Isabel Vogt: A brief history of Homotopy TheoryFriday, April 19 @ 3:30 PM Science Center 530In this talk we will explore the development of the concept of homotopy and homotopy groups. We will start from the origin of the use of continuous deformation in the classic work of Cauchy on the theory of contour integrals. We will then move onto the development of the fundamental group -- whose beginnings trace back to before the introduction of the concept of an abstract group! It will be very apparent that the discovery and standardization of the abstract notion of a group made this task immensely easier and enabled the breakthroughs of Poincare, Cech and Hurewicz in defining the fundamental group and higher homotopy groups in general. We will then move on to motivate the introduction of the higher homotopy groups with a discussion of the history of computation of homotopy groups of spheres, and Eilenberg obstruction theory. The emphasis will be on motivating concepts with both a modern and historical perspective. |
Tony Feng: The Life and Work of RamanujanFriday, April 12 @ 3:30 PM Science Center 530The story of Srinivasa Ramanujan is one of the great gems of mathematical history. Born to a poor family in India, and with no formal training in mathematics, Ramanujan nevertheless made momentous discoveries that have influenced the course of modern mathematics. He rediscovered the prime number theorem, Riemann's zeta function and its functional equation, and also introduced start ling results in such diverse subjects as infinite series, continued fractions, and arithmetic functions. In this talk, we will give an overview of the life, work, and legacy of Ramanujan. After describing his unconventional education and mathematics, we will focus on the story of his famous tau function, and discuss his findings from a modern perspective. |
Lucia Mocz: Emmy Noether and the Dawn of Commutative AlgebraFriday, April 5 @ 3:30 PM Science Center 530The story of Emmy Noether is quite legendary. Born at the end of the 19th century, she became an internationally renowned mathematician in a time when mathematics was only a man's game. While her determination and sheer force of character serve as an inspiration to anyone who studies her life, in this talk we aim to elucidate Noether's contributions to mathematics s and place her work in the context of the mathematics in the forefront of research in her time. We may immediately observe that Noether's work is divided into three epochs, and that there are distinct differences in her mathematical style in each of these time periods. Our focus will be her second epoch, which comprised some of her greatest contributions, not the least among which was her invention of modern-day commutative algebra. We will, however, also discuss how she transitioned into this line of research, as well as how this work gave meaning to the conjectures she formulated in her third epoch. Time permitting, we will also compare her work to that of other mathematicians in her time, namely Hermann Weyl, who she held a close friendship with and whose work, although very disjoint, greatly influenced her own thinking on mathematics. The goal of this lecture will be to come to understand a prominent and often-overlooked mathematician's professional development and influence to mathematics. |
Caitlin Stanton: Evolution of Group TheoryFriday, March 29 @ 3:30 PM Science Center 530The evolution of group theory didn't begin with the formal, abstract definition of a group. Progress in the subject occurred well before the term "group" was used: when Gauss proved that the order of any element in (Z/pZ)^{*} divides (p-1), he didn't have the concept of an abstract group or even a finite Abelian group. In the early days of group theory, mathematicians worked with the properties of specific groups, which invariably involved closure, associativity, identity, and inverses. The development of group theory occurs in four distinct threads: classical algebra, number theory, geometry, and analysis. In this talk, I will narrow our focus to classical algebra and the evolution of permutation groups, starting with early attempts at solving polynomials in the 16th century and ending with the development of Galois theory in the 19th century. |
Levent Alpoge: The Pell EquationFriday, March 8 @ 3:30 PM Science Center 530Studied about a thousand years before it was erroneously named for Pell by Euler, the equation (and its variants) has quite a rich history. Solution methods go back to Brahmagupta and Bhaskara of the 600s, Fermat and Brouncker (and not Pell ...) of the 1600s, and others. I will discuss these methods of solution, and talk a bit about continued fractions. Then I will turn to Gauss and Kronecker, who found expressions for the "minimal" solutions of these equations in terms of trigonometric functions (and a class number) and products of values of Dedekind's \eta function, respectively. I will end with a mention of our collective ignorance: we don't even know the "d" for which (or) has a solution in the integers, though we do have conjectures! |
Anthony Liu: Infinitesimals and the evolution of the calculusFriday, March 1 @ 3:30 PM, Science Center 530Abstract: The development of the calculus in the 17th century introduced just as many questions as it answered. In particular, the new subject's dependence on infinitesimals - quantities simultaneously non-zero and zero - made its logical foundations incredibly unsound. The quest to formalize the fundamental ideas of the calculus spanned centuries, with each generation of mathematicians making their own small adjustments. Slowly, the original geometric conception of the calculus gave way to its modern "analytical" form, and the definitions of functions, limits, and continuity evolved into the concepts we recognize today. In this talk, we will begin by setting the scene for the development of the calculus. Then, in a whirlwind tour, we will visit the work of Leibniz, the Bernoullis, Euler, Cauchy, and Weierstrass, describing in-depth the different ways each conceived of these fundamental notions. |
William Dunham: Heron, Newton, Euler, and BarneyFriday, February 22 @ 3:30 PM, Science Center 507Abstract: Heron's formula, giving the area of a triangle in terms of the lengths of its sides, is one of the great, peculiar results of plane geometry. It is thus to be expected that, over the years, there have been multiple demonstrations of this remarkable formula. Here, I consider four such proofs. Heron's original was a clever if convoluted exercise in Euclidean geometry. Centuries later, Isaac Newton gave a demonstration whose heavy lifting was done by algebra rather than geometry. Leonhard Euler's proof was geometric and exhibited his characteristic flair. Then, in an unsolicited 1990 letter, someone named Barney Oliver shared with me an elegant trigonometric argument where the symmetry of the formula was mirrored by the symmetry of the proof itself. The first two of these Heron's and Newton's, I'll mention only briefly. The last two, Euler's and Barney's we'll prove in detail. The arguments are elementary but not trivial. Taken together, they remind us why the history of our discipline is such a fine source for wonderful mathematics. |
Barry Mazur: History of Mathematics as a ToolFriday, February 15 @ 3:30 PM, Science Center 232Abstract: As an introductory lecture (and, I hope, discussion session) for the seminar, it seems appropriate to ask the question: Why history? Well, a good story is an end in itself. But here - as a sample - are three issues that may motivate us to make fundamental use of the "historical tool." A tool rather than an end. Dealing with ideas in the form they were first discovered often shines a light on the primal motivation for them, and shows them (sometimes, but not always) in their least technical dress. Why did anyone dream up the notion of homotopy, and homotopy groups?Dealing with the evolution of concepts are sometimes the best way of coming to an understanding of those concepts. The concept may be quite a moving target; learning how it came to be what it now is may be the most efficient route for understanding it, and especially for gauging how it may evolve in the future. What was the route that the simple idea of "dimension" took, to include - as it does today - concepts ranging from invariants of C*-algebras to fractals? How did the notions of "topology" and "sheaf" get shaped? How did the notions of "group" and "group representation" evolve, or did they spring fully formed into being? Becoming aware of how certain evident, natural - trivial, it would seem - concepts of mathematics constituted great leaps of insight and intuition when they first emerged on the mathematical scene. This awareness is a great advantage to have, when one teaches! E.g., the notion of linearity is so intuitive to us. But if you could only whisper the one word linear to John Wallis while he was writing his ``The Arithmetic of Infinitesimals" you could have saved him 50 pages of text. In my lecture I'll continue this simple taxonomy of questions that history (of mathematics, of course) may help us answer, and I'll choose one such question and talk about it in more detail. |