Schedule

We plan to have many breaks between talks to promote conversations and academic discourse. All talks will take place in room 1447 of the classroom wing of Klaus building, located at 266 Ferst Dr NW (which is not the math building). There will also be an outing to a local bar or restaurant on Saturday night, more detail to follow.

Saturday

  • 1pm-2pm. Registration.
  • 2pm-3pm. Jennifer Balakrishnan.
  • 3pm-3.30pm. Coffee break.
  • 3.30pm-4.30pm. Sam Payne.
  • 4.30pm-5pm. Coffee break.
  • 5pm-6pm. Eric Larson.
  • 7pm. Social event.

Sunday

  • 9am-9.30am. Breakfast.
  • 9.30am-10.30am. Rohini Ramadas.
  • 10.30am-11am. Coffee break.
  • 11am-12pm. Angelica Cueto.


Titles and Abstracts:

  • Jennifer Balakrishnan. Rational points on the cursed curve.

The split Cartan modular curve of level 13, also known as the "cursed curve," is a genus 3 curve defined over the rationals. By Faltings' proof of Mordell's conjecture, we know that it has finitely many rational points. However, Faltings' proof does not give an algorithm for finding these points. We discuss how to determine rational points on this curve using "quadratic Chabauty," part of Kim's nonabelian Chabauty program. This is joint work with Netan Dogra, Steffen Mueller, Jan Tuitman, and Jan Vonk.

  • Sam Payne. Top weight cohomology of​ Mg,n​​ .

I will discuss applications of tropical geometry to the cohomology of moduli spaces of curves, with and without marked points. Based on joint work with Melody Chan, Carel Faber, and Soren Galatius.

  • Eric Larson. The Maximal Rank Conjecture.

Curves in projective space can be described in either parametric or Cartesian equations. We begin by describing the Maximal Rank Conjecture, formulated originally by Severi in 1915, which prescribes a relationship between the "shape" of the parametric and Cartesian equations --- that is, which gives the Hilbert function of a general curve of genus g, embedded in ℙr​​ via a general linear series of degree d. We then explain how recent results on the interpolation problem can be used to prove this conjecture.

  • Rohini Ramadas. Dynamics on the moduli space of pointed rational curves.

Let f:S2​​→S2​​ be an orientation-preserving branched covering such that the forward orbit of any critical point is finite. Thurston studied the dynamics of f using an induced holomorphic self-map T(f) of the Teichmüller space of complex structures on S2​​ marked by the postcritical set. Koch found that this holomorphic dynamical system on Teichmüller space descends to an algebraic dynamical system on the moduli space M0,n​​. I will introduce these related topological, holomorphic and algebraic dynamical systems, and discuss how the last can be studied on compactifications of M0,n​​.

  • Angelica Cueto. Anticanonical tropical del Pezzo cubic surfaces contain exactly 27 lines.

Since the beginning of tropical geometry, a persistent challenge has been to emulate tropical versions of classical results in algebraic geometry. The well-know statement "any smooth surface of degree three in ℙ​​3​​ contains exactly 27 lines'' is known to be false tropically. Work of Vigeland from 2007 provides examples of tropical cubic surfaces with infinitely many lines and gives a classification of tropical lines on general smooth tropical surfaces in Tℙ3​​

In this talk I will explain how to correct this pathology by viewing the surface as a del Pezzo cubic and considering its embedding in ℙ44​​ via its anticanonical bundle. The combinatorics of the root system of type E_6 and a tropical notion of convexity will play a central role in the construction. This is joint work with Anand Deopurkar.