| Date |
Speaker |
Abstract |
| 1/30 |
Paul Tee |
Which spheres are parallelizable?
In this talk, we'll introduce the concept of a vector bundle and discuss the classic question
of which spheres are parllelizable. We will attack this question by showing the only spheres which are Lie groups have dimensions
building a nonzero de Rham cohomology class in H^3. If time permits, we will give an indication of which spheres are parallelizable
by considering real division algebras.
|
| 2/6 |
|
|
| 2/13 |
Aidan Wood |
What is hyperbolic geometry?
We explore elementary properties of the upper half-plane equipped with a hyperbolic Riemannian metric.
|
| 2/20 |
Zackary Boone |
Rate of decay of probability measures on Cantor sets.
One way to find the “size” of a set is by understanding how quickly the Fourier transform of a
probability measure on that set decays. Along these lines we will introduce a general form of a Cantor set,
how to construct a probability measure on it, then show what the Fourier transform of that probability measure is.
If time permits we will also show how to calculate the Hausdorff dimension of these sets.
|
| 2/27 |
Yaser Monterrey |
Shake concordance.
The main gist is that concordance is a relation that knots have. I will first start out by explaining what
concordance and slice-ness is and then I will go on to describe shake concordance and how it generalizes the
previous notions. I might also bring up the slice ribbon conjecture as an aside.
|
| 3/6 |
Cole Durham |
An introduction to Ricci flow.
Over the past 40 years, Ricci flow has been a topic of great interest in geometric analysis. In this discussion
we will begin with a brief review of the history of the subject, then explain two types of motivating examples, namely
Einstein manifolds and Ricci solitons. Time permitting, we will conclude by stating the curvature evolution equations.
|
| 3/13 |
|
|
| 3/20 |
Michael Albert |
A basic overview of semigroup theory.
The well-posedness of the heat equation with homogeneous Dirichlet boundary conditions on a nice domain
Omega in R^n is well known. We explain how well-posedness in this context arises from the more general theory of
semigroups, and how they allow us to study a wider class of initial/boundary value problems. In particular we
discuss the basics of semigroup theory and will sketch proofs for the famous Hille-Yosida and Lumer-Phillips theorems.
These in tandem are used to prove the well-posedness of the heat equation. We hope to use this talk as a building block
for future discussions about current research in the study of heat kernels, especially in the context of Riemannian and
sub-Riemannian geometry.
|
| 3/27 |
|
|
| 4/3 |
Yaser Monterrey |
Prime knots and concordance.
The talk will complete the previous discussion on knot genus, prime knots,
and Seifert surfaces. It will then briefly touch on slice-ness, concordance,
and the knot concordance group.
|
| 4/10 |
Michael Albert |
Heat Semigroups Part 2.
We will continue our discussion of unbounded operators L on a Banach space,
which we will now take to be a Hilbert space (usually some L^2 space). We will
try to understand the conditions under which L generates a suitable heat semigroup,
in particular when it can be understood as convolution against a heat kernel. We
will introduce some very important concepts from functional analysis. If you want
to preview some of the concepts that we will be talking about, take a look at these
notes from Terry Tao https://terrytao.wordpress.com/tag/essential-self-adjointness/.
|
| 4/17 |
Cole Durham |
Spectrum estimates for stable minimal surfaces.
We introduce a notion of stability for a minimal hypersurface Sigma of M, then
prove estimates for the bottom of the spectrum of the Laplacian on Sigma under
curvature restrictions. Follows the work of Munteanu, Wang, and Sung.
|
| 4/24 |
Logan Borghi |
Towards classification results in Lie theory.
In this talk we will motivate Lie algebras as the analogue of an
inner product space for the endomorphism ring of a vector space. Then we
will develop basic Lie theory including the notion of ideals, quotients, adjoint representations,
and some natural isomorphism results in this context. With this machinery we will prove the first
surprising result which begins the classification story for semisimple Lie algebras: namely that every
simple Lie algebra is isomorphic to a linear Lie algebra.
|
| 5/1 |
Alec Wendland |
Negative Sobolev spaces and a compactness theorem
for a nonlinear approximation of 3D smectic liquid crystals.
Liquid crystal is a state of matter with properties between those of ordinary liquids and solids. In particular, we are
concerned with the free energy of smectic-A liquid crystals, which are liquid crystals found at low temperatures in which
the nematic director of the liquid crystal layers coincides with the unit normal. We first discuss basic embeddings and
compactness results regarding negative-order Sobolev spaces. We then show that we obtain compactness in L^2 for the
gradient u_n of an energy-bounded sequence u_n in H^2 satisfying some additional technical assumptions. If time permits,
we consider a sharp lower bound on the free energies.
|