# Global controllability to trajectories for the viscous Burgers equation and applications * CANCELLED *

# Palindromic and anti-palindromic closures in symbolic dynamics

The palindromic and anti-palindromic closures are used in combinatorics on words in order to generate Sturmian, Thue-Morse or Rote words. Usually, these words are generated either by substitutions or by discrete dynamical systems given by rotations on the torus with a well chosen partition. Thus in the first part of the talk, we will see how to use directive words in order to generate these specific words and an interesting class of words. We will focus also on Justin's formula in order to compute the palindromic closure and we extend this tool to the anti-palindromic case. In a second part, we will investigate geometric palindromic closure to construct finite steps of the famous Rauzy fractal linked to a generalization of the Fibonacci sequence (namely the Tribonacci case). The construction leads to construct Rauzy fractals in all dimensions using geometrical transformations.

# On the Correction for Misclassification Bias in Drug Safety Data Using Validation Sample Approaches

Outcome misclassification in patient health records can bias estimation of adverse drug reaction risk. In this discussion, we will first consider a binary setting and demonstrate the use of internal validation sampling to offset misclassification bias in estimation of the odds-ratio. Investigation of the relative efficiency of odds-ratio estimators arising from the use of conditional versus random validation sampling will be investigated in simulation studies, focusing on differences in the selection of the categorical composition underlying the validation data. A Monte Carlo approximation to validation sample size determination will be recommended. To address the additional influence of confounding, we will introduce an inverse probability weighted approach to rebalance covariates across treatment groups while continuing to mitigate the impact of misclassification bias.

Next, for right censored continuous time survival data, failing to observe the event of interest can introduce misclassification bias in risk estimation. Incorrectly observing cause-specific event types at correctly recorded event times can also introduce bias. An internal validation sampling approach is used to update a set of parametric likelihoods to produce unbiased estimates in scenarios with the presence of either or both of these errors. These approaches are validated through large simulation studies.

# Ricci curvature and geometric analysis on graphs

Ricci curvature lower bound play very important rule for geometric analysis on Riemannian manifold. So it is very interesting to introduce similar concept on discrete setting especially on graphs. We will talk about the Ricci curvature lower bound on graphs where the original idea comes from the Bochner formula on Riemannian geometry. Given the Ricci curvature lower bound on graphs, we will imply some classic results from Riemannian manifold for eigenvalue estimate, gradient estimate, Harnack inequality and heat kernel estimate and so on.

# A rare-variant association test in family-based designs and non-normal quantitative traits

Rare variant studies are now being used to characterize the genetic diversity between individuals and may help to identify substantial amounts of the genetic variation of complex diseases and quantitative phenotypes. Family data have been shown to be powerful to interrogate rare variants. Consequently, several rare variants association tests have been recently developed for family-based designs, but typically, these assume the normality of the quantitative phenotypes. In this talk, we present a family-based test for rare-variants association in the presence of non-normal quantitative phenotypes. The proposed model relaxes the normality assumption and does not specify any parametric distribution for the marginal distribution of the phenotype. The dependence between relatives is modeled via a Gaussian copula. A score-type test is derived, and several strategies to approximate its distribution under the null hypothesis are derived and investigated. The performance of the proposed test is assessed and compared with existing methods by simulations. The methodology is illustrated with an association study involving the adiponectin trait from the UK10K project.

# Algorithms for motion by mean curvature flow of networks of surfaces

I will review some of the most popular numerical methods for simulating the motion of interfaces, including networks of them, by mean curvature and related flows. Motion by mean curvature arises as gradient descent for the sum of areas of surfaces in the network. This energy comes up in a number of applications ranging from image processing, computer vision, and machine learning, to materials science. In image processing, it is used for denoising of images while preserving sharp boundaries of objects (edges). In computer vision, it appears in the Mumford-Shah functional -- one of the most important variational models for image segmentation. In machine learning, it can be used for graph partitioning, e.g. in the context of supervised classification and recognition tasks. Finally, in materials science, the evolution of boundaries of single crystal pieces (grains) that make up a polycrystalline material (e.g. most metals and ceramics) is described by mean curvature flow. In all these applications, numerous topological changes are to be expected, and any relevant numerical method should be prepared to handle them. I will touch on Monte Carlo Potts, level set, phase field, and threshold dynamics methods that have been applied to these problems, and try to give a sense of where things stand.

# Orthogonal polynomials, reproducing kernels, and zeros of optimal approximants

In this talk, I will discuss some polynomials that solve a particular optimization problem in Dirichlet-type spaces. Examples of Dirichlet-type spaces include the Hardy space (analytic functions in the disk whose coefficients are square summable), the Bergman space (analytic functions whose modulus squared is integrable with respect to area measure over the whole disk), and the (classical) Dirichlet space (analytic functions in the disk whose image has finite area, counting multiplicity). The optimal approximants p(z) in question minimize the Dirichlet- type norm of p(z) f(z) - 1, for a given function f(z). I will examine the connections between these optimal approximants, orthogonal polynomials and reproducing kernels, and exploit these connections to describe what is currently known about the zeros.

This work is joint with D. Khavinson, C. Liaw, D. Seco, and A. Sola. In this talk, I will discuss some polynomials that solve a particular optimization problem in Dirichlet-type spaces. Examples of Dirichlet-type spaces include the Hardy space (analytic functions in the disk whose coefficients are square summable), the Bergman space (analytic functions whose modulus squared is integrable with respect to area measure over the whole disk), and the (classical) Dirichlet space (analytic functions in the disk whose image has finite area, counting multiplicity). The optimal approximants p(z) in question minimize the Dirichlet- type norm of p(z) f(z) - 1, for a given function f(z). I will examine the connections between these optimal approximants, orthogonal polynomials and reproducing kernels, and exploit these connections to describe what is currently known about the zeros.

This work is joint with D. Khavinson, C. Liaw, D. Seco, and A. Sola. In this talk, I will discuss some polynomials that solve a particular optimization problem in Dirichlet-type spaces. Examples of Dirichlet-type spaces include the Hardy space (analytic functions in the disk whose coefficients are square summable), the Bergman space (analytic functions whose modulus squared is integrable with respect to area measure over the whole disk), and the (classical) Dirichlet space (analytic functions in the disk whose image has finite area, counting multiplicity). The optimal approximants p(z) in question minimize the Dirichlet- type norm of p(z) f(z) - 1, for a given function f(z). I will examine the connections between these optimal approximants, orthogonal polynomials and reproducing kernels, and exploit these connections to describe what is currently known about the zeros.

This work is joint with D. Khavinson, C. Liaw, D. Seco, and A. Sola.* *

# Gauge Theories: Quivers, Dessins and Calabi-Yau

We discuss how bipartite graphs on Riemann surfaces capture a wealth of information about the physics and the mathematics of large classes of gauge theories, especially those arising from string theory in the context of AdS/CFT.

The dialogue between the physics, the underlying algebraic geometry of Calabi-Yau varieties, the combinatorics of dimers and toric varieties, as well as the number theory of dessin d'enfants becomes particularly intricate and fruitful under this light.

# Introduction to *-Autonomous categories, continued

I will finally get to some of the hard work and show, if time permits, the following two results:

1. Assuming that *V* is the subobject and product closure of a very nice category *S* of topological objects and that *S* contains a very nice injective *K*, then *K* is also injective in *V*.

2. The inclusion of the weak objects in *V* has a left adjoint and the inclusion of the strong objects has a right adjoint.

# Elliptic PDEs in two dimensions (SEMINAR CANCELLED)

I will give a short survey of the several approaches to the regularity theory of elliptic equations in two dimensions. In particular I will focus on some old ideas of Bernstein and their application to the infinity Laplace equation and to the Bellman equation in two dimensions.