The Chern-Weil map gives a description of the characteristic classes of a vector bundle in terms of specific symmetric polynomials. One construction of this map utilizes the machinery of equivariant cohomology and provides an explicit construction of examples such as the Chern Classes. We will review de Rham Cohomology then introduce equivariant cohomology on smooth manifolds. I will finish the talk with the construction of the objects mentioned above.
In this talk I intend to give a brief introduction on what it means for a group to act holomorphically on a Riemann surface. Following this, I would like to present various results regarding such groups. This will include (hopefully), a look at all possible groups which may act holomorphically and effectively on the Torus and the Riemann sphere and certain consequences this yields. To finish, if I have time, I will discuss the different results for compact Riemann surfaces of higher than one genus.
Discrete dynamical systems are often used to model population change over time. One aspect of a system that can be studied is persistence, which basically means indefinite survival of the population. I will define persistence and apply it to a model of two competing species of flour beetle.
A tree that is a subgraph of a graph-theory type of tree can be extended to a spanning tree. A set of linearly independent vectors in a vector space can be extended to a basis. Such similarities are considered in the theory of matroids, which is essentially an abstract theory of dependence. Matroids and oriented matroids will be defined, a topological representation theorem will be presented, and an application or two will be discussed.
In this talk, I will define the Fourier transform on the finite circle and then expand this to finite tori. Time permitting, I will discuss an application to Random walks on Cayley graphs and Markov chains.
Given a domain G in R^2, consider the Laplacian on G with Dirichlet boundary conditions. The eigenvalues of this operator are real and discrete and thus can be ordered to form an increasing sequence. As a result, we can order the eigenfunctions as well. Suppose that u_n is the n-th eigenfunction. Courant's nodal domain theorem states that the zero set of u_n (also called the nodal lines of u_n) splits the domain G into no more than n connected pieces. In this talk, we will prove the theorem after building up the necessary tools. I'll spice it up with some outrageous claims involving world peace and a can of sliced tomatoes.
In a recent RAIRE Saturday Workshop, I gave a presentation to high school students about using the action of permutations to determine a solution to the Rubik's Cube. The solution outlined is straightforward to determine, but difficult and unwieldly to use in practice. In this talk, I will show how a slightly deeper study of the groups involved (as well as some common sense) lead to a much more efficient solution. Further, I will discuss some other methods of solution, including Thistlethwaite's solution using nested subgroups.
I will present some ideas from neuroscience, psychology, computer science and mathematics regarding the functioning and understanding of the brain. I will also conclude my talk with the question/answer what does all this scientific knowledge tell us about us and can it shape our future.
Roughly speaking, moment maps provide a way of organizing the symmetries arising from certain symplectic Lie group actions on symplectic manifolds. When trying to compute a moment map for a specific group action, it can be helpful to know whether or not the map we are looking for is unique. In this talk, I will show how Lie algebra cohomology can be used to determine existence and uniqueness of momentum maps in the context of symplectic group actions. I'll do my best to highlight the main ideas behind the theory and keep technicalities to a minimum.
In this talk, I will show that the sech solution for the focusing Nonlinear Schro"dinger equation is nonlinear stable. I will present the idea of nonlinear stability using an analogy with finding the minimum of a function we learned in vector calculus. Also, I will study the Nonlinear Schro"dinger equation with a potential; this potential will be considered as a perturbation to the NLS, and we will see that its exact solution is also stable.
In this talk I will present the Adler-Kostant-Symes theorem and then I will verify and apply the theorem to the lie algebra sl(2,R) and if time permits I will also discuss how it can be used for infinite dimensional lie algebras and specificely loop algebras.
In the last ten years, Research in Undergraduate Mathematics Education (RUME) has focused on the teaching and learning of many topics in the undergraduate mathematics curriculum, such as functions, calculus, linear algebra, differential equations, writing proofs, etc. In this talk, I will outline recent research concerning students' conceptual understanding of topics in elementary group theory, with a particular focus on the concept of quotient groups.
One of the most interesting problems in Graph Theory today is the Reconstruction Conjecture. First posed almost sixty years ago, the Reconstruction Conjecture is: "A finite simple (unlabelled) graph G on at least three vertices is determined uniquely from its deck of cards." What is all this mumbo jumbo about a graph having a deck of cards? Come find out.
Back in 1959 Stephen Smale found out that you could turn a sphere inside out without puncturing it our by making creases -- a regular homotopy. Unfortunately, he did not have an explicit realization of this process, but later work was able to construct these homotopies. Perhaps the must stunning is William Thurston's for which he made a movie -- "Outside In." I will give some of the mathematical background/history of this problem and then we will watch this short feature. Popcorn will of course be provided.
I will define continuous symmetries for differential equations and show how they can be calculated. I will also show how these symmetries can be used to solve ordinary differential equations. In particular, we will see that most of the tricks we have learnt as undergraduates to integrate second order ODE's can be explained from a group theoretical point of view.