# Scientific

## Polya’s Program for the Riemann Hypothesis and Related Problems

In 1927 Polya proved that the Riemann Hypothesis is equivalent to the hyperbolicity of Jensen polynomials for Riemann’s Xi-function. This hyperbolicity has only been proved for degrees d=1, 2, 3. We prove the hyperbolicity of 100% of the Jensen polynomials of every degree. We obtain a general theorem which models such polynomials by Hermite polynomials. This theorem also allows us to prove a conjecture of Chen, Jia, and Wang on the partition function. This is joint work with Michael Griffin, Larry Rolen, and Don Zagier.

## The orbit intersection problem for linear spaces and semiabelian varieties

We will introduce the Dynamical Mordell-Lang problem by Ghioca and Tucker.

After that, we explain the “orbit intersection problem” for linear spaces and semi-abelian varieties. This is joint work with Ghioca.

## Diophantine equations for fun (and profit?)

Michael Bennett (President, Canadian Mathematical Society; Professor of Mathematics, University of British Columbia)

Diophantine equations are one of the oldest, frequently celebrated and most abstract objects in mathematics. They crop up in areas ranging from recreational mathematics and puzzles, to cryptography, error correcting codes, and even in studying the structure of viruses. In this talk, Dr. Bennett will attempt to show some of the roles these equations play in modern mathematics and beyond.

## Triangular bases of integral closures

Triangular bases of integral closures

## Hybrid Krylov Subspace Iterative Methods for Inverse Problems

Inverse problems arise in many imaging applications, such as image

reconstruction (e.g., computed tomography), image deblurring, and

digital super-resolution. These inverse problems are very difficult

to solve; in addition to being large scale, the underlying

mathematical model is often ill-posed, which means that noise and

other errors in the measured data can be highly magnified in computed

solutions. Regularization methods are often used to overcome this

difficulty. In this talk we describe hybrid Krylov subspace based

regularization approaches that combine matrix factorization methods

with iterative solvers. The methods are very efficient for large scale

imaging problems, and can also incorporate methods to automatically

estimate regularization parameters. We also show how the approaches

can be adapted to enforce sparsity and nonnegative constraints.

We will use many imaging examples that arise in medicine and astronomy

to illustrate the performance of the methods, and at the same time

demonstrate a new MATLAB software package that provides an easy to use

interface to their implementations.

This is joint work with Silvia Gazzola (University of Bath) and

Per Christian Hansen (Technical University of Denmark).

## Rufus Bowen Conference - Lunchtime Speeches

These speeches were given during the remembrance lunch as part of the conference "Current Trends in Dynamical Systems and the Mathematical Legacy of Rufus Bowen".

## Rufus Bowen Conference - Lunchtime Slideshow

This slideshow and the accompanying toasts were given during the remembrance lunch as part of the conference "Current Trends in Dynamical Systems and the Mathematical Legacy of Rufus Bowen".

## The Case for T-Product Tensor Decompositions: Compression, Analysis and Reconstruction of Image Data

Most problems in imaging science involve operators or data that are

inherently multidimensional in nature, yet traditional approaches to

modeling, analysis and compression of (sequences of) images involve

matricization of the model or data. In this talk, we discuss ways in

which multiway arrays, called tensors, can be leveraged in imaging

science for tasks such as forward problem modeling, regularization and

reconstruction, video analysis, and compression and recognition of facial

image data. The unifying mathematical construct in our approaches to

these problems is the t-product (Kilmer and Martin, LAA, 2011) and

associated algebraic framework. We will see that the t-product permits

the elegant extension of linear algebraic concepts and matrix algorithms

to tensors, which in turn gives rise to new, highly parallelizable,

algorithms for the imaging tasks noted above.

## The Geometry of the Phase Retrieval Problem

Phase retrieval is a problem that arises in a wide range of imaging

applications, including x-ray crystallography, x-ray diffraction imaging

and ptychography. The data in the phase retrieval problem are samples of

the modulus of the Fourier transform of an unknown function. To

reconstruct this function one must use auxiliary information to determine

the unmeasured Fourier transform phases. There are many algorithms to

accomplish task, but none work very well. In this talk we present an

analysis of the geometry that underlies these failures and points to new

approaches for solving this class of problems.

## Quantum Graph Theory

Many numerical invariants of a graph, such as the independence number, clique number and chromatic number, have game theoretic descriptions. In these games a referee poses questions to two collaborating non-communicating players and they return answers. Quantum graph theory is concerned with how these graph parameters change when the players are allowed to use the random outcomes of quantum experiments to determine their answers.

In this talk I will explain these concepts, focusing on the chromatic number, survey some of what little is known about the quantum chromatic numbers of graphs, explain the connection between these ideas and famous open conjectures of A. Connes and B. Tsirelson, and introduce an algebra

affiliated with a graph whose representation theory determines the values of these parameters.

Biography:

Vern Paulsen is a Professor of Pure Mathematics and the Institute for Quantum Computing at the University of Waterloo. He was a Professor of Mathematics and John and Rebecca Moores Chair at the University of Houston before moving to Waterloo in 2015. His primary research focus is on the theory of operator algebras and their applications in quantum information theory. He is the author of five research monographs and over 100 research articles. He received his PhD from the University of Michigan.

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