# Algebraic Realization of Complex Equivariant Vector Bundles Over the 2-Sphere

In January of 2014, I gave a short talk concerning my thesis topic at MSRI.

I have notes and slides, and you can watch the talk here. I start about 15 minutes into the video.

I started by giving a brief history of the problem including the Nash-Tognoli Theorem. I also review some of the major publications concerning the algebraic realization problem written by Prof. Heiner Dovermann and others.

# What is… Notices of the AMS?

I really enjoy the Notices of the AMS “What is…?” articles. They are generally brief perfunctory definitions and material aimed at mathematicians outside of the fields described. I enjoy them for gaining some breadth before diving into articles outside of my repertoire. Here are a list of highlights relating to AT, because they’re a fun place to start reading about these topics.

Equivariant Cohomology– Loring W. Tu

an Infinity Category– Jacob Lurie

a Linear Algebraic Group– Skip Garibaldi

Persistent Homology– Shmuel Weinberger

a Projective Structure– William M. Goldman

a Legendrian Knot– Joshua M. Sabloff

a Perverse Sheaf– Mark Andrea de Cataldo and Luca Migliorini

a Stack– Dan Edidin

a Derived Stack– Gabriele Vezzosi

# Women in Math

Our math department may be smaller than many, but we have an active community of women pushing the boundaries of research on many fronts. Since 2000, the U. Hawaii math department has hired three female ten year track professors spanning control theory, number theory, and differential geometry. The women in our department have been getting together outside of the classroom semi regularly during the last year to support one another and talk story.

So far we’ve swapped stories over wine and cheese, Indian food, Thai food, and Crème brûlée. You don’t need a community of women to get through graduate school, but it certainly helps!

# Classify Equiv. Complex bundles over Circle

Aside from a snorkelling break (see picture above), I spent the weekend reading a 2001 paper by the Kyoto group.

Classification of equivariant complex vector bundles over a circle
Jin-Hwan Cho, Sung Sook Kim, Mikiya Masuda, and Dong Youp Suh
J. Math. Kyoto Univ. Volume 41, Number 3 (2001), 517-534.

They wrote a series of papers about classification of vector bundles and the algebraic realization problem during the late 1990’s and early 2000’s. Jin-Hwan Cho was a student of Dong Youp Suh’s and he is now at Suwon University.
A key tool that is used in this and their subsequent complex classification paper is in section 2 when they decompose the G-vector bundles using a result based on a functorial property of a previously defined induced bundle.

# Final Comprehensive

In August of 2013, I finally presented on Milnor’s Characteristic Classes for my final comprehensive exam. I am officially a PhD candidate now!
I focused on his proof of the polynomial ring structure of the cohomology ring of the complex Grassmann manifold. I assumed everyone already had working definitions of the Euler class and Stiefel-Whitney classes. I presented Milnor’s definition of Chern classes based on the Gysin sequence, and then powered through the notationally heavy proof.

# Becker-Gottlieb Transfer

I gave a talk during the June 2013 algebraic topology summer course at MSRI. Can you see how nervous I am? notes

I spoke about the Becker Gottlieb transfer map as described in

J. Becker, D. Gottlieb, “The transfer map and fiber bundles” Topology , 14 (1975) pp. 1–12 link

In Gottlieb’s 1975 paper, he proved the existence of a transfer homomorphism involving the total space and the base space of a fiber bundle. The transfer homomorphism is between the nth cohomology of the total space, and the nth cohomology of the base space.

The transfer map is defined s.t. the composition of the induced pullback of the projection and the transfer map is just multiplication by the Euler characteristic of the fiber.
$\hat{\tau} p*: H*(B) \rightarrow H*(B)$

A natural question to pose is whether or not this transfer homomorphism was induced by some map from $B$ to $E$. This would make the transfer homomorphism geometric.

The paper ends with a proof of the Adams conjecture. It is an important result dealing with real vector bundle isomorphism classes and cohomology operations in K-Theory. The original proof used algebraic geometry. Quillen then proved the Adams Conjecture using the Brauer Induction Theorem. Becker and Gottlieb’s proof is considered exceptional, because it only uses algebraic topology.

Their proof reduces the problem to elements of $KO(X)$ which are $2n$-dimensional vector bundles. It uses the transfer map by applying a splitting principal derived from the transfer map.

notes

# NSF Grant

During my first two years of grad school, I was an NSF graduate fellow through the GK-12 grant held here at U. Hawaii. I worked with both a Hawaiian immersion school Halau Lokahi and a more traditional public high school on island, Kailua High School. My favorite projects included teaching middle schoolers how to turn squares into spheres, mobius bands, and tori. I had high schoolers explore surfaces with connected and disconnected boundaries by cutting up large quantities of paper.

Super M Website

Mahalo nui loa NSF!