Lonseth Lectures

Abstract: Inverse problems arise in all fields of science and technology where
causes for a desired or observed effect are to be determined. By
solving an inverse problem is in fact how we obtain a large part of
our information about the world. An example is human vision: from the
measurements of scattered light that reaches our retinas, our brains
construct a detailed three-dimensional map of the world around us. In
the first part of the talk we will describe several inverse problems
arising in different contexts.

In the second part of the lecture we will discuss invisibility. Can we
make objects invisible? This has been a subject of human fascination
for millennia in Greek mythology, movies, science fiction, etc
including the legend of Perseus versus Medusa and the more recent Star
Trek and Harry Potter. In the last 20 years or so there have been
several scientific proposals to achieve invisibility. We will describe
in a non-technical fashion a simple and powerful proposal, the
so-called transformation optics, and some of the progress that has
been made in achieving invisibility.

Biography: Gunther Uhlmann is the Robert R. and Elaine F. Phelps Endowed Professor at the University of Washington, where he has been since 1984. He has received Sloan and Guggenheim fellowships, and he is a Fellow of the American Academy of Arts and Sciences. Uhlmann's research focuses on inverse problems, applying mathematics to determine internal properties of various mediums. His work spans fields like medical imaging, geophysics, and material science. He has been recognized with prestigious awards such as the Bôcher Memorial Prize and the Kleinman Prize.

Abstract: We explore the benefits of partnerships between governments and private insurers in the context of inclusive insurance (also referred to as microinsurance), as powerful and cost-effective tools for achieving poverty reduction. To explore these ideas, we model the capital of a household from a ruin-theoretic perspective to measure the impact of microinsurance on poverty dynamics and the governmental cost of social protection. We compare a few scenarios: uninsured, insured without subsidies and insured with various subsidies. Although insurance alone (without subsidies) may not be sufficient to reduce the likelihood of falling into the area of poverty for specific groups of households, since premium payments constrain their capital growth, our analysis suggests that subsidised schemes can provide maximum social benefits while reducing governmental costs.

Biography: Dr. Constantinescu received her PhD in Mathematics from Oregon State University in 2006. Prior to graduate studies, Corina worked as an actuary and led the life insurance department of one of the first private Romanian insurance companies. During 2013-2016 she coordinated the EU-funded RARE network, connecting 12 prominent international institutions to work on the theoretical side of the analysis of ruin probabilities in case of disasters or extreme shocks for insurance-like risk pools. The RARE research led to the introduction and analysis of new risk measures and the (asymptotic) quantification of aggregated risks. Since 2018, Dr. Constantinescu is regularly teaching and supervising MSc students from the African Institute of Mathematical Science (AIMS) network. In 2020, she was one of the two academics named on the 100 Women to Watch list of the Cranfield University’s School of Management. Dr. Constantinescu publishes in both actuarial and applied probability journals. She serves as associate editor in a number of actuarial journals and is part of the publicity team of Bernoulli Society for Mathematical Statistics and Probability. Her expertise is in modeling insurance portfolios' dynamics and their response to solvency requirements. Her current interest is in the mathematics of insurance for (low-income) populations that cannot afford insurance. During her sabbatical year, 2022-23, she explores these financial inclusion aspects as a visiting scholar within the Social Finance Programme, International Labour Organization, United Nations.

On Friday, April 19, 2019, Mai Gehrke from Laboratoire J.A. Dieudonné, Université Côte d’Azur, France joined us to present "Using abstract mathematical structures to study algorithmic complexity questions."

Automata are very simple computational models. They are important in applications of computer science but also serve as a laboratory for studying the complexity of algorithms. In this talk we introduce automata and show how finite monoids, certain very abstract algebraic structures, may be assigned as invariants of automata. We illustrate how these invariants are powerful enough to make deep computational questions decidable. Finally we give a glimpse of an idea how this can be generalized to provide sophisticated mathematical tools for the study of computational complexity classes.

On Thursday, May 3, 2018, Henri Berestycki from EHESS, France joined us to present "Of Predator and Prey."

The classical Lotka-Volterra system that describes predator – prey interaction and is one of the cornerstones of mathematical ecology. An extension of the original Lotka- Volterra system that aims at showing how territories are formed as a result of strong competition between packs of predators.

Professor Berestycki defines himself as a mathematician in dialogue with other fields of knowledge. He has worked in several areas motivated by physics, in particular scalar field equations, reactive flows and combustion. More recently, he has been involved in research motivated by questions in social sciences, such as economics and finance and collective behaviors. Professor Berestycki’s Lonseth Lecture reflects his interest in modeling problems in ecology and reaction-diffusion equations.

On Tuesday, May 2, 2017, William Yslas Vélez from University of Arizona joined us to present the "Lonseth Lecture Series."

Many different events impact your decisions (note the plural) to continue your studies in mathematics. The solitary work that you perform when you do homework, the interactions with peers and the feedback from your instructors all play a role in your self-assessment. These self-assessments continue through life. You find that you have a fondness for one subject but not another and decisions are governed by your own personal skill sets and passion. Yes, passion!

At the undergraduate level, mathematics is presented as discrete subjects with little connecting ideas, like algebra and analysis. Yet this is not how mathematical research is done. In this presentation I will spend most of the time giving an example where ideas from algebra, number theory, analysis all played a vital role in providing insight into factoring primes (What?). But, I also want to talk a bit about your role in mathematics.

Your self-assessment can be a boon or a bane. It can promote your mathematical growth or hinder it. I will give examples of what I used in my almost fifty years in the mathematical profession to give me an exciting mathematical career. I invite you to think about how you have arrived at this wonderful period in your lives of being mathematics majors, what the barriers you have surmounted. I invite you to write down your thoughts on why you became a mathematics major and why you have continued and email your thoughts to me at the above address.

On Tuesday, May 10, 2016, Richard Tapia from Rice University joined us to present "The Remarkable Journey of Isoperimetric Problem: From Euler to Steiner to Weierstrass."

Dr. Richard Tapia, professor in the Department of Computational and Applied Mathematics at Rice University, will present his talk, "The Remarkable Journey of Isoperimetric Problem: From Euler to Steiner to Weierstrass,” an overview of the history of the impactful isoperimetric problem. He will identify three distinct classes of solution approaches that have been used throughout history: the Cartesian coordinate representation approach of Euler, the synthetic geometry approach of Steiner, and the parametric representation approach of Weierstrass. Euler incorrectly believed that he had established sufficiency, when in reality he had not even established the necessity that he has been credited with by mathematical historians. This failure led Steiner in 1838 to propose his approach which gave only necessity and not sufficiency as he believed.The Steiner path was completed by Lawlor in 1998. Euler’s and Steiner’s failures led Weierstrass in 1879 to propose his approach, which did indeed lead to sufficiency but was excessively long. The Weierstrass approach was completed in 1934 by Littlewood, Hardy, and Polya. Prof. Tapia will present a completion of Euler’s approach in a surprisingly elementary proof.

On Tuesday, May 12, 2015, Harold R. Parks from Oregon State University joined us to present "Plateau's Problem and the Geometry of Soap Films."

By dipping a wire frame in a soap solution, one can often produce a soap film spanning the frame. That soap film will have the smallest area among nearby surfaces that also span the frame. The mathematical model for this type of surface is a minimal surface. Plateau's problem is to show the existence of a minimal surface with a given boundary.

Plateau's problem, which dates back to 1873, turned out to be quite challenging, and it inspired some beautiful mathematics. The problem is also sufficiently open to interpretation and generalization that the 1936 awarding of a Fields Medal for its solution did not end the story. This lecture will discuss the history of Plateau's problem, its solution, and the people involved in solving it.

On Tuesday, May 20, 2014, David Pengelley from New Mexico State University joined us to present "Sophie Germain’s grand plan to prove Fermat’s Last Theorem."

Sophie Germain (1776-1831) is the first woman known to have created important new mathematical research. She is best known in number theory for the first general result aimed at proving Fermat's Last Theorem, finally proven only 20 years ago. However, unpublished manuscripts, and a letter to Gauss, reveal that for her this result was only minor fallout from a multifaceted grand plan she pursued for proving the theorem outright, emphasizing new theoretical techniques of broad applicability. The scope of Germain's attempt remained unequaled for half a century, until her path was unknowingly trodden again by others. We will explore her grand plan and its side thrusts, including remarkable lower bounds on the size of possible solutions to Fermat's equation. Her work has likely lain unread for nigh 200 years. We argue for a substantial elevation of her stature as a number theorist.

On Tuesday, May 21, 2013, Dr. Paul Zorn from St. Olaf College joined us to present "Extreme Calculus."

There is more to elementary calculus than may first meet the eye, even to those of us who have learned (or taught it) at various levels. With appropriate help from graphical, numerical, and algebraic computing, well-worn calculus techniques and topics---polynomials, optimization, root-finding, methods of integration, and more---often point to deeper, more general, more interesting, and sometimes surprising mathematical ideas and techniques. I'll illustrate my thesis with figures, examples, and a lot of e-calculation, aiming to take elementary calculus to its interesting extremes.

On Tuesday, May 15, 2012, David Bressoud from Macalester College joined us to present "Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture."

What is the role of proof in mathematics? Most of the time, the search for proof is less about establishing truth than it is about exploring unknown territory. In finding a route from what is known to the result one believes is out there, the mathematician often encounters unexpected insights into seemingly unrelated problems. I will illustrate this point with an example of recent research into a generalization of the permutation matrix known as the "alternating sign matrix." This is a story that began with Charles Dodgson (aka Lewis Carroll), matured at the Institute for Defense Analysis, drew in researchers from combinatorics, analysis, and algebra, and ultimately was solved with insights from statistical mechanics.

On Tuesday, May 24, 2011, J. Michael Shaughnessy from Portland State University joined us to present "Favorite Chance Encounters."

Dr. Shaughnessy has taught mathematics content courses and directed professional development experiences for mathematics teachers at all levels, K–12, as well as community college and university. He has authored or coauthored more than 60 articles, books, and book chapters on issues in the teaching and learning of mathematics. From 1996 to 2008 Dr. Shaughnessy served as the director of the doctoral program in mathematics education at Portland State University, Portland, Oregon. Throughout his career, his principal research interests in mathematics education have been the teaching and learning of statistics and probability and the teaching and learning of geometry. Shaughnessy served as a member of the Board of Directors of the National Council of Teachers of Mathematics (NCTM) from 2001 to 2004, and in April 2010, he began a two-year term as NCTM President. Dr. Shaughnessy received his Ph.D. in mathematics education from the Department of Mathematics at Michigan State University in 1976. He taught in the Department of Mathematics at Oregon State University from 1976 until 1991, and at Portland State University in the Department of Mathematics and Statistics from 1991 to 2008.

On Tuesday, May 11, 2010, Keith Devlin from Stanford University joined us to present "When Mathematics Changed Us."

At four distinct stages in the development of modern society, a mathematical development changed — in a fundamental, dramatic, and revolutionary way — how people understand the world. (A fifth such change may be taking place during our lifetime, but only history will say if this is really the case.) Those advances occurred around 5,000 B.C.E. and in the 13th, 16th, and 17th centuries. I will look at how human life and cognition changed on each of those four occasions.

Based in part on Devlin's recent book The Unfinished Game: Pascal, Fermat and the Seventeenth Century Letter that Made the World Modern, Basic Books 2008.

On Tuesday, May 19, 2009, Robert Daverman from University of Tennessee, A.M.S. joined us to present "Mysteries of the Cantor Set."

The Cantor set exhibits captivating and, occasionally, bizarre phenomena in diverse branches of mathematics. And it is a fundamentally important object -- anyone who completely understands the Cantor set is assured of mathematical success. This talk will describe some beguiling Cantor set properties and will conclude with several questions about it which the speaker wishes someone would/could answer.

On Tuesday, May 27, 2008, John W. Lee from Oregon State University joined us to present "Weierstrass Approximation Theorems."

Weierstrass published his celebrated approximation theorems in July of 1885. I will start with brief speculations on the antecedents of Weierstrass' work and move on to a selective survey of results and/or proofs related to Weierstrass' original theorems. The survey will include results and/or proofs of Lebesgue, Landau, de la Vallee Poussin, Bernstein, Korovkin, and Stone, as time permits.

On Tuesday, May 8, 2007, Professor Jim Douglas, Jr. from Purdue University joined us to present "The Role of Capillarity in Multiphase Flow in Porous Media."

Professor Lonseth was very interested in seeing that mathematics interact with other disciplines to improve the understanding of phenomena in these disciplines, and this lecture will be devoted to showing by three examples the importance of including the effects of capillarity in approximating multiphase flows in porous media. The first example involves a simple laboratory experiment and was responsible for reorienting an experimental procedure in a major petroleum research laboratory. The second relates to the mathematical description of multiphase flow in fractured media, where omitting capillarity leads to a seriously incorrect model. The third example, which concerns three-phase flow, exhibits a nonstandard shock that is incorrectly simulated (or not found) without capillarity. The presentation will not require expertise in simulating flows in porous media.

On Tuesday, May 2, 2006, Peter Lax from New York University joined us to present "Degenerate Symmetric Matrices."

If a finite group G acts on a set X in such a way that each non-trivial element of G fixes a unique point, then they all must fix the same point (i.e. G has a global fixed point which is necessarily unique). We will cover the proof of this result as given in a paper by Max Forester and Colin Rourke.

On Tuesday, May 10, 2005, Doug Arnold from Institute for Mathematics and its Applications, University of Minnesota joined us to present "The New Mathematical Gravitational Astronomy."

Contemporary understanding of the cosmos is based on on Einstein's amazing insight that gravity is simply a manifestation of curvature. One ineluctable, though subtle, consequence of this theory of general relativity, is that violent cosmic events--imagine two black holes wildly orbiting around each other in the moments before they merge--emit gravitational signals that propagate off into space. The nascent field of gravitational astronomy seeks to use these tiny ripples on surface of spacetime as our first window to the universe looking outside of the electromagnetic spectrum. The technological and scientific challenges of detecting gravity waves are immense, but the mathematical difficulties which must to be overcome to interpret these signals through computer simulation of general relativity may be the greatest of all. This lecture, held during the centenary of Einstein's annus mirabilis and on the heels of 2005 Mathematics Awareness Month dedicated to the theme Mathematics and the Cosmos, will discuss the fascinating emerging science of gravitational astronomy and the mathematics and mathematical challenges at its heart.

On Tuesday, May 11, 2004, Steven G. Krantz from Washington University in St. Louis joined us to present "A Matter of Gravity."

It is a standard topic in any multivariable calculus course to develop the concept of "centroid" or "center of gravity", and to teach the student to calculate this center. Rarely is there any further investigation into properties of the center of gravity. Nonetheless, there are interesting questions about the center of gravity that could have been asked three hundred years ago, but evidently were not addressed until recently. We consider some new features and properties of the concept of center of gravity. Both topological and geometrical aspects will be examined. Stability results are proved.

On Tuesday, April 29, 2003, John H. Ewing, Executive Director American Mathematical Society, joined us to present "The Mathematics Inside Your Computer."

Computers don't operate using only bits and bytes to perform logic and arithmetic. They use sophisticated mathematics to perform many of the routine tasks you take for granted every time you turn on your machine. This talk will survey a small sample of that sophisticated mathematics, from an unsophisticated point of view.

On Tuesday, May 28, 2002, Colin Adams from Williams College joined us to present "Real Estate in Hyperbolic Space: Investment Opportunities for the New Millennium."

Have you found the new investment climate a bit on the chilly side? Nervous about stocks, bonds and mutual funds? Afraid of risky investments in Euclidean space? Then real estate in hyperbolic space is for you. We will discuss the enormous potential of this new investment opportunity and describe the many fascinating properties of hyperbolic space that make it such an attractive place to live. This is the financial equivalent of the 1980's junk bond. Don't miss it. Bring your checkbook and credit references! No previous math or real estate background assumed! Recommended for students and faculty alike! Roger Ebert says, "Two fingers up!"

On Tuesday, May 16, 2000, noted Mathematical Historian and Biographer Constance Reid joined us to present "The Improbable Life of Richard Courant."

Almost thirty years after his death, Richard Courant remains a highly controversial figure in mathematics, complex and contradictory; but the message he emphasized throughout his long career was one that he had absorbed in his youth in Gottingen from David Hilbert and Felix Klein--the underlying unity of all the mathematical sciences, pure and applied.

On Tuesday, May 18, 1999, Kenneth A. Ross from University of Oregon joined us to present "The Mathematics of Card Shuffling."

How many times do you have to shuffle a deck of cards before it is well mixed? What do we mean by well mixed? Questions like this will be discussed and seen to lead to the study of random walks on certain finite groups. This is an expository talk on work by Persi Diaconis and his colleagues, though a colleague of mine and I have obtained some related but more technical results.

On Tuesday, May 5, 1998, Philip A. Anselone from Oregon State University joined us to present "The Power of Calculus: The Legacy of Newton."

Isaac Newton developed calculus and used it to derive universal laws of motion and gravitation that apply not only on Earth but also to the planets and stars. His laws justify and explain the pervious discoveries of Galileo and Kepler. Newton's laws, particularly force equals mass times acceleration and the universal law of gravitation, are introduced in calculus classes, but usually there isn't enough time to deal adequately with their all-important consequences. The result is that students do not fully appreciate the extraordinary magnitude of Newton's accomplishments. In this lecture we shall discuss three topics that stem directly from Newton's laws: 1. Escape velocity of a projectile launched from the Earth; 2. The derivation of Kepler's laws of planetary motion; 3. The representation of solid bodies as point masses. Even today some of the details of Newton's analysis have to be sketched in order to make the arguments reasonably accessible to calculus students.

On Tuesday, May 6, 1997, Margaret Wright from Bell Laboratories and Lucent Technologies joined us to present "Model, Speed up, Optimize, Remodel: Fun and Profit for Mathematics and It's Friends."

Mathematics plays a major role in formulating and modeling real-world problems--but models are never right the first time. So mathematics also enters in speeding up complicated calculations, optimizing whatever the current model may be, figuring out its defects, and then producing a more realistic model. This talk will describe how mathematicians and computer scientists have worked with experts in radio engineering and user interface design to produce not only a useful product for Lucent Technologies (a software tool for designing wireless communication systems), but also original mathematical research in optimization and computational geometry.

On Tuesday, May 7, 1996, Robert Osserman from Stanford University joined us to present "The Shape of the Universe."

Mathematics plays a major role in formulating and modeling real-world problems--but models are never right the first time. So mathematics also enters in speeding up complicated calculations, optimizing whatever the current model may be, figuring out its defects, and then producing a more realistic model. This talk will describe how mathematicians and computer scientists have worked with experts in radio engineering and user interface design to produce not only a useful product for Lucent Technologies (a software tool for designing wireless communication systems), but also original mathematical research in optimization and computational geometry.

On Thursday, May 25, 1995, Ronald L. Graham from AT&T Bell Labs joined us to present "Mathematics and Computers: Recent Successes and insurmountable Challenges."

There is no question that the recent advent of the modern computer has had a dramatic impact on what mathematicians do and how they do it. However, there is increasing belief that many apparently simple problems may in fact be forever beyond any conceivable computer approach. In the talk I will describe a variety of mathematical problems in which computers either have had, may have or will probably never have a significant role in their solutions.

On Tuesday, May 24, 1994, Tsit-Yuen Lam from University of California-Berkeley joined us to present "Mistakes We all Made: How Error-Free is Mathematics?"

Mathematics, as a subject, derives its beauty from its internal consistency and sound logic. It is thus axiomatic that the proofs and argumentations used in the development of mathematics be absolutely accurate and error-free. Yet the history of mathematics is replete with instances of false starts, half-truths, and incomplete or downright erroneous arguments. Even the greatest of mathematicians are known to have erred in their proofs. In the talk, Professor Lam will give a light-hearted view of some of the famous (or infamous) errors made in the long history of mathematics. Along the way, he will also comment on the pedagogical values of mistakes in mathematics, and discuss ways by which we may try to minimize our mistakes.

On Tuesday, April 27, 1993, Mary Ellen Rudin from University of Wisconsin-Madison joined us to present "Dimension."

When dealing with topological spaces which are note necessarily metric, we run into a variety of questions. We will discuss several rather nice classes of such spaces as well as some conjectures. We will prove, with the aid of one rather special example, that three of the conjectures are false.

On Tuesday, May 19, 1992, John Horton Conway from Princeton University joined us to present "On the Shape of Things."

Conway is recognized for his studies in combinatorics and group theory, which is the branch of algebra that studies the properties of symmetries of figures, and how you can go from one symmetry to another. Conway has made some major an fundamental discoveries in this field.

On Tuesday, May 14, 1991, Ian Stewart from University of Warwick joined us to present "Four Encounters With Sierpinski's Gasket."

Sierpinski's gasket is a fractal, obtained by repeatedly deleting the middle section of a triangle. It shows up in a number of different areas of mathematics, with surprising cross-connections. The talk will describe four occurrences of the gasket: 1. What Sierpinski originally invented it for; 2. Parity of binomial coefficients; 3. The Tower of Hanoi puzzle; 4. Michael Barnsley's Chaos Game.

On Tuesday, May 1, 1990, Serg Lang from Yale University joined us to present "A B C Conjecture."

Recently, there have been some deep new insights into classical old problems, like Fermat's last theorem. Some of these insights can be expressed in terms of fairly elementary mathematics involving polynomials and numbers. I will describe some of these insights.

On Tuesday, May 16, 1989, George Andrews from Pennsylvania State University joined us to present "Ramanujan's Lost Notebook."

The "Lost" Notebook provides us with a record (probably incomplete) of Ramanujan's discoveries during the last year of his life. A number of his formulas from this document have been proved and analyzed; however, many remain unproved and totally mysterious. We shall survey some of the topics covered by the "Lost" Notebook, and we shall consider some of those formulas which are still open.

On Tuesday, May 17, 1988, G. D. Chakerian from the University of California-Davis joined us to present "Cantor Dust Under a Binary Tree."

This lecture will deal with some of the more paradoxical properties of the real numbers, from a geometrical point of view. In particular, the famous Cantor ternary set will be used to illustrate the idea of a fractal, a set of fractional dimension.

On Tuesday, May 19, 1987, Gilbert Strang from MIT joined us to present "Chaos: Strange Attractors and Fractuals."

Professor Strang is noted for his illuminating lectures on a wide variety of mathematical topics. His talk should appeal to students and former students of mathematics and also to teachers of mathematics from high school through graduate school.

On Tuesday, May 20, 1986, Ivan Niven from University of Oregon joined us to present "Some Surprising Results in Elementary Mathematics."

Although the background assumed is modest, the results are ingenious and not widely known. Professor Niven is noted for his lucid presentations of mathematical ideas. His lecture should appeal to students and former students of mathematics and also to teachers of mathematics from high school through graduate school.