Mentoring

Laboratory of Geometry at Michigan, LoG(M)

• Haoyu Du and Madhav Krishna, "Sets of skew lines on extremal surfaces", Winter 2022, Co-mentored with Janet Page, Graduate mentors Anna Brosowsky and Sandra Nair

  • These students studied the sets of skew lines on extremal surfaces. In particular, they related this question to the ``clique listing'' problem in combinatorics/computer science. They introduced a novel version of that problem and gave an algorithm to solve it, which in this setting allows one to compute the maximal sets of skew lines on extremal surfaces. Computationally they applied this algorithm for extremal surfaces of degrees 3, 4, and 5. Preprint available upon request.

• Henry Fleischmann, Christina Greenfeld, Christina Jiang, and Aelita Klausmeier, "Visualizing Gerrymandering", Fall 2020, Co-mentored with Samuel Hansen

  • These students used an existing code base from the MGGG redistricting lab to examine if the Michigan congressional maps from the 2010 census were gerrymandered. This work led to a follow-up project, see "Michigan redistricting 2021" below.

• Joseph Donato, Faustas Udrenas, and Zhijian Zhang, "Smooth Hilbert Schemes", Winter 2020, Graduate mentors Monica Lewis and Zhan Jiang

  • These students studied the cohomology of smooth Hilbert schemes on projective space via combinatorial methods. In particular, they were able to compute the sum of the Betti numbers for six of the seven families of smooth Hilbert schemes. This work was published in the Journal of Algebraic Combinatorics, see Research.

Undergraduate Research Opportunity Program at UM

• Lawrence Wang and Ziyu Yang, "Michigan redistricting 2021", Winter 2021

  • These students, along with Henry Fleischmann (see below), are helping to provide in-state support to the "Quantifying Gerrymandering" group at Duke University in their work for the Michigan Independent Citizens Redistricting Commission who are responsible for setting Michigan's congressional districts for the next decade.

UM Mathematics Summer REU Program

• Marianne DeBrito and Meixuan Sun, "Symbolic Hilbert (quasi-)polynomials of square-free monomial ideals", Summer 2020, Co-mentored with Janet Page

  • These students worked on understanding the relationship between the Hilbert polynomial and the symbolic Hilbert (quasi-) polynomial for that class of ideals.

Independent Study at Stony Brook University

• Eric Schieferstein, "Effective cones of smooth nested Hilbert schemes of points on the plane", Fall 2018-Spring 2019

  • Eric first took a reading course on algebraic geometry with me. In the second semester, he worked on understanding nested Hilbert schemes and computed the effective cones of the first several examples of smooth nested Hilbert schemes of points on the plane.