Invited Addresses

• Earl Raymond Hedrick Lecture Series

  • Classical structure in modern geometry, or modern structure in classical geometry

    Ravi Vakil, Professor of Mathematics, Stanford University

    One of the beauties of mathematics is the fact that many themes run through the subject, over many centuries. Many classical ideas continue to bear fruit in modern contexts, and modern ideas can still shed new light on classical problems. The Hedrick series will explore this in geometry. This series is intended for a general mathematical audience, and the talks will be independent.

    • Lecture 1: The mathematics of doodling

      Thursday, August 6, 10:30 – 11:20 am

      Doodling has many mathematical aspects: patterns, shapes, numbers, and more. Not surprisingly, there is often some sophisticated and fun mathematics buried inside common doodles. I'll begin by doodling, and see where it takes us.

    • Lecture 2: Murphy's Law in geometry

      Friday, August 7, 9:30 – 10:20 am

      When mathematicians consider their favorite kind of object, the set of such objects often has a richer structure than just a set – often some sort of geometric structure. For example, it may make sense to say that one object is “close to” another. As another example, solutions to equations (or differential equations) may form manifolds. These “moduli spaces” often are hoped to behave well (for example be smooth). I'll explain how many ones algebraic geometers work with are unexpectedly as far from smooth as they possibly can be.

    • Lecture 3: Generalizing the cross ratio: the space of n points on the projective line

      Saturday, August 8, 9:30 – 10:20 am

      Four ordered points on the projective line, up to projective equivalence, are classified by the cross ratio, a notion introduced by Cayley in the nineteenth century. This theory can be extended to more points, leading to one of the first important examples of an invariant theory problem, studied by Kempe, Hilbert, and others. Instead of the cross ratio (a point on the projective line), we get a point in a larger projective space, and the equations necessarily satisfied by such points exhibit classical combinatorial and geometric structure. For example, the case of six points is intimately connected to the outer automorphism of S_6. Much of the talk will be spent discussing the problem, and an elementary graphical means of understanding it. This is joint work with Ben Howard, John Millson, and Andrew Snowden.

  Biography: Ravi Vakil is a Professor of Mathematics and the David Huntington Faculty Scholar at Stanford University. He was born in Toronto, Canada, and studied at the University of Toronto, where he was a four-time winner of the Putnam competition (“Putnam Fellow”). He received his Ph.D. from Harvard in 1997, and taught at Princeton and MIT before moving to Stanford in 2001. He is an algebraic geometer, and his work involves many other parts of mathematics, including topology, string theory, applied mathematics, combinatorics, number theory, and more. His awards include the Alfred P. Sloan Research Fellowship, the National Science Foundation CAREER Award, the American Mathematical Society Centennial Fellowship, and the Presidential Early Career Award for Scientists and Engineers. He is the Robert K. Packard University Fellow in Undergraduate Education, and has won the Dean's Award for Distinguished Teaching. He works extensively with talented younger mathematicians at all levels, from high school (through math circles, camps, and olympiads), through recent Ph.D.'s.

• MAA Invited Addresses

  • Predicting Values of Arbitrary Functions

    Alan Taylor, Union College

    Thursday, August 6, 8:30 – 9:20 am

    To what extent is a function's value at a point x of a topological space determined by its values in an arbitrarily small (deleted) neighborhood of x? For continuous functions, the answer is typically “always” and the method of prediction of ƒ(x) is just the limit operator. Chris Hardin and I generalized this observation on limits to the case of an arbitrary function mapping a topological space to an arbitrary set, and showed that the best one can ever hope to do is to predict correctly except on a scattered set. Moreover, we produced a predictor whose error set is always scattered. In this talk, we outline the proofs of these two theorems and then derive some of the main results from our two earlier papers, “An introduction to infinite hat problems” (Mathematical Intelligencer, 2008) and “A peculiar connection between the axiom of choice and predicting the future” (American Mathematical Monthly, 2008). In particular, we show that given the values of a function on an interval (-∞, t), the strategy produces a guess for the value of the function at t and this guess is correct except for a countable set that is nowhere dense. In this sense, if time is modeled by the real line, then the present can almost always be correctly predicted from the past.

    Biography: Alan Taylor is the Marie Louise Bailey Professor of Mathematics at Union College, where he has been since receiving his PhD in mathematics from Dartmouth College in 1975. Trained as a logician, Taylor spent the first fifteen years of his career in the area of infinitary combinatorics, with particular interests in ideals, ultrafilters, and certain aspects of Ramsey theory. By 1990, Taylor's interests had shifted to the mathematical study of fair division and voting, where the next fifteen years saw the publication of five books in those areas. For the past three years, Taylor has somewhat returned to infinitary combinatorics, and this will be the focus of his MAA lecture. Taylor was a Sigma Xi Distinguished Lecturer and won the Union College Stillman Prize for Teaching and the Clarence F. Stephens Distinguished Teaching Award from the Seaway Section of the MAA.

  • The mathematics of collective synchronization

    Steven Strogatz, Jacob Gould Schurman Professor of Applied Mathematics, Cornell University

    Thursday, August 6, 9:30 – 10:20 am

    Every night along the tidal rivers of Malaysia, thousands of male fireflies congregate in the mangrove trees and flash on and off in silent, hypnotic unison. This display extends for miles along the river and occurs spontaneously; it does not require any leader or cue from the environment. Similar feats of synchronization occur throughout the natural world, whenever large groups of self-sustained oscillators interact. This lecture will provide an introduction to the Kuramoto model, the simplest mathematical model of collective synchronization. Its analysis has fascinated theorists for the past 35 years, and involves a beautiful interplay of ideas from nonlinear dynamics, statistical physics, and fluid mechanics. Classic results, recent breakthroughs, and open problems will be discussed, and a video of synchronous fireflies will be shown.

    Biography: Steven Strogatz is the Jacob Gould Schurman Professor of Applied Mathematics at Cornell University. He is fascinated by nonlinear dynamics and complex systems, especially as they manifest themselves in everyday life. For example, he and his students have explored the wobbling of London's Millennium Bridge on its opening day; the network math behind “six degrees of separation” and the Kevin Bacon game; the rhythms of our sleep-wake cycles; the ups and downs of love affairs; the geometry of DNA; and the synchronous flashing of fireflies. He has received numerous awards for his research, teaching, and public service, including: a Presidential Young Investigator Award from the National Science Foundation (1990); MIT's highest teaching prize, the E. M. Baker Award for Excellence in Undergraduate Teaching (1991); the Tau Beta Pi Teaching Award from Cornell's College of Engineering (2006); and the Communications Award from the Joint Policy Board for Mathematics (2007), a lifetime achievement award for the communication of mathematics to the general public. His books include the textbook Nonlinear Dynamics and Chaos (Perseus, 1994); the popular science book Sync (Hyperion, 2003), which was chosen as a Best Book of 2003 by Discover magazine and has been translated into six languages; and The Calculus of Friendship (Princeton University Press), scheduled for publication in July 2009.

  • Statistics in algebraic combinatorics

    Greg Warrington, University of Vermont

    Saturday, August 8, 8:30 – 9:20 am

    A central tension in mathematics is knowing how much to forget. Retain too many properties and the conjecture is not true. Lose too much structure and there is nothing meaningful to say. A variation of this balance is especially evident in algebraic combinatorics; oftentimes the objects of study are shadows of deep algebraic and geometric constructs.

    The association of statistics (i.e., weights) to simple combinatorial objects lets us recover some of the deeper structure. For example, permutations index a class of geometric objects known as Schubert varieties. By recording the number of inversions of a permutation we obtain the dimension of the corresponding variety.

    In this talk I describe some statistics on familiar combinatorial objects such as permutations, lattice paths and partitions. These statistics can be appreciated for the beautiful identities they satisfy and the surprising relationships among them. I will illustrate both qualities with examples. However, such statistics can also serve to illuminate the theory of symmetric functions. I will describe several situations where the underlying algebra suggests we should be able to find statistics satisfying certain properties. In a few cases such statistics have been found/invented; in other cases we are still looking.

    Biography: Greg Warrington is an assistant professor of mathematics at the University of Vermont. He received his Ph.D. in mathematics from Harvard University and his B.A. from Princeton University. Prior to moving to Vermont this year, Greg held a postdoctoral position at the University of Massachusetts at Amherst, an NSF postdoctoral fellowship at the University of Pennsylvania, and an assistant professorship at Wake Forest University. He was also an AMS Project NExT Fellow (Forest Dot).

    Greg's research is in algebraic combinatorics. Specifically, he is interested in how concrete combinatorial structures can illuminate underlying algebraic and geometric ideas.

  • Cryptography: How to Keep a Secret

    Alice Silverberg, Professor of Mathematics and Computer Science, University of California at Irvine

    Saturday, August 8, 10:30 – 11:20 am

    When you send your credit card number over the Internet, cryptography helps to ensure that no one can steal the number in transit. Julius Caesar and Mary Queen of Scots used cryptography to send secret messages, in the latter case with ill-fated results. More recently, cryptography is used in electronic voting, and it is also used to “sign” documents electronically. While cryptography has been used for thousands of years, public-key cryptography dates only from the 1970's. Some recent exciting breakthroughs in public-key cryptography include elliptic curve cryptography, pairing-based cryptography, and identity-based cryptography, all of which are based on the number theory of elliptic curves. This talk with give an elementary introduction to cryptography, including elliptic curve and pairing-based cryptography.

    Biography: Alice Silverberg is a Professor of Mathematics and Computer Science at the University of California, Irvine. Her research interests include number theory and cryptography. She graduated summa cum laude in mathematics from Harvard, and earned a Certificate of Advanced Study from Cambridge and a PhD and Master's degree in mathematics from Princeton. Gender equity issues are a long-standing concern of hers, as an outgrowth of her time spent studying at traditionally male institutions. She serves on the Executive Committee of the AWM, and she has served on the AMS Council and a number of AMS committees. She was awarded Humboldt, Bunting, Sloan, IBM, and NSF Fellowships and an MSRI Research Professorship, and has held a number of visiting or consulting positions in the US and abroad, including at DoCoMo USA Labs, Xerox PARC, Bell Labs, the University of Erlangen, MSRI, IHES, the Max Planck Institute, Macquarie University, and IBM. Silverberg consulted for the TV show NUMB3RS, and occasionally writes mathematically-inspired Scottish country dances.

• MAA Lecture for Students

  Mathemagic with a Deck of Cards on the Interval Between 5.700439718 and 806581751709438785716606368564

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  Colm Mulcahy, Spelman College

  Thursday, August 6, 1:00 – 1:50 pm

  Some unavoidable coincidences – as well as some truly surprising ones – will be explored as we survey 21st century mathemagical creations/entertainments with a deck of cards, touching on topics in combinatorics, algebra, and probability.

  Biography: Colm Mulcahy earned a B.Sc and M.Sc. in Mathematical Science from University College Dublin, in Ireland, in the late 1970s. A few years after getting his PhD from Cornell University in 1985, based on research in the higher level theory of quadratic forms conducted under Alex F.T.W. Rosenberg, he joined the mathematics faculty at Spelman College, in Atlanta, Georgia. He's been there ever since, and served as department chair 2003-2006.

  Over the years, his interests have expanded to include number theory, geometry, geometric design, image processing and data compression, and he loves to engage undergraduate students in research in these and other areas. His expository Mathematics Magazine paper “Plotting and Scheming with Wavelets” earned him a Carl B. Allendoerfer Award from the MAA in 1997. He got hooked on mathematical card tricks a decade ago – arguably as a delayed reaction to Martin Gardner's extensive writings on the subject – and since 2004 has explored many mathemagical principles, both old and new, in the bimonthly “Card Colm” for MAA Online.

  He enjoys music, eating, cooking, running and cycling, though not necessarily in that order. He is married to mathematician Vicki Powers of Emory University.

• James R. Leitzel Lecture

  Communicating Among Communities and Calling for Change: Continuing the Improvement of Mathematics Education

  Joan Ferrini-Mundy, Michigan State University and the National Science Foundation

  Friday, August 7, 10:30 – 11:20 am

  Almost two decades ago Jim Leitzel's vision for the continued improvement of mathematics education called for communication among mathematicians, educational researchers, teacher educators, and others. Collaborations among stakeholders with diverse perspectives are central to many of today's major mathematics education initiatives. What shared commitments have emerged as most promising for improving mathematics learning? What is the role of undergraduate mathematics education, mathematics education research, and the mathematical education of teachers in addressing problems of national scope and urgency? A discussion of the challenges and opportunities in the current federal policy climate for continuing to call for change in mathematics education.

  Biography: Dr. Joan Ferrini-Mundy is the Director of the Division of Research on Learning in Formal and Informal Settings in the Directorate for Education and Human Resources at the National Science Foundation. While at NSF she holds her faculty position at Michigan State University where she is a University Distinguished Professor of Mathematics Education. Ferrini-Mundy has been a member of the Board of Directors of the National Council of Teachers of Mathematics, the Board of Governors of the Mathematical Association of America, and the Mathematical Sciences Education Board. She served as an ex officio member of the President's National Mathematics Advisory Panel in 2007-2008. Her research interests include calculus teaching and learning, the assessment of secondary school teachers' mathematical knowledge for teaching, K-12 mathematics and science educational improvement, and educational policy.

• NAM David Blackwell Lecture

  Why Should I Care About Elliptic Curves?

  Edray Goins, Purdue University

  Friday, August 7, 1:00 – 1:50 pm

  An elliptic curve possessing a rational point is an arithmetic-algebraic object: It is simultaneously a nonsingular projective curve with an affine equation , which allows one to perform arithmetic on its points; and a finitely generated abelian group , which allows one to apply results from abstract algebra.

  In this talk, we discuss some basic properties of elliptic curves, and give applications along the way.

  Biography: Edray Herber Goins received his doctorate in 1999 from Stanford University under the supervision of Daniel Bump. Dr. Goins works in the field of number theory, as it pertains to the intersection of representation theory and algebraic geometry. He has held positions at the world's premiere research institutions, including the National Security Agency in Ft. Meade, Maryland; the Mathematical Sciences Research Institute in Berkeley, California; the Institute for Advanced Study in Princeton, New Jersey; the Max Planck Institute for Mathematics in Bonn, Germany; Harvard University in Cambridge, Massachusetts; and the California Institute of Technology in Pasadena, California.

  In January 2004 he was featured in Black Issues in Higher Education as one of the “2004 Emerging Scholars of the Year.” The issue featured nine “young educators bring[ing] their passion and excitement for teaching, research, and training to the forefront of the academy.” In March 2006 he gave the annual Bharucha-Reid Lecture during the Faculty Conference on Research and Teaching run by the National Association of Mathematicians (NAM).

• Pi Mu Epsilon J. Sutherland Frame Lecture

  The Mathematics of Perfect Shuffles

  Persi Diaconis, Mary V. Sunseri Professor of Statistics and Mathematics, Stanford University

  Friday, August 7, 8:00 – 8:50 pm

  Magicians and gamblers can shuffle cards perfectly (demonstrations provided). Understanding what can (and cannot) be done with shuffles leads to math problems, some beyond modern mathematics. The math is also useful for describing all sorts of computer algorithms.

• AWM-MAA Etta Z. Falconer Lecture

  The sum of squares of wavelengths of a closed surface

  Kate Okikiolu, Professor of Mathematics, University of California at San Diego

  Friday, August 7, 8:30 – 9:20 am

  For the Laplacian on a closed manifold, we define a spectral invariant which is heuristically the sum of squares of the wavelengths which is a regularized trace of the inverse of the Laplacian. On a technical level, this is an analog for surfaces of the ADM mass from general relativity. We discuss a negative mass theorem for surfaces of positive genus, and give a probabilistic interpretation.

  Biography: Okikiolu earned her B.A. in Mathematics from Cambridge University in England before coming to the United States in 1987 to attend graduate school mathematics at UCLA (the University of California, Los Angeles). There, she worked with two mentors, Sun-Yung (Alice) Chang and John Garnett, and was able to solve a problem concerning asymptotics of determinants of Toeplitz operators on the sphere and a conjecture of Peter Jones, characterizing subsets of rectifiable curves in Euclidean n-space. She earned her Ph.D. at UCLA in 1991, and she has been exhibiting first-rate mathematical abilties.

  After her doctorate, Kate went, in 1993, to Princeton University where she was an Instructor and an Assistant Professor until 1995. From 1995 until 1997 she was a visiting Assistant Professor at MIT. She became a resident status in the U.S. at this time. Since 1997, she has been on the faculty in the Mathematics Department of the University of California, San Diego (UCSD), first as an Assistant Professor. Also in 1996, Dr. Okikiolu spoke as part of the twenty-fifth anniversary celebration for Association of Women in Mathematics (AWM). In 2002, she gave the Claytor-Woodard lecture at the NAM meeting at the Joint Mathematics Meetings.

  In June 1997, Kate Okikiolu was the first black mathematician to win the most prestigous award for young mathematics researchers in the United States, a Sloan Research Fellowship. In 1997, UCSD promoted her to Associate Professor. Recently, Dr. Okikiolu has been researching the "spectral determinant" of three-dimensional drums, which is essentially the number obtained by multiplying all the individual sound pitches made from a drum note. In a separate project, Okikiolu also studies linear distortions of drum notes and other types of signals. Research in this area may have implications for problems in quantum physics. For her work aiding inner-city children, Okikiolu plans to make a series of videos depicting model teaching lessons that emphasize real-world perspectives.