Homotopy Day 2025

Homotopy Day 2025 will take place on Tuesday, February 4, 2025, at the Weizmann Institute of Science, Rehovot. The event will be held in Room 155 of the Ziskind Building.

The day will feature talks on homotopy theory and related topics, given by students, postdocs and faculty. Additionally, there will be free time for discussions to foster collaborations and informal exchanges. The event is open to all mathematics students and researchers.

Organizers: Shay Ben-Moshe, Shachar Carmeli and Lior Yanovski.
For inquiries, please contact Shay at shaybm9@gmail.com.

Registration

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Schedule

Abstracts

Asaf Yekutieli – On the rationalization of the K(n)-local category

The homotopy theory of K(n)-local spectra is intimately related to the arithmetic of height n formal groups. Scholze and Weinstein have studied the moduli problem of height n formal groups through the lens of perfectoid geometry, to great effect, establishing a connection with a different moduli problem.
Barthel, Schlank, Stapleton and Weinstein have exploited this connection to study the homotopy groups of the K(n)-local sphere, culminating in a computation of its rational homotopy type, verifying the conjectural description. The crucial observation underlying their work is that this connection facilitates circumventing a certain issue that has thwarted more direct computational approaches to K(n)-local homotopy theory.
This observation suggests a novel strategy to gain insights into monochromatic homotopy theory, and possibly beyond. I shall discuss this strategy, focusing on work in progress carried out jointly with Ben-Moshe, where we extend the description obtained by BSSW to encompass the rationalization of (a large chunk of) the K(n)-local category.

Oren Ben-Bassat – Derived Analytic Geometry

I will give a messy, brief, and disorganized overview of the book draft: arXiv:2405.07936. This includes some of the work that led up to it (starting around 2013) and some questions I have been thinking about. I will try to emphasize the parallels with homotopy theory.

Aziz Kharoof – Topological and Homotopical Methods for Analyzing Contextuality

Simplicial distributions offer a topological framework for analyzing contextuality for collections of probability distributions. This contextuality, which often arises from physical systems, can be studied by examining the set of joint distributions associated with a Bell-type scenario. These distributions form a polytope, whose structure is key to distinguishing contextual and non-contextual distributions. Researchers aim to understand this polytope by identifying its vertices and the hyperplanes that separate contextual from non-contextual simplicial distributions.
The mentioned approach is based on simplicial sets, so an intriguing question arises: How can we leverage the topology and homotopy theory of simplicial sets to address these challenges?
This talk introduces several of topological tools and results that provide insights into these problems:

1. Collapsing Map: A method for simplifying simplicial structures while preserving essential properties, aiding in the analysis of contextuality.
2. Homotopical Characterization of Strong Contextuality: We present a novel homotopical characterization of strongly contextual simplicial distributions with binary outcomes, specifically those defined on1-dimensional space.
3. The Gluing Lemmas: Results that enable the reconstruction of global properties of simplicial distributions from local data, offering a powerful way to handle complex scenarios.

Beckham Myers – Power operations and redshift

The redshift phenomenon in chromatic homotopy theory is about the nature of (iterated) K-theory and chromatic height. It can be interpreted as an instance of a broader redshift philosophy which relates categorification and (semiadditive) height. I will describe the theory of power operations (in height 1) as a concrete instance of this.

Amos Kaminski – Derived Stone Embedding

A classical result, the Stone embedding, characterizes profinite sets as totally disconnected, compact Hausdorff spaces. In this talk, I will discuss a derived version of this result, building on the framework introduced in "Pyknotic objects, I. Basic notions," which embeds the pro-category of π-finite spaces into pyknotic spaces. I will outline a partial characterization of the essential image of this embedding, extending the classical Stone embedding to a derived context.

Leor Neuhauser – The multiplicative structure of cobordisms

Cobordism, a fundamental concept in differential geometry, plays a deep role in both homotopy theory and higher category theory. In homotopy theory, (framed) n-manifolds up to cobordism are isomorphic to the n-th stable homotopy group of spheres. This isomorphism preserves the graded ring structure, with addition corresponding to disjoint union of manifolds and multiplication corresponding to multiplication of manifolds.
In the context of higher categories, we can define the (∞,n)-category Bord_n of (framed) 0-manifolds, with 1-bordisms between them, 2-bordism between bordisms, and so on up to dimension n. The cobordism hypothesis posits that Bord_n is the free fully-dualizable (∞,n)-category on a single generator. While the additive structure of disjoint union naturally lifts to a symmetric monoidal structure on Bord_n, lifting the multiplicative structure is more subtle.
In this talk, I will describe how to define this multiplicative structure, and, assuming the cobordisms hypothesis, I will show that the (∞,∞)-category Bord_∞ is an idempotent algebra. This result is part of joint work with Shai Keidar and Lior Yanovski.

Anton Khoroshkin – Contractads – algebraic structures behind generalized configuration spaces

For each simple connected graph $\Gamma$, one can associate a generalized configuration space consisting of a collection of points on $X$, indexed by the vertices of the graph, where the edges of the graph dictate the rule that certain points cannot coincide.
In this talk, I will first introduce a new algebraic structure called a "contractad," defined on the union of these spaces for $X = \mathbb{R}^n$. This structure extends the concept of the little discs operad and is based on graph contractions. Second, I will demonstrate how this algebraic framework can be used to extract information about the Hilbert series of cohomology rings. Interestingly, the same approach can also be applied to derive generating series for various combinatorial data associated with graphs, such as the number of Hamiltonian paths, Hamiltonian cycles, acyclic orientations, and chromatic polynomials.
This talk is based on joint work with my student D. Lyskov (arXiv:2406.05909).

Arye Deutsch – Floer homotopy theory old and new

Floer's idea of computing generalized homology from geometric data laid the groundwork for what is now known as Floer homotopy theory. This talk traces the evolution of this idea, starting with Morse homology and advancing through the flow categories of Cohen, Jones, and Segal, leading to the spectral model developed by Abouzaid and Blumberg.
We will explore how gradient flow lines and Morse-theoretic data encode homotopical information, culminating in the construction of a spectral sequence derived from the geometric structures underlying flow categories. The talk aims to provide an accessible overview of the key ideas and motivations shaping Floer homotopy theory today.