forces can effectively generate the curved tissue profile, while extracellular matrix is necessary for preserving the bent shape even in the absence of subcellular mechanical forces once the shape is generated. The developed integrated multiscale modelling environment can be readily extended to generate and test hypothesized novel mechanisms of developmental dynamics of other systems, including organoids that consist of several cellular and extracellular matrix layers.

## No IET is Mixing

In 1980, Katok proved that no interval exchange transformation (IET) is mixing for any Borel invariant measure. The same holds for any special flow constructed by mean of any IET and a roof function of bounded variation. In this talk, I aim to explain to you the proof of these results.

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## Statistical Estimation with Differential Privacy

Naively implemented, statistical procedures are prone to leaking information about their training data, which can be problematic if the data is sensitive. Differential privacy, a rigorous notion of data privacy, offers a principled framework to dealing with these issues. I will survey recent results in differential private statistical estimation, presenting a few vignettes which highlight novel challenges for even the most fundamental problems, and suggesting solutions to address them. Along the way, I’ll mention connections to tools and techniques in a number of fields, including information theory and robust statistics.

## Furstenberg's topological x2 x3 result

In 1967 Furstenberg proved that any closed subset of the one dimensional torus R/Z, invariant under the two maps x -> 2x mod 1, x -> 3x mod 1, is either finite or the entire torus. I will explain a proof of this result due to Boshernitzan (1994). Furstenberg's proof is slightly longer but perhaps more conceptual. I will explain the main steps in Furstenberg's approach and their connection to joinings.