“Real Smirnov Functions and Applications”, Stephan Garcia (Pomona College)
Abstract: Smirnov function is a holomorphic function (satisfying certain natural growth conditions) on the open unit disk whose boundary values on the unit circle are real almost everywhere. Although these functions have many interesting algebraic and analytic properties, they have not been as well-studied as other natural classes (e.g. inner functions, functions with positive real part). We will discuss a method for factoring real Smirnov functions into simple building blocks whose structure can be easily understood. Real Smirnov functions have more applications than one might initially suspect. Some applications we will discuss are: (1) "extending the Riesz projection operator" to Lp for 0 < p < 1, (2) parameterizing the kernels of Toeplitz operators, and (3) parameterizing the backward shift invariant subspaces of the Hp spaces for 1 < p < 1.
Student Colloquium: Thursday, September 6, 12:00 in Olin 268
"Bagels, beach balls, and the Poincare Conjecture", Emily Dryden (Bucknell University)
Abstract: In 1904, the French mathematician Henri Poincare made a conjecture about which three-dimensional objects could be stretched and bent into spheres. Many mathematicians worked on this conjecture and its generalizations over the next century, and in 2000, a prize of one million dollars was offered for its solution. Shortly thereafter, the conjecture was finally solved by an enigmatic Russian mathematician. The journal Science named this the scientific breakthrough of the year in 2006.
We'll talk about the general background and history of the problem and get our hands on various shapes to better understand the topological ideas.
Curiosity is the only prerequisite.
DVP Seminar: Wednesday, September 11, 4:00 in Olin 372
"On the path component of composition operators on the Hardy Space", Eva Gallardo-Gutierrez (University of Zaragoza)
Student Colloquium: Wednesday, September 19, 4:00 in Olin 268
"Euler’s Amicable Numbers", William Dunham (Koehler Professor of Mathematics, Muhlenberg College)
Abstract: A pair of positive integers is called amicable if each is the sum of the proper divisors of the other. The smallest example is the pair 220 and 284, because the sum of the proper divisors of 220 is 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284 and the sum of the proper divisors of 284 is 1 + 2 + 4 + 71 + 142 = 220. This pair was known to the Greeks, and two others were found prior to the 18th century, when Leonhard Euler (1707 – 1783) arrived on the scene. In an awesome display of mathematical power, he found 59 new pairs! In this talk we see how Euler increased the world’s supply of amicable numbers twenty-fold. The argument is easy to follow for anyone with some college mathematics (but not necessarily number theory) under their belts. It should be a fitting topic as we celebrate this, Euler’s 300th birthday.
Student Colloquium: Thursday, October 4, 12:00 in Olin 268
"Euler’s formula for polyhedra and the birth of topology", Dave Richeson (Dickinson College)
Abstract: In 1751 Euler discovered that any polyhedron with V vertices, E edges, and F faces satisfies V-E+F=2, but the proof he gave was flawed. Many rigorous proofs followed, but it took 150 years for mathematicians to fully understand this simple formula. Today Euler's formula is held aloft as one of the most beautiful theorems in all of mathematics and the first great theorem of topology. In this talk we will give the history of Euler's theorem and give a sampling of its many surprising applications.
DVP Seminar: Tuesday, October 9, 4:00 in Olin 372
Abstract: The interplay between geometry and combinatorics through the study of moment maps and the geometry of toric varieties is a well known and fertile theme in mathematics. Here we will see how Witten's non-abelian localization principle in equivariant cohomology for the norm-square of the moment-map in the context of toric varieties motivates new polytope decompositions into cones. Some of these will generalize both the Brianchon-Gram and the Lawrence-Varchenko decompositions. This is joint work with J. Agapito.
DVP Seminar: Thursday, October 11, 4:00 in Olin 372
"Intersection numbers of polygon spaces"
Abstract: We study the intersection ring of the space of polygons in R^3. We will find homology cycles dual to generators of this ring and prove a recursion relation in the number of steps for their intersection numbers. This result is analog of the recursion relation appearing in the work of Witten and Kontsevich on moduli spaces of punctured curves and on the work of Weitsman on moduli spaces of flat connections on two-manifolds of genus g with m marked points. Based on this recursion formula we obtain an explicit expression for the computation of the intersection numbers of polygon spaces. This is joint work with J. Agapito.
DVP Seminar: Thursday, October 18, 4:00 in Olin 372
"Modular forms in number theory"
Abstract: There is a remarkable array of number theoretic functions which are related to the Fourier coefficients of modular forms. For example, Euler's famous partition generating function is the first step in showing that many functions from additive number theory are modular in this sense. In this talk we will discuss applications of modularity to several number theoretic functions, including restricted partition functions and Ramanujan's mock theta functions. We will also touch on the modularity of some objects arising in arithmetic geometry, such as Calabi-Yau varieties
DVP Seminar: Monday, October 22, 4:00 in Olin 372
"Arithmetic of curves and surfaces"
Abstract: The study of rational points on varieties is an enduring theme of research in number theory. Two of the most beautiful results in this area are Wiles' proof of Fermat's Last Theorem and Mazur's characterization of the torsion subgroups of elliptic curves over the rationals. In this talk we will discuss some related results for curves and surfaces which arise in number theory.
DVP Seminar: Tuesday, October 2, 4:00 in Olin 372
"To Form a More Perfect Union of Cones"
(only Multivariable Calculus required)
Abstract: We will consider quadratic formss, their meaning and many usages.
In particular, we will consider the space of all positive definite quadratic forms and its nice decomposition into cones via "perfect" forms. Connections are made with the speaker's research.
"Tiling from the floor up", Peter McNamara (Bucknell University)
Abstract: It is not hard to see that we can use 32 dominos to cover the squares of a chessboard, where each domino covers two adjacent squares. But in how many ways can a chessboard be tiled in this way? Can we still do it if we remove a corner square and the square from the opposite corner? We will talk about these questions and many others that highlight the fun and intrigue of tilings.
DVP Seminar: Tuesday, October 30, 4:00 in Olin 372
"Not your grandfather's Nullstellensatz"
Abstract: Hilbert's Nullstellensatz is one of the most celebrated theorems
of commutative algebra and the jumping off point for Algebraic Geometry. In
this talk we endeavor to convince the audience that the Nullstellensatz is
really a very simple theorem in non-commutative algebra!
DVP Seminar: Thursday, November 1, 4:00 in Olin 372
"Generalizing the notion of Koszul Algebra"
Abstract: A Koszul Algebra is a finitely generated graded quadratic algebra
with very nice cohomological properties. Koszul algebras play a pivotal
role in algebra, topology and combinatorics. We introduce a generalization
of Koszul and prove that our generalization has many of the same good
properties as the class of Koszul algebras.
DVP Seminar: Tuesday, November 13, 4:00 in Olin 372
"Middle School Teachers' use of Curricular Reasoning in a Collaborative Professional Development Project"
Description: Curricular knowledge was first discussed by Lee Shulman as a form of knowledge teachers use when they work with instructional materials and curricular goals. In this study, I examined how teachers employed their curricular knowledge as they reasoned with and about high quality instructional materials. I explored: how teachers used curricular reasoning in planning, implementing, and reflecting on mathematics lessons; and how a collaborative, curriculum-based professional development experience supported the teachers’ use of curricular reasoning. Implications for the ways in which they reasoned and how the experience supported their reasoning will be discussed.
DVP Student Seminar Series: Thursday, November 15: 12:00 in Olin 268
"Preservice Teaches' Understandings for and Learning about Data Representation through Connections between Mathematics and Science Methods Courses"
Session on PSTs’ Understandings about Data Display for Students
Description: Preservice elementary teachers (PSTs) often lack experiences and understandings for working with and displaying data, and yet they will be expected to teach data analysis and display to their future students. I will present a data project I developed with a science education colleague that aimed to strengthen PSTs’ mathematics content knowledge for working with data and provide an experience in which they had opportunities to meaningfully connect mathematics and science inquiry. The project revealed gaps in PSTs’ understandings as well as ways in which they can develop stronger understandings about data. Strengths and challenges of the project will be discussed.
The Mathematics Department
& The Psychology Department
“Predictive Mechanisms in the Brain”
Keith L. Downing ‘82
Norwegian University of Science and Technology
Wednesday --- November 28
Abstract: Many natural scientists believe that motion control is the prime impetus for the evolutionary emergence of the brain, and basic cybernetics shows that motion control for all but the simplest movements requires the ability to predict the consequences of one's actions. Hence, many prominent neuroscientists view prediction as one of the core brain functions - one originally evolved to support sophisticated movement but later reused for cognitive purposes. However, there is little consensus as to the exact nature of predictive information and processes, nor the neural mechanisms that realize them. This talk reviews a host of neural models believed to underlie the learning and deployment of predictive knowledge in a variety of brain regions: neocortex, hippocampus, thalamus, basal ganglia and cerebellum. These are compared and contrasted in order to codify a few basic aspects of neural circuitry and dynamics that appear to be the heart of prediction.
Student Colloquium: Thursday, November 29, 12:00 in Olin 268
" The Freaky Side of Math" Nate Brown (Pennsylvania State University)
Abstract: Math is filled with mind boggling truths, things that violate our most basic intuitions. This is the good stuff, in my opinion, and this talk will focus on some of my favorite freaky facts.