Writing Milestone Seminar

The Writing Milestone Seminar is an opportunity for PhD students in the Mathematics department to present on their second year writing milestone. It's a chance to talk about the piece of mathematics that we have learned to love and treasure, and a space for discussing our experiences as members of this very first iteration of a new department tradition.

The seminar meets on Thursdays 4:30-5:30pm in PDL C401. Most days will be 50 minute seminar talks, but some days will feature round table discussions on students' experience with the writing milestone. Students from all years are encouraged to attend!

For Spring 2025, the Writing Milestone Seminar is organized by Micheal Zeng and Zawad Chowdhury. Please contact us if you would like to present or have any questions!

Spring 2025 Schedule

Week 1 - April 3

Speaker: Micheal Zeng
Title: Flags, spanning lines, and Grothendieck shenanigans
Abstract: The flag variety is a well-studied object in combinatorial algebraic geometry. It is a celebrated theorem by Borel that the cohomology ring of the flag variety is isomorphic to the coinvariant algebra. Pawlowski and Rhoades (2017) constructed the space of spanning line configurations X_{n,k}, a quasiprojective variety whose cohomology ring is isomorphic to a generalized coinvariant algebra R_{n,k}. The space X_{n,k} provides a geometric foundation for the combinatorics of Fubini words.
The Grothendieck group K_0 classifies vector bundles over an algebraic variety and has close connections to cohomology. In this talk, we show that the Grothendieck group K_0(X_{n,k}) is isomorphic to the generalized coinvariant algebra R_{n,k}. Along the way, we discuss Schubert and Grothendieck polynomials, Fubini words, and the relevant combinatorics.

Week 2 - April 10

Speaker: Tyson Klingner
Title: Yau's proof of the Calabi Conjecture
Abstract: Given a compact complex M manifold there is in general no natural choice metric. A natural question is to ask what is the "best" metric we can endow on M. In the 50's, the differential geometer Eugenio Calabi studied this question and through considering various curvature notions made a conjecture on the existence of a special class of metrics called Kähler-Einstein metrics. In 1978, Yau definitively proved the conjecture using geometric analysis. This talk will be expository and cover this beautiful topic.

Week 3 - April 17

Speaker: Wolfgang Allred
Title: Support Theory through Tensor Triangulated Geometry -- or -- Three Fourths of the Right Half of a Playstation controller
Abstract: By fertilizing the poor arid desert of triangulated categories with a monoidal structure we can grow a virile forest of tensor triangulated categories. But beware: it is a haunted forest full of spectres. Will we brave the dark to claim the rumored treasure of cohomological support varieties? Or will we turn tail and run home? Tune in this week to find out.

Week 4 - April 24

Speaker: Dhruv Bhatia
Title: Tropical curves and the top weight cohomology of M_{g, n}
Abstract: A tropical curve is a graph (+ some extra data) that occurs as combinatorial information associated with the limiting object of a family of smooth curves. I will tell the story about how understanding the moduli space M^{trop}_{g, n} of these gadgets can help us understand a small part of the cohomology of M_{g, n}, the moduli space of smooth projective curves. Along the way, we'll do some mixed Hodge theory and also meet this little guy:

Week 5 - May 1

Speaker: Luz Grisales Gomez
Title: On the master theorem connecting incidences and tilings
Abstract: Linear incidence geometry is perhaps one of the oldest subjects studied in Mathematics. It is concerned with configurations of points, lines, and hyperplanes that are given by incidence constraints. Despite the beauty of the proofs and the valuable insights that they offer to any geometer, this subject has been dormant for a while. This is because, in recent decades, new algorithmic approaches to verifying such theorems have been proven to be highly effective to the point that theorems of this kind can now be checked with minimal human input. However, a new set of questions has now arisen with Fomin and Pylyavskyy's Master Theorem, which not only gives a completely new approach to the subject by associating tilings of a surface to incidence theorems but also is a useful tool for proving and generating incidence theorems in all dimensions.

Week 6 - May 8

Speaker: Cordelia Li
Title: Minimal transitive factorizations supported on quasi-threshold graphs
Abstract: Given a permutation in S_n, one can ask how many ways to decompose it into a minimal-length, transitive product of transpositions. We show that the number of minimal transitive factorizations of the identity permutation in S_n into transpositions supported on a given quasi-threshold graph G is always divisible by (2n-2)!/n!, generalizing results of Hurwitz for the complete graph and Pak for the star graph. We give a combinatorial formula for the quotient as a sum over certain weighted spanning trees of G called factorization trees.

Week 7 - May 15

Panelists include: Spencer Catron, Zawad Chowdhury, Arkamouli Debnath, Michael Zeng
Title: Writing Milestone Experience Roundtable
Abstract: A roundtable discussion about people's experiences with the writing milestone so far. We'll discuss how people found professors to review their milestone, how they chose their topic and how the writing process went. First years are encouraged to attend with their questions about the writing milestone, and older students are invited to come share their experience!

Week 8 - May 22

Speaker: Arkamouli Debnath
Title: Geometric Invariant Theory and Derived Categories
Abstract: Taking quotients in algebraic geometry is difficult, mainly because you don't just want a topological quotient but also a variety/scheme structure on it. Geometric Invariant Theory (GIT) gives one possible way to form quotients. On the other hand, derived categories are homological tools that one associates to algebro-geometric objects (such as schemes or stacks) in a similar way as one associates various algebraic invariants (such as the homotopy groups, singular cohomology, etc) to topological spaces. In my writing milestone I have explored derived categories that show up in the context of geometric invariant theory, which will be the content of my talk.

Week 9 - May 29

Speaker: Jay Reiter
Title: Chromatic homotopy theory and the moduli of formal groups
Abstract: In the 1960s, Novikov noticed that descent along the unit map for MU, the ring spectrum representing complex cobordism, was a particularly fruitful method for computing the homotopy groups of spheres. Within a few years, Quillen observed a connection between MU and geometric objects called formal groups. This was the beginning of chromatic homotopy theory and, among many other discoveries, revealed new Bott-periodicity-like symmetries in the homotopy groups of spectra. Since the 2000s, many of these results are best conceptually understood through the language of stacks in which they take on a rich arithmetic flavor. In this talk, we'll catch a glimpse of how the moduli stack of formal groups controls stable homotopy theory, and in particular how understanding its geometry offers conceptual explanations for these periodicity phenomena in the homotopy groups of spectra.

Week 10 - June 5

Speaker: Zawad Chowdhury
Title: Graphical Designs of the Hypercube Graph
Abstract: Graphical Designs are subsets of a graph that perfectly average a suitable set of eigenvectors of the graph Laplacian. In this talk we'll discuss these designs for the hypercube graph. Some surprising connections will show up, including Hadamard Matrices and the Radon Transform!