# Seminars & Events for PACM/Applied Mathematics Colloquium

##### From Stochastic Modeling to Fractional Modeling - New Tools in Computational Science & Engineering

##### Asymptotics beyond all orders: the devil's invention?

"Divergent series are the invention of the devil, and it is shameful to base on them any demonstration whatsoever." --- N. H. Abel. The lecture will introduce the concept of an asymptotic series, showing how useful divergent series can be, despite Abel's reservations. We will then discuss Stokes' phenomenon, whereby the coefficients in the series appear to change discontinuously. We will show how understanding Stokes phenomenon is the key which allows us to determine the qualitative and quantitative behaviour of the solution in many practical problems. Examples will be drawn from the areas of surface waves on fluids, crystal growth, dislocation dynamics, localised pattern formation, and Hele-Shaw

##### Applications of diffusion maps in dynamical systems

There is great current interest in the use of diffusion maps for dimension reduction. We discuss some examples of diffusion methods applied to understanding dynamical data, in particular combining spectral approaches with delay coordinates. In addition, we extend the usual diffusion map construction by introducing local kernels, a generalization of the standard isotropic kernel. In fact, for data lying on a manifold, any Riemannian geometry can be generated by an appropriate local kernel. This work is joint with Tyrus Berry.

##### Community detection with the non-backtracking operator

Community detection consists in identification of groups of similar items within a population. In the context of online social networks, it is a useful primitive for recommending either contacts or news items to users. We will consider a particular generative probabilistic model for the observations, namely the so-called stochastic block model and prove that the non-backtracking operator provides a significant improvement when used for spectral clustering. joint work with C. Bordenave and L. Massoulie.

##### Physics-inspired algorithms and phase transitions in community detection

Detecting communities, and labeling nodes, is a ubiquitous problem in the study of networks. Recently, we developed scalable Belief Propagation algorithms that update probability distributions of node labels until they reach a fixed point. In addition to being of practical use, these algorithms can be studied analytically, revealing phase transitions in the ability of *any* algorithm to solve this problem. Specifically, there is a detectability transition in the stochastic block model, below which no algorithm can label nodes better than chance. This transition was subsequently established rigorously by Mossel, Neeman, and Sly, and Massoulie.I'll explain this transition, and give an accessible introduction to Belief Propagation and the analogy with free energy and the cavity method of statistical physics.

##### Exact post-selection inference

**This is a joint PACM/ORFE Colloquium: **We describe a framework for exact post-selection inference.At the core of our framework is what we call selective inferencewhich allows us to define selective Type I and II errors for hypothesis tests and selective coverage for intervals. Several examples are discussed including the LASSO, forward stepwise, change point detection. This is joint work with several: Will Fithian, Jason Lee, Richard Lockart, Joshua Loftus, Dennis Sun, Yuekai Sun, Ryan Tibshirani, Rob Tibshirani.

##### Special PACM Seminar: Critical points and integral geometry of smooth Gaussian random functions

**Please note special day, time and location. **In this survey talk, we describe some what might be described as the geometric theory of smooth (marginally stationary) Gaussian random functions. Beginning at the local level, the celebrated Kac-Rice formulacan be used to derive accurate approximations to the distribution of the maximum of such random functions. From this local calculation,global Riemannian integral invariants appear. Zooming out to a global level,the appearance of these integral invariants can be explained by appealing to the Kinematic Fundamental Formula.

##### Set Oriented Numerical Methods for Dynamical Systems and Optimization

Over the last two decades so-called *set oriented* numerical methods have been developed in the context of the numerical treatment of dynamical systems. The basic idea is to cover the objects of interest - for instance *invariant **sets* or *invariant measures *- by outer approximations which are created via multilevel subdivision techniques. At the beginning of this century these methods have been modified in such a way that they are also applicable to the numerical treatment of *multiobjective optimization **problems*. Due to the fact that they are set oriented in nature these techniques allow for the direct computation of the entire so-called *Pareto set*.

##### The physical and mathematical structure of images

Images are both maps of continuous physical phenomena and discrete mathematical objects. While Shannon established the fundamental relationship between physical and mathematical images over half a century ago, considerable further progress in understanding this relationship has been achieved in the past quarter century through the development of wavelets and compressive measurement. This talk reviews simple physical models for the physical to mathematical transformation and discusses strategies for coding the physical interface to increase measurement efficiency. We specifically discuss novel sampling strategies for x-ray tomography, diffraction tomography and focal imaging.

##### Conservation of knottiness in real and idalized fluids

**This is a joint talk with PACM/MAE. ** To tie a shoelace into a knot is a relatively simple affair. Tying a knot in a field is a different story, because the whole of space must be filled in a way that matches the knot being tied at the core. The possibility of such localized knottedness in a space-filling field has fascinated physicists and mathematicians ever since Kelvin’s 'vortex atom' hypothesis, in which the atoms of the periodic table were hypothesized to correspond to closed vortex loops of different knot types. More recently, knottiness (Helicity) has re-emerged as a conserved quantity in manyidealized situations (such as Euler fluids and ideal plasmas).

##### Eigenvector Centralities for Temporal Networks

Motivated by a series of examples (the U.S. Ph.D. exchange in mathematics, co-starring relationships among top-billed actors in Hollywood, and co-citations between U.S. Supreme Court decisions), we study the general problem of calculating eigenvector centralities as generalized to temporal directed network data. We investigate a parsimonious extension of hub and authority scores to "multilayer" situations where multiple slices of network data connect similar groups of agents in different settings. By varying the coupling between layers, we develop a family of centrality measures whose values are localized in time in one extreme (weakly coupled) and constant across time in the other (strongly coupled).

##### When Exactly Do Quantum Computers Provide a Speedup?

Twenty years after the discovery of Shor's factoring algorithm, I'll survey what we now understand about the structure of problems that admit quantum speedups. I'll start with the basics, discussing the hidden subgroup, amplitude amplification, adiabatic, and linear systems paradigms for quantum algorithms. Then I'll move on to some general results, obtained by Andris Ambainis and myself over the last few years, about quantum speedups in the black-box model. These results include the impossibility of a superpolynomial quantum speedup for any problem with permutation symmetry, and the largest possible separation between classical and quantum query complexities for any problem.

##### Belief Propagation Algorithms: From Matching Problems to Network Discovery in Cancer Genomics

* This is a PACM Distinguished Lecture.* We review a certain class of algorithms, belief propagation algorithms, inspired by the study of phase transitions in computationally difficult problems. We show how these algorithms can be used both in the mathematical analysis of relatively simple problems like matching, and in the heuristic analysis of more complex problems. In particular, we show how particular forms of these algorithms can be used to discover pathways in cancer genomics, and to suggest possible drug targets for cancer therapy. These methods give us the ability to share information across multiple patients to help reconstruct highly patient-specific networks and potential treatments.

##### Efficient numerical methods for wave scattering in periodic geometries

A growing number of technologies (communications, imaging, solar energy, etc) rely on manipulating linear waves at the wavelength scale, and accurate numerical modeling is key for device design. I will focus on diffraction problems where time-harmonic scalar waves scatter from piecewise-uniform periodic media. After reviewing the integral equation method, I explain two innovations that allow our solvers to be high-order, robust, and optimal O(N) complexity: 1) new surface quadrature schemes in 2D and 3D compatible with the fast multipole method, and 2) a simple way to "periodize" the integral equation so that the unknowns live on only a single period of the geometry. The resulting linear systems are solved iteratively or in a fast direct fashion. I will present efficient solvers for gratings with up to a thousand periodic interface

##### Disordered Hyperuniform Materials: New States of Matter

PLEASE CLICK ON COLLOQUIUM TITLE FOR COMPLETE ABSTRACT. While there are four commonly observed states of matter (solid crystal, liquid, gas, and plasma), we have known for some time now that there exist many other forms of matter. For example, both quasicrystals and liquid crystals are states of matter that possess properties that are intermediate between those of crystals and conventional liquids.

##### Hypothesis Testing on Random Networks

I will discuss some new hypothesis testing problems on random graph models. A popular example is the problem of detecting community structure in a graph. Here we will consider more exotic situations, such as testing one of the basic assumption in social network analysis: whether the graph has "geometric structure". We will also talk about dynamical models of random graphs (such as preferential attachment), and how to test different hypotheses on the "history" of these graphs.

##### Is there Hope for Structure in Network Science?

The rapid growth and availability of data is changing the face of science. One particularly challenging area has been understanding and analyzing complex networks, which naturally model relationships in many types of data (e.g. friendships in a social network or protein/gene interactions in a biological one).

##### Manifolds on the verge of a regularity breakdown

Invariant manifolds are landmarks that organize the long term behavior of dynamical systems with complicated trajectories. From the point of view of applications, it is interesting to know that they persist under perturbations or that one can validate the the numerical calculations finding them. There are two main theorems in this direction: Normal hyperbolicity and KAM theory. We will present numerical explorations and rigorous results on what happens on the boundary of validity of these theorems. There are some surprising regularities and scaling relations, some of which can be proved rigorously. This is joint work with many people including R. Calleja, A. Haro, T. Blass.

##### Computational imaging for phase, 3D and super-resolution imaging

Computational imaging involves the joint design of optical systems and post-processing algorithms, such that computation replaces optical elements, enabling simple experimental setups. This talk will describe new optical microscopes that employ simple experimental architectures and efficient nonlinear inverse algorithms to achieve high-resolution 3D and phase images. By leveraging recent advances in computational illumination, we achieve brightfield, darkfield and phase contrast images simultaneously, with extension to 3D and gigapixel phase imaging. We discuss unique challenges for large-scale real-time imaging of biological samples in vitro and in vivo.

##### Minimax Rates for Poisson Compressed Sensing

Sparse inverse problems in the presence of Poisson noise with physical constraints arise in a variety of applications, including photon-limited imaging systems based on compressed sensing. The performance of compressed sensing in these settings can be markedly different from classical settings. Prior results on Poisson compressed sensing provided upper bounds on mean squared error performance; however, it was unknown whether those bounds were tight or if other estimators could achieve significantly better performance. This work provides minimax lower bounds on mean-squared error for sparse Poisson inverse problems under physical constraints. The lower bounds are complemented by minimax upper bounds which match the lower bounds for certain problem sizes and noise levels.