University of Cape Town

Event Poster

## 2018 | Interlacing of Zeros and Wendroff's Theorem

**Speaker**: Dr. Kathy Driver

**ABSTRACT:**

Suppose x_1,..., x_n and y_1,..., y_{n-1} are two lists of real, distinct, points satisfying the INTERLACING property x_1 < y_1 < x_2 < y_2 < · · · < x_{n-1} < y_{n-1} < x_n. In 1961, Wendroff proved that if P_n(x) = (x - x_1) · · · (x - x_n) and P_{n-1}(x) = (x - y_1) · · · (x - y_n) are the monic (highest coefficient equal to 1) polynomials with simple zeros at x_1, x_2,..., x_n and y_1, y_2,..., y_{n-1} respectively, there exist infinitely many sequences P_0, P_1,... of orthogonal polynomials with P_n(x) and P_{n-1}(x) as above. We explain what sequences of orthogonal polynomials are and why they are important and we give the original straightforward beautiful elegant proof of Wendroff's Theorem which shows that the interlacing of the zeros of P_n and P_{n-1} is crucial in the construction of orthogonal sequences.

University of St Andrews

Event Poster

## 2016 | Mathematical Problems that Cannot be Solved

**Speaker**: Dr. James Mitchell

**ABSTRACT:**

Many early experiences of mathematics involve learning methods for solving certain types of problems, such as quadratic equations, finding derivatives in calculus, and so on. A conclusion that is often drawn from this is that every problem in mathematics can be resolved by means of some mechanical method. At the start of the 20th century this view was shared by many mathematicians. They sought to remove human ingenuity from mathematics and to find ''a universal procedure for determining the truth or falsity of any mathematical statement. ''In this talk I will address the question of whether or not all of mathematics can be reduced to a mechanical procedure, and what precisely a mechanical procedure is in the first place.

University of Waterloo

## 2014 | Game Theory and the Mathematics of Altruism

**Speaker**: Dr. Jason Bell

**ABSTRACT:**

Game theory is a branch of mathematics that deals with strategy and decision making and is applied in economics, computer science, biology, and many other disciplines as well. We will discuss some of the basic points of game theory and discuss the so-called iterated prisoner dilemma, a game that is of central importance in the study of cooperation between individuals. We will then describe various strategies to this game and explain why altruism is something that can evolve naturally.

Read more about Dr. Bell's talk in the UCCS Communique.

University of Arizona

## 2013 | Secrets From Deep Human History

**Speaker**: Dr. Joseph Watkins

**ABSTRACT:**

Spectacular advances in the technologies that produce genetic data and continued advancement in probability theory and statistical inference have combined to dramatically alter our understanding of deep human history. For example, the sequencing of ancient DNA from Neanderthal bones in Croatia and in Denisova Cave in the Altai Mountains came with new inferential strategies that led to the finding of their genetic signatures in modern humans. In this talk, we will explore the extent to which similar events took place in Africa where we have not yet found any ancient human remains to be sequenced. Comparisons with primate genomes provide a second look at deep human history. Thus, we will spend a few minutes to introduce the steps our group is taking to examine how evolutionary pressures have impacted the human genome.

## 2012 | Mathematics and the Ocean

**Speaker: **Dr. Jerry L. Bona

**ABOUT THE SPEAKER:**

Jerry L. Bona received his PhD in 1971 from Harvard University under supervision of Garrett Birkhoff and then worked at the Fluid Mechanics Research Institute at University of Essex. Subsequently he was faculty member at the University of Chicago, Pennsylvania State University and the University of Texas at Austin, before joining the University of Illinois at Chicago.

He is well-known for his contributions to the fields of fluid mechanics, partial differential equations and computational math and has been active in other branches of pure and applied mathematics, ocean engineering and economics.

Read more about Br. Bona's talk in the UCCS Communique.

**ABSTRACT:**

Describing various aspects of the Earth's oceans using mathematics goes back to the 17th century. Some of the world's greatest mathematicians and physicists have been involved in this enterprise. The lecture will begin with a cursory sketch of some of the more important milestones in the mathematics of the ocean. We will then move on to indicate briefly an example taken from water wave theory of how mathematical models are created. We then turn to some of the more spectacular applications of the theory. This will involve us in tsunami propagation, rogue waves and near-shore zone sand bars and beach protection, as time permits.

## 2011 | Random Planar Maps

**Speaker**: Dr. Omer Angel

**ABSTRACT:**

A planar map is a planar graph embedded in the plane, considered up to continuous deformations. These object have been studied extensively in combinatorics, physics (as discrete random surfaces) and more recently probability theory. Much progress has been made in recent years in understanding the typical structure of these objects, and glimpses of a deeper theory are visible, particularly connecting the scaling limit of random planar maps with conformally invariant models of statistical physics. I will survey some results and conjectures concerning these objects, and discuss some recent progress in understanding percolation and random walks on these random graphs.

Northwestern University

## 2010 | Computational Modeling of Neurons

**Speaker**: Dr. William L. Kath

**ABSTRACT:** With its approximately 100 billion neurons and 200 trillion connections, the human central nervous system is astoundingly complex. Nevertheless, experimental advances are rapidly revealing new insights about the workings of neurons and the networks in which they are connected. Simultaneously, computational models of neurons have grown swiftly in terms of both their capability and utility. When constrained by experimental data, such models greatly enhance the observations and provide tools to construct new experimentally testable predictions. In this talk I will describe how this two-pronged approach has helped explain some of the function of hippocampal CA1 pyramidal neurons, a group of principal cells in a region of the brain that is important for the formation of new memories. The models and experiments indicate that these relatively large neurons integrate and process their inputs in a two-stage manner, in that they first combine inputs in localized parts of the dendritic tree before making an ultimate determination whether or not to signal downstream neurons with an action potential.

**ABOUT THE SPEAKER:**

Bill Kath is a professor in the Departments of Engineering Sciences and Applied Mathematics & Neurobiology and Physiology. From 2005-2010 he was the Co-Director of the Northwestern Institute on Complex Systems at Northwestern University. His research interests include computational neuroscience, nonlinear optics, linear and nonlinear wave propagation and nonlinear dynamics. He received the NSF Presidential Young Investigator Award in 1985, was elected a Fellow of the Optical Society of America in 2007, and elected a Fellow of the Society for Industrial and Applied Mathematics in 2010. He has over 150 peer reviewed publication and 4 US patents.

University of Colorado at Boulder

## 2009 | Extraordinary Waves: From Beaches to Lasers

**Speaker**: Dr. Mark J. Ablowitz

**ABOUT THE SPEAKER:**

Mark Ablowitz is considered a pioneer in the field of applied mathematics, and his work in the field is among the most highly cited in the world. He is best known for his landmark contributions to the "inverse scattering transform," or IST, a method used to solve nonlinear wave equations. Mathematicians and physicists have used the IST to gain a better understanding of phenomena such as water waves. Ablowitz joined the CU-Boulder faculty in 1989.

**ABSTRACT:**

Waves are fascinating. There are a class of extraordinary localized waves, called solitary waves or solitons which were first documented 175 years ago. This lecture will trace the history of these waves, their associated mathematics and will explain why mathematics played a crucial role in both historical and modern developments. Applications range from water waves to giant internal ocean waves to long distance communications, lasers and Bose-Einstein condensation and more. The discussion will be general and will leave all equations behind.