Speaker: Torgeir Aambø - Norwegian University of Science and Technology 

Abstract: The classical Barr-Beck theorem gives sufficient criteria for an adjunction being monadic. In a monoidal setting this gives criteria for checking when a category is equivalent to modules over a ring object. During the talk we will explore some adjacent results, more specifically a local, co-local and dual version of the theorem, as well as some consequences and examples. 

Recording: Here!

Notes: Here!

Speaker: Eleftherios Chatzitheodoridis - University of Virginia

Abstract: Rational homotopy theory is a computationally friendly tool in the computationally unfriendly world of homotopy theory by discarding all torsion information in higher homotopy groups by tensoring with the divisible additive abelian group of the rational numbers in fashion informed by topology. This motivates us to study the constructions of rational homotopy theory, namely rational spaces and the rationalization of a space, using the construction of the rational n-sphere as a concrete running example throughout our study.

Recording: Here!

Notes: Here!

Speaker: Bridget Schreiner - University of Notre Dame

Abstract: Homological stability and representation stability give us ways to understand when a sequence of mathematical objects stabilizes (e.g. a sequence of groups, spaces, algebras). This talk will briefly review the notions of primary stability, and then introduce the concept of secondary stability. I will focus on two key examples, secondary stability of ordered configuration spaces (due to Miller and Wilson), and secondary stability of mapping class groups (due to Galatius, Kupers, and Randall-Williams), focusing on the similarities between the approaches, and the potential difficulties that arise when looking at even higher levels of stability. 

Recording: Here!

Speaker: Sandip Samanta -  Indian Institute of Science Education and Research Kolkata 

Abstract: We will start with the definition of fibrations and examples. Then given a fibration F → E → B, we will discuss the associated based loop space fibration ΩF → ΩE → ΩB. Then we will define H-space and H-splitting in a fibration with some known related results. Finally will introduce the brace product in a fibration with a section and state some results regarding it. 

Recording: Here!

Speaker: Baylee Schütte - University of Aberdeen

Abstract: A projective k--frame on a smooth manifold M is a smoothly varying assignment to each point x in M of a k-tuple of linearly independent lines in TxM. The projective span of M, denoted pspan(M), is the maximum k such that M admits a projective k-frame. We give a brief (and accessible) overview of the techniques we have used to determine necessary and sufficient conditions for pspan(M) >/= 2. Such techniques include Moore-Postnikov obstruction theory and McClendon's theory of higher order twisted cohomology operations. This research may have applications in "skew-framed cobordism" and the theory of "projective characteristic classes.” 

Recording: Here!

Slides: Here!

Speaker:  Justin Barhite - University of Kentucky 

Abstract: The familiar trace of a linear map generalizes to a notion of trace for any endomorphism of a dualizable object in a symmetric monoidal category. These traces often provide information about fixed points. Moreover, by recognizing traces in different categories as instances of this same abstract formulation, we can compare them functorially. We will introduce the notions of dualizability and trace, look at some motivating examples, and obtain the Lefschetz-Hopf theorem as an application of functoriality of trace.

Recording: Here!

Speaker:  Georgy C.  Luke - Indian Institute of Science Education and Research, Tirupati

Abstract: Colored links are links whose components are labeled with colors. The isotopy of colored links is an ambient isotopy of links that preserves the colors and orientation of the link. We will construct colored invariants inspired by the theory of quandles

Recording: Here!

Speaker: Dezhou Li - Northeastern University

Abstract: In Cohen's famous calculation of the mod p cohomology of configuration spaces, the key ingredient was a complete description of the Cartan–Leray spectral sequence for the configuration space of k=p points. I will discuss work in progress aimed at giving a complete description of this spectral sequence for arbitrary k. This work may help to shed light on the question of torsion in the cohomology of configuration spaces of graphs.

Recording: Here!

Speaker: Willow Bevington -  Edinburgh University

Abstract: Stratifications can been used to study homotopy invariants on manifolds and varieties with singularities, it is desirable then, to have an ∞-categorical description of stratified spaces. I will discuss some of the ways of getting a homtopy theory of stratified spaces pioneered by Peter Haine and discuss their relation to eachother and, time permitting, discuss the roles of stratifications in Exodromy in Haine's work with Clark Barwick and Saul Glasman.

Recording: Here!

Speaker: Uğur Bektaş Cantürk  - Middle East Technical University. 

Abstract: The amount of the existing data is increasing rapidly and any way to understand data is welcomed. A relatively new branch of mathematics, called Topological Data Analysis (TDA), proposes to use topology to study the shape of data. In this talk, we will see one of the most widely studied method in TDA which is Persistent Homology (PH).

We will first give the motivation and define basic structures of 1-parameter PH such as filtrations and persistence modules. Then, we will distinguish given modules with the help of invariants and stable distances. After showing 1-parameter is not enough for each case, we will try to extend structures to multiparameter, and observe differences in their theories. 

Recording: Here!

Slides: Here!

Speaker: Vladislav Zemlyanoy  - Higher School of Economics (Moscow, Russia) 

Abstract: The strong shape category of compact metrizalbe spaces is known to have multiple equivalent definitions, and many properties and results are known for it, such as the invariance of Steenrod-Sitnikov homology and Čech cohomology. Fine shape, as defined by Melikhov, is an extension of strong shape to noncompact metrizable spaces that keeps invariance of both; its definition is also far simpler that that of most noncompact shape theories. This raises the question of extending other known results for compact strong shape to (noncompact) fine shape. In that vein, representing fine shape through approaching maps of absolute retracts, we can show it to be a specific left fraction category, thus extending Cathey's definition of compact strong shape. In the process, we raise two more questions yet to be answered fully: one about noncompact version of Mrozik's mapping cylinder of an approaching map, and one about finding a model structure for which fine shape is the homotopy category. The latter in particular could hardly be done for compact spaces only, as the arising path spaces would in all probability be noncompact. 

Recording: Here!

Speaker: Wenwen Li  - University of Central Oklahoma

Abstract: Configuration space is a concept in the field of motion planning. In the case of arranging n robots in a warehouse, a configuration records one possible arrangement of those robots when we treat robots as points. The space that contains all possible configurations is called the n-th configuration space (of the warehouse). In the real-world scenario, however, there is a minimal distance allowed between each pair of robots because the robots are thick, and we cannot treat them as points. The space that contains all possible configurations is called the n-th configuration space with the restraint parameter r, where r is an n(n-1)/2-dimensional vector. In this talk, we will explore the configuration spaces that correspond to two robots moving on connected metric graphs with or without restraint parameter r and discuss connections between the configuration spaces and the (2-parameter) persistence theory.

Recording: Here!

Speaker: Grisha Taroyan  - University of Toronto and HSE 

Abstract: The talk is based on my recent paper (https://arxiv.org/abs/2303.12699) and essentially provides a version of the monoidal Dold--Kan correspondence for arbitrary Fermat theories in characteristic zero. From the geometric standpoint, this incarnates an equivalence between simplicial and differential graded versions of derived analytic geometry. Required background is purely homotopical, all definitions related to Fermat theories and stacks will be given on the spot.

Recording: Here!

Slides: Here!

Speaker: Julie Rasmusen - University of Warwick

Abstract: In recent years work by Calmés–Dotto–Harpaz–Hebestreit–Land–Moi–Nardin–Nikolaus–Steimle has moved the theory of Hermitian K-theory into the framework of stable ∞-categories. I will introduce the basic ideas and notions of this new theory, but as it is often the case when working with K-theory in any form, this can be very hard to describe. I will therefore introduce a tool which might make our life a bit easier: Real Topological Hochschild Homology. I will explain the ingredients that goes into constructing in particular the geometric fixed points of this as a functor, generalising the formula for ring spectra with anti-involution of Dotto–Moi–Patchkoria–Reeh.

Speaker: Juan Moreno - University of Colorado Boulder 

Abstract: Higher real K-theory spectra are spectra obtained from Morava E-theory by taking homotopy fixed-points with respect
to finite subgroups of the Morava stabilizer group. They are so-named because taking the homotopy fixed-points of height 1
E-theory by the central C2 gives the ordinary real K-theory spectrum, KO. As we will review in the beginning of the talk,
these spectra are central objects in chromatic homotopy theory. 
Work of Beaudry-Goerss-Hopkins-Stojanoska shows that taking the K(n)-local Spanier-Whitehead dual of these spectra 

essentially amounts to suspending by a certain representation sphere before taking homotopy fixed-points.

We use their work to identify the representation shift in certain cases, then focus on some examples in which the result can be simplified further to an integer shift. 

Speaker: Anton Engelmann -  University of Bonn

Abstract: With the discovery of a connection between complex oriented cohomology theories and elliptic curves a whole new branch opened for algebraic topologists. One of the new objects arising from this connection are topological modular forms which have important applications to the computation of homotopy groups of spheres and number theory, but they are also interesting in its own right. This talk will focus on the definition of the slightly different versions of topological modular forms that are around and on some of their properties.

Slides: Here!

Annotated Slides: Here!

Speaker: Sofía Martínez Alberga  - Purdue University  

Abstract: In general the objective of algebraic topology is to classify spaces using some algebraic invariants or up to some notion of equivalence. In the area of equivariant homotopy theory the goal is the same but now spaces equipped with a group action are considered and algebraic invariants of choice are homotopy groups. It turns out there is an analogous version of Whitehead's theorem in the equivariant setting which in some sense motivates studying weak homotopy equivalences over homotopy equivalences. This talk will review some of these homotopical notions and introduce Elmendorf's theorem. Proved in the eighties, this theorem sheds some light on how one can better understand equivariant homotopical notions as functors from the orbit category of the group to the category of topological spaces. Also this talk hopes to address how this perspective is used more modernly to better understand equivariant notions in other categories (namely simplicial sets and enriched categories) which will give rise to a theorem similar to Whitehead's. 

Speaker: Marwa  Mosallam - Binghamton University and Cairo University

Abstract: We will generalize the notion of the classical discrete Fourier transform, known in mathematical analysis, in chromatic homotopy theory by developing a general theory of higher semiadditive Fourier transforms that includes both the classical discrete Fourier transform for finite abelian groups at height n =0 as well as a certain duality for the E_n-(co) homology of \pi-finite spectra, established by Hopkins and Lurie, at heights n \geq 1. In particular, we will explain the notions of higher roots of unity, the categorical Fourier transform and the higher cyclotomic extensions. If time permits, we will lift the chromatic Fourier transform to the telescopic world which might shed light on the failure of the telescope conjecture.   

References:

1. "The Chromatic Fourier Transform " by Tobias Barthel, Shachar Carmeli, Tomer M. Schlank and Lior Yanovski.

2. "Chromatic Cyclotomic extensions" by Shachar Carmeli, Tomer M. Schlank and Lior Yanovski.

3. "The Chromatic Nullstellensatz" by Robert Burklund, Tomer M. Schlank and Allen Yuan.

4. "Ambidexterity and height" by Shachar Carmeli, Tomer M. Schlank and Lior Yanovski.

5. "Ambidexterity in chromatic homotopy theory" by Shachar Carmeli, Tomer M. Schlank and Lior Yanovski.

6. "Chromatic Structures in Stable Homotopy Theory" by Tobias Barthel and Agnés Beaudry .