Algebraic Geometry: Tropical, Convex, and Applied Bernd Sturmfels, University of California, Berkeley
In tropical arithmetic, the sum of two numbers is their maximum and the product of two numbers is their usual sum. Many results familiar from algebra and geometry, including the quadratic formula, the fundamental theorem of algebra, and Bezout's theorem, continue to hold in the tropical world. In this lecture we learn how to draw tropical curves and why evolutionary biologists might care about this.
The method immediately leads to questions about curves in the plane of higher degree, and in particular, to elliptic curves.
This lecture concerns convex bodies with an interesting algebraic structure. A primary focus lies on the geometry of semidefinite optimization. Starting with elementary questions about ellipses in the plane, we move on to discuss the geometry of spectrahedra, orbitopes, and convex hulls of real varieties.
The central curve of a linear program is the algebraic curve along which the interior point algorithms travel. We determine the degree, genus, and defining ideal of this curve. These invariants, as well as the total curvature of the curve, are expressed in the combinatorial language of matroid theory. This is joint work with Jesus De Loera and Cynthia Vinzant.
In the mid-20th century, pure and applied math split, and, in spite of the financial pressure for collaboration exerted by the NSF, they still largely go their own ways. I believe this is damaging to both. In my own experience, math comes alive through an exciting dialectic between theory on the one hand and examples, applications, and experiments on the other. The fantasy of a pecking order topped by the most abstract pure math was canonized by Bourbaki and, as I learned from critical emails last year, is accepted by large segments of the public. I will discuss how I see this affecting both K-12 instruction and the excessive specialization of all branches of math research.
Viewed from various perspectives, the evolution of a dynamical system over time can appear both orderly and extremely disordered. I will describe some mechanisms behind chaos and stability in dynamics and how in certain contexts this intermixing of behaviors is to be expected.
We discuss how limit shapes and facets form in simple models of random discrete interfaces. In particular, the "lozenge" tiling model is a model of random stepped surfaces; one can write down and solve a PDE that describes the limiting surface (when the mesh size tends to zero) for given boundary frame. The solutions are parametrized by complex analytic functions, in a similar manner to Weierstrass's parametrization of minimal surfaces (soap films) using conformal mappings.
Mathematics implicates motions and machines; computations and colorings; the strings and arrows of life. Perhaps the grandest expression of the beauty and power of mathematics is revealed in the quantification and qualification of that which is not there: holes. Topology-the mathematics of holes-will be surveyed with a fresh look at the many ways in which topology is used in data management, networks, and optimization.
Because I Love Mathematics: The Role of Disciplinary Grounding in Mathematics Education
Much like my mentor, Etta Falconer, I enjoy mathematics but have devoted a career to ensuring that students of all walks of life have opportunities to learn important mathematics. The role of the discipline of mathematics, mathematicians' ways of reasoning, and participation in the mathematical community have been a clear part of this work. In this talk, I discuss the recent focus in K-12 mathematics education on the Standards for Mathematical Practice in the Common Core State Standards in Mathematics and the need for teachers and students to be grounded in the reasoning habits of mathematics to ensure opportunities for future students to learn mathematics at the highest levels. Drawing on research and stories of future high school teachers and current middle and high school students, I discuss the ways in which mathematics as a discipline shapes teachers' views of teaching and students' opportunities to learn. I conclude with points mathematicians and mathematics educators should consider in discussing the important role of mathematics in mathematics education.
Project NExT (New Experiences in Teaching) demonstrated the role of mentoring in a select segment of the mathematics community at a critical point in the career development of those who participated. The Project's success confirmed the value of mentoring and its effectiveness, even when mentoring expands beyond the one-to-one form. This presentation will explore some of the many forms and benefits of structured mentoring, particularly for students at all levels and for young faculty. It will point out the uses of mentoring, along with other activities, to expand the mathematics community, increase its diversity, and enhance the development of its members.
We are familiar with the prime numbers as those integers that cannot be factored into smaller integers, but if we consider systems of numbers larger than the integers, the primes may indeed factor in those larger systems. We discuss various questions mathematicians ask about how primes may factor in larger systems; talk about both classical results and current research on the topic; and give a sense of the kind of tools needed to tackle these questions.
The concept of threshold or tipping point, a mathematical dimensionless quantity that characterizes the conditions required for the occurrence of a drastic transition between states, is central to the study of the transmission dynamics and control of diseases such as dengue, influenza, SARS, malaria, and tuberculosis, to name a few. The quantification of tipping point phenomena goes back to the modeling and mathematical work of Sir Ronald Ross (second Nobel laureate in medicine, 1911;) and his "students" (Kermack and McKendrick, 1927, 1932). Ross, in fact, proceeded to confront the challenges associated with understanding and managing malaria patterns at the population level right after the completion of his scientific malaria discoveries. The quantification of the concept of tipping point, in the context of epidemiology, has found countless applications directly tied in to the design, development, and implementation of public health policy. Ross's writings emphasized the value of mathematical models as integrators of multilevel information and processes, and his mathematical framework led to the development of a mathematical theory of infectious diseases (an outstanding review of the field can be found in Hethcote, SIAM Review, 2000). The overview in this lecture provides a personal perspective on the role of mathematical models in the study of the dynamics, evolution, and control of infectious diseases over multiple scales.
Few people expect to encounter mathematics on a visit to an art gallery or even a walk down a city street (or across campus). When we explore the world around us with mathematics in mind, however, we see the many ways in which mathematics can manifest itself, in streetscapes, sculptures, paintings, architectural structures, and more. This illustrated presentation offers illuminating glimpses of mathematics, from Euclidean geometry and normal distributions to Riemann sums and Möbius strips, as seen in a variety of structures and artworks in such cities as Washington, D.C.; Philadelphia; Toronto; Montreal; New Orleans; andMadison, Wisconsin.