Research

I am interested in o-minimal and real algebraic geometry, and specifically the role of symmetry (very broadly defined) in these contexts. Please consult the documents below for a summary of my projects established and ongoing. 

• Research Statement (overview for a broader audience)

• Research Statement (detailed version)

Papers

• Equivariance in Approximation by Compact Sets (with Saugata Basu)

• On the Topology of Symmetric Semialgebraic Sets (dissertation)

Talks

• Paper Snowflakes. Also, Vandermonde Varieties in Type B (Wabash College Mathematics and Computer Science Colloquium, December 2024; Ides of August, Wabash College, August 2024)
  Abstract: Paper snowflakes work because of symmetry. You cut holes in a folded piece of paper, and then unfold to view the snowflake-like design you have created. Here, instead of creating snowflakes, we would like to count how many holes they have. Can we make that task easier by counting while the snowflake is in its folded state?
  
  Mathematically, this is precisely what Basu and Riener do for one particular type of symmetry in their paper, "Vandermonde Varieties, Mirror Spaces, and the Cohomology of Symmetric Semialgebraic Sets." They develop an algorithm for counting the "holes" of symmetric sets which requires far fewer steps than the best known algorithms without symmetry. Now, we would like to see if this algorithm works for the next major class of symmetry. Towards that aim, we provide an introduction to certain prototypical symmetric sets known as Vandermonde varieties, and discuss what we now know about them in this next type of symmetry.
  
  Slides (Math and CS Colloquium)

• Monotonicity and Totally Nonnegative Spaces: An Update (Purdue Model Theory and Applications Seminar, February 2023)
  Abstract: Fomin and Shapiro conjectured that the link of the stratified space formed by the totally nonnegative part of the unipotent radical of a split semi-simple algebraic group over R is a regular cell complex. Though resolved by Hersch, interest remains in alternate proof routes. The concept of monotonicity may provide a means to a simpler proof, as graphs of monotone maps are regular cells. In type A, examples suggest that the strata in question may be graphs of monotone maps, though the symbolic manipulation grows difficult for n even as small as 4.
  Meanwhile, Davis, Hersh, and Miller have shown that the contractibility of the fibers of maps associated to a given totally nonnegative space would imply the regularity of the decomposition of the original space. The fibers are somewhat simpler to describe, and in type A we had hoped to prove this conjecture by demonstrating that the fibers stratify into monotone cells. This is indeed true for n = 4, but counterexamples to monotonicity (though not regularity) begin to arise in the n = 5 case. We present the successes, counterexamples, and current state of our investigation of the monotonicity of the strata of the fibers of maps to totally nonnegative spaces.
  
  Slides

• Vandermonde Varieties in Type B (New Directions in Real Algebraic Geometry, Oberwolfach workshop, March 2023; Purdue Model Theory and Applications Seminar, February 2023)
  Abstract: This talk concerns the topology of Vandermonde varieties in the setting of Type B symmetry. Recent results of Basu and Riener leverage symmetry relative to the action of the symmetric group S_n on in the study of the cohomology of semialgebraic sets. Vandermonde varieties, which are classically defined by the first several generators of the ring of S_n-symmetric polynomials, play a key role in these arguments. With an eye towards extending the results of Basu and Riener to broader classes of symmetry, we examine Vandermonde varieties defined relative to the next major type of reflection symmetry. This talk will describe Vandermonde varieties within this setting and present new results (joint with Dr Saugata Basu), including the topological regularity of the intersection of a Type B Vandermonde variety with a fundamental region of the group's action.
  
  Slides (MTA seminar) 

• A Tourist's Guide to O-minimality (Rose-Hulman Mathematics Seminar, October 2022; MAA Indiana Section Meeting, September 2021; Purdue Student Colloquium, March 2019)
  Abstract: Come, step away from the chaos of the real numbers to the idyllic world of o-minimality, where sets and functions behave nicely. This expository talk will take you on a tour of a few of the theorems within walking distance of the definition of an o-minimal structure, with brief glimpses of some proofs. Specifically, we will visit the Monotonicity Theorem, Cell Decomposition, and the Curve Selection Lemma, on the way to arguing that definable subsets of R^n are connected iff they are path connected. You may even leave with a souvenir proof that all groups are Abelian.
  
  Slides (Rose-Hulman Mathematics Seminar)
  Slides (MAA Indiana Section Meeting)
  Slides (Purdue Student Colloquium) 

• Monotonicity and Totally Nonnegative Spaces (Graduate Research Day, Purdue University, November 2019)
  Abstract: This talk investigates the structure of the space of totally nonnegative n by n upper triangular matrices with 1's on the diagonal. This space can be decomposed into strata which are images of certain maps associated to the elements of the symmetric group . It has been shown that these strata are (topologically) regular cells. However, a shorter proof may be possible. We will describe the notion of monotone maps, the graphs of which are regular cells, and explain how an application of this concept to the fibers of the aforementioned maps could provide an alternate proof. This is joint work with Dr. Saugata Basu.
  
  Slides

• Introduction to Model Theory (Purdue O-minimal Geometry/Hodge Theory Seminar, August 2019)
  This lecture was intended to provide the seminar participants with some initial background in model theory. Topics included languages, structures, formulas, theories, consistency, completeness, definable sets, the compactness theorem (ultrafilters and ultraproducts), quantifier elimination, strongly minimal sets and structures, and o-minimal structures.