Posts

Speaker: Sergei Burkin – Pontifical Catholic University of Rio de Janeiro Abstract: Several well-known categories, including simplex category $\Delta$, Joyal’s categories $\Theta_n$, Segal’s category $\Gamma$ and Moerdijk-Weiss dendroidal category $\Omega$, allow to encode homotopy coherent structures via Segal conditions. We show that most of these categories arise from operads in a canonical way, namely, via the only natural generalization of Quillen’s twisted arrow category construction to operads. This construction gives a good context for these categories....

Speaker: Julius Frank – University of Aberdeen Abstract: Differential graded algebras whose homology is a graded polynomial algebra in one variable are surprisingly sparse. One example can be obtained by taking the derived $E_1$-quotient of a commutative ring by some prime $p$. This construction also shows up in some algebraic K-theory calculations, which motivates investigating this algebra with trace methods. Examples are necessarily non-commutative: I will explain how all $E_2$-DGAs with polynomial homology over a fixed perfect ring are equivalent as ring spectra....

Speaker: Julie Rasmusen – University of Warwick Abstract: In recent years work by Calmés-Dotto-Harpaz-Hebestreit-Land-Moi-Nardin-Nikolaus-Steimle have moved the theory of Hermitian K-theory into the framework of stable $\infty$-categories. I will introduce the basic ideas and notions of this new theory and introduce a tool which can help us understand this better: Real Topological Hochschild Homology. I will then explain the ingredients that goes into constructing the geometric fixed points of this THR as a functor, generalising the formula for ring spectra with anti-involution of Dotto-Moi-Patchkoria-Reeh....

Speaker: Sarah Petersen – University of Colorado Boulder Abstract: In this largely expository talk, we will overview some of the motivation for and constructions of Brown-Gitler spectra. We will discuss their use in homotopy computations and indicate some areas of current research interest. body {text-align: justify}

Speaker: Ben Spitz – University of California, Los Angeles Abstract: Mackey and Tambara functors are equivariant generalizations of abelian groups and commutative rings, respectively. What this actually means is that, in equivariant homotopy theory, Mackey functors appear wherever one would expect to find abelian groups, and Tambara functors appear wherever one would expect to find commutative rings. However, the theories of Mackey and Tambara functors are comparatively much less developed than those of abelian groups and commutative rings....

Speaker: Torgeir Aambø – Norwegian University of Science and Technology Abstract: Chromatic homotopy theory views the stable homotopy category as certain nicely behaved layers glued together along formal neighborhoods. These are respectively described by the famous Morava E-theories $E(n)$ and Morava K-theories $K(n)$. We can single out one of these layers – in some sense reducing the entire colorful spectrum down to a single chroma – giving us monochromatic homotopy theory....

Speaker: Bastiaan Cnossen - Hausdorff Research Institute for Mathematics Abstract: Categories of equivariant objects (representations, equivariant spectra, equivariant Kasparov categories, etcetera) often exist globally for all finite groups, interrelated by restriction functors which restrict the group action. Frequently, there are also induction​ functors in the other direction, which are both left and right adjoint to the restriction functors. One may see this as an equivariant analogue of the existence of biproducts in a category....

Speaker: Christian Carrick - Utrecht University Abstract: If $X$ is a spectrum, then the cohomology groups $H^*(X;\mathbb{F}_ 2)$ form a module over the $\text{mod }2$ Steenrod algebra $\mathcal{A}$. Via the Adams spectral sequence, this algebraic structure almost completely determines the homotopy type of $X$ in many cases. However, $\mathcal{A}$ is a non-Noetherian algebra, so it is often convenient to study $H^*(X;\mathbb{F}_2)$ instead as a module over finite subalgebras of $\mathcal{A}$. When $H^*(X;\mathbb{F}_2)$ is a finitely presented $\mathcal{A}$-module, this results in no loss of information, and Mahowald-Rezk demonstrated that such spectra $X$ - “fp spectra” - admit a host of interesting categorical properties, including a chromatic version of Brown-Comenetz duality....

Speaker: Sebastian Chenery - University of Southampton Abstract: We will discuss recent work inspired by a paper of Jeffrey and Selick, where they ask whether the pullback bundle over a connected sum can itself be homeomorphic to a connected sum. We provide a framework to tackle this question through classical homotopy theory, before pivoting to rational homotopy theory to give an answer after taking based loop spaces. body {text-align: justify}

Speaker: Alice Hedenlund - Uppsala University Abstract: Twisted stable homotopy theory was introduced by C. Douglas in his 2005 PhD thesis, meeting a particular need in Floer homotopy theory to deal with infinite-dimensional manifolds that are “non-trivially polarised”. Roughly, one could think of twisted spectra as arising as sections of a bundle of categories whose fibre is the category of spectra. There are multiple ways of rigorously making sense of this: using sheaves of categories, local systems of categories, or modules over Thom spectra....