The Stanford University Mathematical Organization recently began a speaker series. The goal of the series is to let students listen to professors discuss their backgrounds and research, as well as to let students hear some interesting problems or ideas.
"Modern Search Engines must automatically map documents into an index, and allow users to search this index for the information they need. These challenges have encouraged exciting combinations of engineering, linguistics and mathematics.
We count the number of times different words occur in each document, giving a "word-document table" or matrix. Comparing a query with each document is achieved by adding up the numbers in the table for each query-word and returning the documents that get the highest score. If we think of the numbers in the word-document matrix as coordinates, we can think of each row (or column) as a vector, and so the words (or documents) are mapped to points in a linear vector space. This idea proves to be very fruitful. We can analyze the vectors using clustering and compression algorithms which map the model into a more compact, lower-dimensional space. (This is a vibrant area of current research in statistics and computation with potential applications in diverse areas including medicine, biology, image processing --- and information retrieval!)
We can use the standard scalar product of vectors to compare words with one another and build an 'automatic thesaurus'."
"For many years, topologists and geometers have studied spaces of
loops in a surface or other region as a way of detecting
interesting topological properties. More recently, physicists
have studied spaces of loops in connection with String theory. In
this talk I will discuss how groups of braids can be used to 'tie
together' the geometry of loop spaces with the algebra of rational
Ralph Cohen is a professor in the Stanford Math Department whose research interests include algebraic and differential topology.
Which altered chessboards can be tiled with dominos?
Which have Hamiltonian Cycles? (In other words, when can a king taking only horizontal and vertical steps visit each square on the chessboard exactly once and return to the square he started on?)
What about these?
Bring your answers to the next SUMO talk, where Scott Sheffield will speak on combinatorial "height functions" as a tool for studying tilings and cycles in planar graphs. Results will include existence theorems, sampling algorithms, and a "variational principle" relating the asymptotic behavior of typical tilings on large grid graphs to the solutions of a differential equation with boundary conditions. Applications of this theory and its generalizations are diverse, including protein folding, dimer molecule statistics, and the thermodynamics of phase changes. This talk will be accessible to undergraduates!!!
Scott Sheffield is a graduate student in the Stanford Mathematics Department.
"I will give an elementary introduction to the beautiful mathematical models of soap films, including a hands-on demonstration. It is useful (but not necessary!) to know some calculus for this talk. My goal is to teach you a bit of the mathematics involved in the field of minimal surfaces. By the end of the talk I will be able to explain some current research in this field."
Helen Moore graduated from the North Carolina School of Science and Math, and the University of North Carolina at Chapel Hill. She received her PhD in mathematics from the State University of New York in 1995, where she received a university-wide teaching award. She taught at Bowdoin College in Maine for three years before coming to Stanford University. She currently holds a lecturer teaching position in the Stanford Math Department and a National Science Foundation POWRE grant for her research. Professor Moore has given more than 40 invited talks in 4 countries. She loves to play Ultimate frisbee and acoustic guitar. Her research involves the mathematics of soap films.
"How can we approximate PI as a rational number? One easy way is to truncate the decimal expansion of PI by taking 31/10 = 3.1 or 3,141,592/1,000,000 = 392,699/125,000 = 3.141592. On the other hand, there are GOOD approximations with SMALL denominators, such as 22/7 = 3.14... and 355/113 = 3.141592.
Do these GOOD approximations exist for any irrational number? How do we find them? How closely can they approximate an irrational number? And why do we care about the denominators? With these questions in mind, we will discuss continued fractions and diophantine approximations. We will also describe various results -- some as old as 250 years and some as new as 3 months.
Alex Pekker is an undergraduate in the math department, as well as a very active member of SUMO!
"There are elliptic curves in cell phones, handheld computers, internet browsers, wireless modems, and smart cards. Ministers in the European Union use elliptic curves to communicate with each other. There will soon be elliptic curves in the euro, the new European currency. Elliptic curves helped solve Fermat's Last Theorem. What are these mysterious curves, and why are they so powerful? Find out the inside scoop at this talk."
Alice Silverberg works on number theory and algebraic geometry. Her mathematical interests have taken her to Japan, Australia, and Europe. In her spare time she composes mathematically-inspired Scottish Country Dances, and she dances and plays keyboard accompaniment for Scottish and English Country Dances. She is a professor at Ohio State University, and is currently visiting the Mathematical Sciences Research Institute in Berkeley. Her webpage is here.
"One of the driving forces of a researcher is to prove and understand something you're sure is true, but can't see your way around. I will illustrate by explaining some of the amazing empirical findings connecting the Riemann hypothesis with the Eigenvalues of typical large matrices."
Persi Diaconis has led a double career in magic and mathematics. He left home at age fourteen and made his living as a magician until age twenty-four, having started out as assistant to the greatest sleight-of-hand worker of the era, Dai Vernon. Following extensive training in the applied probability of card shuffling, he went back to college to learn the theory behind it, and continued with a Ph.D. in mathematical statistics at Harvard University. He was awarded a MacArthur Fellowship in 1982, and held positions in statistics at Stanford and in mathematics at Harvard. He is currently a professor of mathematics and statistics at Stanford.