# Graduate student homotopy theory seminar

Time: Monday 3pm
Location: 441 Altgeld Hall

Talks:

• February 3: Ningchuan Zhang
• Title: Exotic elements in Picard groups
• Abstract: In this talk, I will discuss the subgroup of exotic elements in the $K(h)$-local Picard groups. We will first show this subgroup is zero when $p\gg h$ and then focus on the $(h,p)=(1,2)$ and $(2,3)$ cases.
• February 10: Ningchuan Zhang
• February 17: Brian Shin
• Title: A geometric perspective on the foundations of modern homotopy theory
• Abstract: Homotopy theorists have always been interested in studying spaces. However, the meaning of the word space'' has evolved over the years. Whereas one used to say space to mean a topological space, it seems the modern stance is to view a space as an $\infty$-groupoid. In this expository talk, I would like to connect the modern stance back to geometry. In particular, I will demonstrate how the $\infty$-category of spaces can be built out of the category of manifolds. As an application, we will use this connection to give a geometric perspective on infinite loop space theory.
• Februrary 24: Brian Shin
• Title: An introduction to motivic homotopy theory
• Abstract: Motivic homotopy is often thought of as the homotopy theory of algebraic varieties. In this expository talk, we'll see exactly what that means. In particular, we'll see how the construction of the category of motivic spaces is a direct algebro-geometric analog of that of the category of spaces. More interestingly, we'll also see how the analogy breaks down.
• March 2: Liz Tatum
• Title: Relations between Spectral Sequences
• Abstract: Consider a ring spectrum E and a spectrum X. The E-based Adams Spectral Sequence is a tool for approximating the homotopy groups $\pi_{*}X$. Depending on the choice of ring spectrum E, the Adams spectral sequence might be easier to compute, but might give a weaker approximation to $\pi_{*}X$. One could ask “If A, B are two different ring spectra, what can an A-based Adams spectral sequence tells us about a B-based Adams spectral sequence”?
In the paper “On Relations Between Adams Spectral Sequences, With an Application to the Stable Homotopy of a Moore Space”, Miller proves a theorem addressing this question. In this talk, I’ll introduce some of the tools Miller uses to formulate and prove this theorem, and outline the previously mentioned application.
• March 9: Tsutomu Okano
• Title: Some applications of tangent categories.
• Abstract: The cotangent complex formalism is a useful framework for developing obstruction theoretic tools such as Andre-Quillen cohomology. I will present a theorem that identifies the tangent categories of Cat_S, where S is some symmetric monoidal infinity-category. Some more example applications of this formalism will follow.
• March 23: Doron Grossman-Naples
• Title: Finite Spaces and Finite Models
• Abstract: When we try to model simplicial complexes using posets, finite spaces arise as a natural bridge between these two categories. In this talk, I will describe the theory of these spaces and the nature of this correspondence, and discuss the resulting theory of finite models.
• March 30: William Balderrama
• April 6: Heyi Zhu
• April 13: Johnson Ga Jun Tan
• April 20: Joseph Rennie
• April 27: Abhra Kundu
• May 4: Venkata Sai

 Department of Mathematics 273 Altgeld Hall, MC-382 1409 W. Green Street, Urbana, IL 61801 USA Telephone: (217) 333-3350    Fax: (217) 333-9576     Email: math@illinois.edu