Nate will discuss the definition of a mathematical knot, the knot group, Fox calculus, and topics related to the Alexander polynomial. Some familiarity with the fundamental group will be assumed.
I will talk about PDEs in general from the analysis point of view. I
will talk about different methods of how people tried to show the
well-posedness locally/globally in time (LWP, GWP respectively). Here,
the well-posedness means existence, uniqueness, and stability of the
solutions to a PDE. Although there have been geometric approaches to
the LWP/GWP of the problem (such as inverse scattering method, etc.), I
will talk about the analytical methods, using Functional Analysis and
Fourier Analysis. I will also try to show the idea of the
well/ill-posedness, using ODEs as examples. If time permits, I'll talk
about how the number theory comes into to play a crucial role in the
periodic setting.
Jennie scanned the notes
from Hiro's talk.
Constant mean curvature is the differential property which characterizes surfaces which locally, as exemplified by the sphere, have the least area per unit volume they enclose. The study of CMC surfaces goes back at least to the late 19th century, as evidenced by the work of Delaunay. In this talk, I will: explain what mean curvature is, describe some of the classification results that have been obtained so far, and describe some of my own work on this topic. I will not get involved with the gritty details of proofs so even those who have not taken "manifolds" should find a good bit of it understandable. You can find some pretty pictures of CMC surfaces at http://www.gang.umass.edu/gallery/cmc/.
I will be giving a TWIGS-style talk about sheaves and sheaf cohomology. After giving basic definitions, I will briefly discuss some examples of where these objects appear. We will then use sheaf cohomology to understand the relationship between the seemingly unrelated Cech, deRham, and Singular cohomology theories which arise in the disparate contexts of sheaf theory, algebraic topology, and differential geometry.
Given a simple d-polytope P in Rd, we can associate a canonical, finite dimensional real graded vector space A(P), where dimR(Ak(P))=hk(P) is an important combinatorial invariant of P. Then one can show an analogue of the hard Lefschetz theorem holds on A(P). This, together with some easy observations, give a proof of the necessity of McMullen's conditions, a complete characterization of the f-vector of a simple polytope. I will give all definitions necessary to state McMullen's conditions, and mainly concentrate on the construction and properties of A(P), following a beautiful paper by Timorin using volume polynomials. This talk should be very accessible to all.
I will give a very concrete introduction to semisimple Lie Algebras and root systems through analyzing the simplest type of Lie Algebra. Lie Algebras are complicated objects that have an amazingly simple underlying structure called a root system. We will explore one type of Lie algebra, find it's root system, and talk about what happens in other types. The talk should be very accessible to anyone with a good knowledge of linear algebra. (And will be a different viewpoint from that in Paul Gunnell's Lie Algebra class this semester, so those of you in it have no excuse not to come.)
First, we will consider the Fourier series on the circle. In the elementary PDE class, the exponentials appear as the eigenfunctions of the Laplacian. I will give other group-theoretic characterization of them: they are the basis of the dual group (set of the characters) on the circle group, they span the minimal translation invariant subspaces, they are the homomorphisms of the algebra of summable functions. Then, I will discuss the Fourier series on finite commutative groups. As an example, I will talk about the Poisson summation formula in this context. At the end, I will talk about how one needs to proceed in the case of non-commutative groups, which do not have enough "characters". This leads us to the group representation. If time permits, I will discuss briefly about the representation of G=SO(3) by first considering the representation of its double cosets K/G/K. (I will need to skip the details and the arguments are formal.)