Mathematics

Operadic Monoidal Structures

Tsutomu Okano (UIUC Math)

Abstract: In this talk I will discuss how (higher) operads help us encode monoidal structures in (higher) categories. I will also discuss how to generalize this to parametrized settings and hope to convey the usefulness of such formalism in equivariant and motivic homotopy theories.

Mapping space spectral sequences

William Balderrama (UIUC Math)

Abstract: The classical story of obstruction theory for computing maps into a space Y involves lifting maps up the Postnikov tower of Y. In this talk, I will introduce a form of this obstruction theory for computing maps between highly structured objects in homotopy theory. Along the way, we will see why Quillen cohomologies show up in homotopy theory, take derived categories of derived categories, and take multiplicative Postnikov towers of nonconnective ring spectra.

Group Theory for Homotopy Theorists

Brian Shin (UIUC Math)

Abstract: In this expository talk, we'll introduce a model structure on the category of groups and demonstrate how to effectively study groups using this model. This model has the technical advantage of avoiding the overly abstract definition of a group via sets with binary operation. It also allows for clean definitions of colimits and free objects. If time permits, we'll discuss monoidal structures for a certain localization of this model structure. This is based on a short article by Krause-Nikolaus.

Complex structures on Real vector bundles

Abhra Abir Kundu (UIUC Math)

Abstract: In this talk, I will state the first and the second obstruction to having a stable complex structure on a real vector bundle. I will then show how one can go from stable complex structure to complex structure. And, if time permits, I will try to sketch how the second obstruction can be expressed as a secondary cohomology operation.

Crystalline period map

Venkata Sai Bavisetty (UIUC Math)

Abstract: In Chromatic homotopy theory, one tries to understand the homotopy groups of spheres using the height filtration on formal group laws. This way at each height we get a spectral sequence whose $E_2$ term is the group cohomology of the Morava stabilizer group with coefficients in the Lubin-Tate ring. In this talk, I hope to explain a conceptual way to figure out the action of the Morava Stabilizer group on the Lubin-Tate ring.

Organizational Meeting

Brian Shin (UIUC Math)

Algebraic theories and homotopy theory

William Balderrama (UIUC Math)

Abstract: In this talk, I will motivate and introduce algebraic theories as a category-theoretic approach to finite product theories. I will then talk about a well-behaved notion of an infinitary algebraic theory, and the introduction of homotopy-theoretic structure, which can be used to define notions of homology and cohomology for the models of an algebraic theory. This is the first of two talks; the second will use these ideas to produce applications in stable homotopy theory.

Modeling higher algebra with product-and-loop theories

William Balderrama (UIUC Math)

Abstract: In this talk, I will introduce the extra homotopical properties of a (suitably infinitary) algebraic theory that make it suitable for modeling spectral, or otherwise higher, algebra, rather than merely derived forms of ordinary algebra. To illustrate the utility of this viewpoint, I will indicate some of the computational tools that can be constructed and understood from this perspective. Time permitting, I will discuss some applications to chromatic homotopy theory.

Splitting $BP\langle 1\rangle \wedge BP\langle 1\rangle$ at odd primes

Liz Tatum (UIUC Math)

Abstract: The Adams Spectral Sequence is a tool for approximating $\pi_{*}X$, where $X$ is a connective spectrum. If $E$ is a ring spectrum satisfying certain properties, then we can define an $E$-based Adams spectral sequence converging to $\pi_{*}\hat{X}$, where $\hat{X}$ is the $E$-completion of $X$. When $E_{*}E$ is flat over $E_{*}$, the $E^{2}$-page of the spectral sequence can be described as $Ext_{E_{*}E}(E_{*}, E_{*}X)$. But if $E_{*}E$ is not flat over $E_{*}$, then there is no such description. Instead, we must study $E\wedge E$ to understand the spectral sequence. The Brown-Peterson spectra $BP\langle n \rangle $ are an example of such spectra. One approach is to split the product $E \wedge E$ into more manageable pieces. When $n=1$, we can construct a splitting $BP\langle 1 \rangle \wedge BP\langle 1 \rangle$ as $\vee_{k=0}^{\infty}\Sigma^{2k(p-1)} BP\langle 1 \rangle 1 \wedge B(k)$, where $B(k)$ is the $k^{th}$ integral Brown-Gitler spectrum. We give a sketch of Kane's construction of this splitting for odd primes.

The Recognition Principle for Infinite Loop Spaces

Brian Shin (UIUC Math)

Abstract: In this expository talk, I'd like to discuss the infinite loop space recognition principle. In particular, I'd like to examine Boardman-Vogt's infinite loop space machine from a modern view point.

Motivating Higher Toposes: Geometric Characteristic Classes

Joseph Rennie (UIUC Math)

Abstract: This will be part one of two talks aimed at motivating higher topos theory from physics. In this talk we will give a brief history of classical obstructions for manifolds, then suddenly find ourselves naturally requiring tools from higher topos theory. In the end, we shall see how working with simplicial sheaves on Manifolds allows us to define (but more importantly compute) geometric characteristic classes.

Motivating Higher Toposes: Higher Bundle Theory

Joseph Rennie (UIUC Math)

Abstract: In this (self-contained) talk, I will begin with a quick recap of the motivation for higher bundle theory from the first talk. I will then say a few words about Toposes, and proceed to spend the majority of the talk attempting to develop a general theory of higher bundles. Along the way, we will see how the necessary properties for this development (almost) force higher topos structure. (Technical details will be sacrificed for intuitive clarity. No particular model of higher categories will be imposed.)

Topological Data Analysis: Theory and Applications

Daniel Carmody (UIUC Math)

Abstract: There are two algorithms which form the backbone of many applications of modern topological data analysis: the Mapper algorithm (Singh, Memoli, Carlsson), and persistent homology (Zomorodian, Carlsson). In this talk I'll introduce both algorithms, talk about the homotopy theory behind them, and give an application of each.

Global homotopy groups and global functors

Heyi Zhu (UIUC Math)

Abstract: The 0th equivariant homotopy group of an orthogonal $G$-spectrum defines a $G$-Mackey functor with restriction and transfer functors out of the the orbit category of $G$ and when $G$ is finite, the interactions between restrictions and transfers is given by a double coset formula. In the global setting, we define an orthogonal spectrum as a functor out of inner product spaces and will see that its 0th global homotopy group defines a "global functor" out of the global Burnside category so that restriction generalizes naturally for arbitrary continuous maps of groups and transfer along inclusion of closed subgroups. In this sense, the global functors generalize $G$-Mackey functors by allowing $G$ to vary. This talk follows the treatment in Schwede's book Global homotopy theory.

Atiyah-Segal completion theorem

Tsutomu Okano (UIUC Math)

Abstract: In this talk I will walk through Atiyah and Segal's "Equivariant K theory and completion". The main result is a nice proof of the so-called Atiyah-Segal completion theorem, which relates the equivariant K theory of a G-space with the K theory of its homotopy orbit. Towards the end, I will also discuss the algebraic analogue of this result.

Introduction to Picard groups

Venkata Sai Narayana Bavisetty (UIUC Math)

Abstract: Picard groups have been classically studied in commutative algebra and algebraic geometry. These can be suitably generalised to define picard groups of $\mathbb{E}_{\infty}$ ring spectra. In this situation a natural question to ask is "How is the picard group of $R$ related to the picard group of $R_*$?". In this expository talk, I will explain some methods which have been used to answer the above question.

Splitting induced from transfer

Abhra Abir Kundu (UIUC Math)

Abstract: Associated to a finite covering map is a map in homology and cohomology, known as transfer, which goes in the opposite direction. This map usually doesn’t come from a map at the level of spaces but interestingly enough it does when we look at its version in spectra. This often leads to interesting splittings in spectra after appropriate localisation. In this talk, I will explain how one gets this map. Then I will concentrate on one such transfer and the splitting induced from it. I will show a few pictures of algebraic objects associated to this transfer which will illustrate this splitting and then end by observing some facts about the splitting from those pictures.

The Gross-Hopkins duality

Ningchuan Zhang (UIUC Math)

Abstract: In this talk, we introduce the Gross-Hopkins duality in chromatic homotopy theory, which relates the the Spanier-Whitehead duality and the Brown-Comenetz duality in the $K(n)$-local category. We will mainly focus on the underlying algebraic geometry of the duality phenomena and work out some explicit examples at height 1.

The Gross-Hopkins duality (part 2)

Ningchuan Zhang (UIUC Math)

Abstract: Following last week’s talk, we will sketch the proof of the Gross-Hopkins duality following Neil Strickland’s paper. In addition, we will explain how the Gross-Hopkins period map is related to the duality phenomenon.