It is well known that a real number is badly approximable if and only if the partial quotients in its continued fraction expansion are bounded. Motivated by a recent wonderful paper by Ngoc Ai Van Nguyen, Anthony Poels and Damien Roy (where the authors give a simple alternative solution of Schmidt-Summerer's problem) we found an unusual generalization of this criterion for badly approximable d-dimensional vectors.
On badly approximable numbers
Tue, Feb. 11 12:10pm (MATH …
Maria Gillespie (CSU)
Given four lines in three-dimensional space, how many lines pass through all four of them? Enumerative questions like these that arise in geometry can be surprisingly difficult to solve using elementary methods. In this talk, we describe the modern method of using moduli spaces, cohomology rings, and combinatorics to answer such problems. With this framework in mind, we will then describe recent joint work with Renzo Cavalieri and Leonid Monin, in which the combinatorics of "parking functions" was used to calculate the degree of a certain embedding of the moduli space of genus-zero stable curves into a product of projective spaces.
How to park your car on a moduli space
Tue, Feb. 11 1pm (MATH 220)
Grad Algebra Seminar
Peter Mayr (CU Boulder)
We give an explicit example of a Promise Constraint Satisfaction Problem PCSP for Boolean structures that can be reduced to a tractable CSP for a structure of size 3 but not for any smaller structure. This was found in an REU in Summer 2019 with Deng, El Sai, Manders, Nakkirt and Sparks.
Tropical curves are piecewise linear objects arising as degenerations of al gebraic curves. The close connection between algebraic curves and their tro pical limits persists when considering moduli. This exhibits certain spaces of tropical curves as the tropicalizations of the moduli spaces of stable curves. It is, however, still unclear which properties of the algebraic mod uli spaces of curves are reflected in their tropical counterparts. In my ta lk, I will report on joint work in progress with Renzo Cavalieri and Hannah Markwig, in which we define tropical psi classes and study relations betwe en them. I will explain how some of the expected identities cannot be recov ered from a purely tropical perspective, whereas others can, revealing the tropical nature they have been of in the first place.
Moduli spaces of tropical curves and tropical psi classes