## Student Research

Student research projects in mathematics are a great way for young mathematicians working toward their bachelor degrees to experience what real research in mathematics is like, and gives opportunities for interaction with faculty outside of the classroom setting. If you are a junior or senior math major and interested in a project with me in the area of topology/geometry, or topological applications to data, please contact me.

If you are a Masters student interested in topology/geometry, and you would like to do a Plan B paper, then I typically have a few possible projects in mind at any given time. In particular, I plan on having a series of Plan B papers that altogether explore the classification of (low-dimensional) manifolds. Let me know if you'd like to do one.

If you are a Masters student or a PhD student interested in topology/geometry, and you would like to do a thesis/dissertation in this area, please contact me.

## Past Projects:

### Fall 2023 - Baire's Theorem on the Real Line

For Honors Credit in my Advanced Calculus class, Michael Leiby presented and wrote a brief report on Baire's Theorem on the Real Line.

### Spring 2023 - Metrization Theorems in Topology

For his Plan B paper towards a Master's degree, Kodjo Youto proved the Smirnov-Nagata and Smirnov Theoerms, which give necessary and sufficient conditions for a topological space to be metrizable.

### Fall 2022 - The Stone-Čech Compactification

For her Plan B paper towards a Master's degree, Eva Anderson gave the proof of the existence of, as well as all necessary preliminaries for, the Stone-Čech compactification β(*X*) of a completely regular topological space *X*. Such a compactification is the unique one (up to homeomorphism) for which any continuous *f* from *X* to a compact Hausdorff space *C* has a unique continuous extension from β(*X*).

### Spring 2022 - Dimension in Diffeology

For his Plan B paper towards a Master's degree, Bishal Panthee summarized an important result in the theory of diffeological spaces: that the closed half-line identified with each orbit space of *n*-dimensional Euclidean space modulo rotations and reflections, yields a distinct diffeological structure. That is, we have a countable family of non-diffeomorphic diffeological spaces. Moreover, identifying the closed half-line with the corresponding subset of the real line also yields another non-diffeomorphic diffeological space.

### Fall 2021 - Diffeomorphisms of *C*^{r} and Analytic Manifolds

^{r}

For his Plan B paper towards a Master's degree, Evan Miller read "Differentiable Manifolds" by Hassler Whitney (*Annals of Mathematics, Second Series* **37** (1936), 645-680), and redid the proof that any *C ^{r}*-manifold admits a real analytic structure (

*r*>0). This foundational paper uses now-standard results such as the Tubular Neighbourhood Theorem, analytic approximation of differentiable maps, and embedding theorems to construct a tubular neighbourhood around an embedded manifold; identifying this with an open neighbourhood of the zero-section of the normal bundle, this fibre bundle has analytic total space, and a clever argument allows for a partition of it into analytic fibres over an analytic base

*C*-diffeomorphic to the original.

^{1}### Summer/Fall 2021 - Animated Clustering of U.S. Precipitation Data

For her Honors Capstone Project, Maija Rettelle clustered precipitation data from 1950 onwards in ten year intervals with the goal of capturing shorter-term regional phenomena in precipitation-related climate. By comparing the clusters created from such intervals, one can watch and study the evolution of precipitation regions over the last 70 years.

### Fall 2020 - Topological Data Analysis on U.S. Precipitation Data

For his Honors Capstone Project, Evan Miller compared Fourier analysis results with a periodicity measurement called Sw1pers, that uses persistent homology and sliding windows to detect cyclic patterns within data, using precipitation data from across the continental U.S., obtained from the Global Historical Climate Network. The conclusions drawn from the analysis are interesting, and future projects will be based on them.

- The paper can be found here.
- A poster for a SCREE poster session.

### Spring 2020 - The Quotient Manifold Theorem

In this Masters Plan B paper, Dylan Dowrick proves that the topological quotient of a smooth manifold by the action of a compact Lie group admits a unique smooth structure so that the quotient map is a submersion; that so-called Quotient Manifold Theorem.

### Spring 2020 - Homotopy Invariance of Homology

In this Masters Plan B paper, Bridgette Russell proves that given two homotopic continuous maps between two topological spaces, their induced maps on singular homology are equal.

### Summer 2016 - Invariants of Quotients by Circle Actions (University of Colorado Boulder)

My paper "The differential structure of an orbifold" shows that the underlying (local) semi-algebraic variety of an orbifold, equipped with its ring of smooth functions, contains a complete set of invariants of the orbifold; that is, an entire orbifold atlas can be reconstructed from the variety with its differential structure. Can something similar be done for spaces that (locally) are quotients of manifolds by circle actions? If the differential structure is not sufficient, what other pieces of information are required? This was an REU project with three undergraduate students (Natalie Downey, Lucas Goad, and Michael Mahoney) and one graduate student (Suzanne Craig). The results have appeared in a paper in the undergraduate research journal, Involve.

- Paper
- A poster for an REU poster session.
*Mathematica*program for computing invariant polynomials of circle actions (torus actions, in fact)

### Spring 2015 - Evolution of Curves and Surfaces (University of Illinois)

A recent project that I have been interested in is using the power of *Mathematica* to explore evolutionary techniques for (so-far simple) optimisation problems. This was a project under the umbrella of the Illinois Geometry Lab in Spring 2015, with students Roger Burt, Yingqiu Huang, and Michael Schirle, along with graduate student assistant Brian Collier.

- Project Website (developed by Yingqiu Huang)