# Mathematics Colloquium

The Mathematics Department Colloquium is a quarterly series of invited speakers from all mathematical fields and geared toward a broad audience. Archives can be found here.

The talks are generally every Tuesday from 4-5 PM, with an informal tea beforehand. For the duration of COVID-19, it will be held via Zoom, with a meet-and-greet 30m before the talk-- bring your own beverage.

**Winter 2021 Math Colloquium**

***** dates and times as given below *****

**"tea" (meet-and-greet) @ ~30m before the talk**

**Schedule ***(click dates for title and abstract)*

Tues 1/19 | @ 4pm PST | Emily Clader | SF State University |

Thurs 1/21 | @ 9am PST | Tam Nguyen Phan | Institute of Algebra and Geometry; Karlsruhe Institute of Technology |

Fri 1/22 | @ 2pm PST | Nick Salter | Columbia University |

Tues 2/2 | |||

Thur 2/4 | |||

Tues 2/9 | |||

Thur 2/11 | |||

Tues 2/16 | |||

Thur 2/18 | |||

Tues 2/23 | hold for other event |
||

Tues 3/2 | |||

Tues 3/9 | @ 4pm PST | Alexandra Kjuchkova | U Penn |

**Emily Clader, SF State University**

*Permutohedral Complexes and Curves With Cyclic Action*

There is a beautiful combinatorial story connecting a polytope known as the permutohedron, the algebra of the symmetric group, and the geometry of a particular moduli space of curves first studied by Losev and Manin. I will describe these three seemingly disparate worlds and their connection to one another, and then I will discuss joint work with C. Damiolini, D. Huang, S. Li, and R. Ramadas that generalizes the story to a family of "permutohedral complexes", a family of complex reflection groups, and a new family of moduli spaces.

**Thursday, January 21, at 9am**

**Tam Nguyen Phan, Institute of Algebra and Geometry; Karlsruhe Institute of Technology**

**Flat cycles in the homology of congruence covers of SL(n,ℤ)\SL(n,ℝ)/SO(n)**

The locally symmetric space *SL(n,ℤ)\SL(n,ℝ)/SO(n)*, or the space of flat *n*-tori of unit volume, has immersed, totally geodesic, flat tori of dimension (*n − 1*). These tori are natural candidates for nontrivial homology cycles of manifold covers of *SL(n,ℤ)\SL(n,ℝ)/SO(n)*. In joint work with Grigori Avramidi, we show that some of these tori give nontrivial rational homology cycles in congruence covers of *SL(n,ℤ)\SL(n,ℝ)/SO(n)*. We also show that the dimension of the subspace of the *(n − 1)*-homology group spanned by flat *(n − 1)*-tori grows as one goes up in congruence covers. The prerequisite for this talk is very basic linear algebra.

**Nick Salter, Columbia University**

**Families of Riemann surfaces and higher spin structures**

Riemann surfaces are central objects in mathematics, bringing complex analysis, algebraic geometry, topology, group theory, dynamics (and more) into close conversation. In many situations, Riemann surfaces occur in *families* that parameterize some additional algebraic or geometric structure that can be placed on a fixed underlying topological surface. The first part of this talk will be an introduction to families of Riemann surfaces, with an emphasis on the topological aspects of the theory. In the second part, I will discuss some of my own contributions (in collaboration with Aaron Calderon and Pablo Portilla Cuadrado), concerning families of Riemann surfaces equipped with so-called *higher spin structures*, which arise in a surprising diversity of settings (linear systems on algebraic surfaces, singularity theory, Teichmüller dynamics).

**Alexandra Kjuchkova, U of Pennsylvania**

*title*

abstract