Abstract: In this video I describe my research and two future projects. In my research, I approach manifold topology by examining configurations of finite subsets of manifolds within the homotopy-theoretic context of ∞-categories by way of stratified spaces. Through these higher categorical means, I identify the homotopy types of such configuration spaces in the case of the circle and Euclidean space. These results naturally extend to two projects, one towards the study of finite type knot invariants and the other towards understanding the combinatorics of (∞, n)-categories.

Abstract: Unlike singular cohomology, which is graded by dimension, equivariant cohomology is graded by the representation ring RO(G). Computations in RO(G)-graded cohomology have led to important results, including the Hill-Hopkins-Ravenel solution to the Kervaire invariant one problem. Yet computations can be challenging, even for G=C2, the cyclic group of order two. In my thesis, I proved a structure theorem for RO(C2)-graded cohomology that simplifies computations. I continue to build on that work individually and in collaborations, including generalizing to larger groups, classifying certain module spectra, and using algebraic models to study commutative equivariant ring spectra.

Abstract: Hello, I'm Tasos Moulinos, currently a postdoc at the University of Toulouse in France. My research broadly lies in the areas of algebraic topology and algebraic geometry. Using the framework of higher algebra and derived algebraic geometry, I study cohomological structures arising from topology, geometry, and arithmetic.

Abstract: Chromatic homotopy theory defines approximations to the stable homotopy category, called the K(n)-local categories, that isolate certain periodic phenomena in stable homotopy groups. These K(n)-local categories can be studied using algebraic geometry and profinite group cohomology. I discuss my work on several problems in K(n)-local stable homotopy theory and its relation to algebraic geometry, including cooperations for the K(1)-local topological modular forms spectrum, obstruction theory for highly structured ring spectra, and duality decompositions for the K(2)-local sphere.

Abstract: My research applies the computational methods of homotopy theory to answer questions about the modular representation theory of finite groups. One problem of interest is in computing the group of endo-trivial modules. In homotopy theory, this group is known as the Picard group of the stable module category.

This group was originally computed by Carlson-Thévenaz using the theory of support varieties. However, work of Mathew, and Mathew-Stojanoska introduced a new, homotopical approach to this computation using descent methods and the homotopy fixed point spectral sequence. I extend these methods to provide new computations for (generalized) quaternion groups.

Abstract: I study a variety of problems in topology and geometry, all of which are closely related to classical problems of algebraic topology: vector fields on spheres, the generalized vector field problem, the existence of nonsingular bilinear maps, immersions and embeddings of projective spaces, the Borsuk-Ulam theorem, and more.

Abstract: My name is Viktoriya Ozornova and this video is about a joint project with Julie Bergner and Martina Rovelli on the comparison of certain models of (infty,2)-categories. I explain what an (infty,2)-category should be and how it can be implemented in two particular ways. Finally, I talk about the equivalence of these implementations.

Abstract: Hi! I'm Maru Sarazola, a PhD student at Cornell University working with Inna Zakharevich. I'm mainly interested in applying categorical methods to study problems in algebraic K-theory and homotopy theory. If you would like to get in touch, or read more about me and my projects, you can find more info on my webpage: http://pi.math.cornell.edu/~maru/

Abstract: In this talk I give an overview of my past and present research. My focus is on algebraic K-theory of singular schemes, which I study using topological Hochschild homology and p-adic cohomology theories. I mention results pertaining to three classical questions in K-theory: nil-invariance, excision, and A^1-invariance.

Abstract: In this video, Mingcong Zeng talks about his research in homotopy theory, which relates the Segal conjecture, Real bordism spectra and Lubin-Tate E-theories together.

Website: https://sites.google.com/view/mingcongzeng/

Abstract: We describe a general framework that makes it easy to produce obstruction theories in any setting with a good notion of homotopy operations. By applying this to the study of K(n)-local E-infinity algebras over a Lubin-Tate spectrum, we obtain new E-infinity complex orientations at heights 1 and 2.

Abstract: Not much is known about the ring of stable homotopy groups of spheres which is otherwise notorious for its complexity. In this video, the speaker explains why this ring is important in mathematics, how this problem motivated his research work, modern approaches to this problem and some of his contributions. The speaker also discusses specific open problems that he hopes to study in near future.

Abstract: I discuss some problems related to homological residues fields, which give a way of working with residue fields in tensor triangulated geometry. I discuss how these relate to tensor triangulated residue fields that exist in some examples, and in the case of the stable homotopy category state how these homological residue fields can be describe in terms of Morava K-theories. This is part of joint work with Paul Balmer and with Greg Stevenson. There are open problems in this area that go in many directions; I am particularly interested in some problems in representation theory and equivariant homotopy theory.

Abstract: In this video research statement, I describe two projects in equivariant and motivic stable homotopy theory. I also talk briefly about the ongoing and future work.

1. We construct the C_2-effective spectral sequence using the motivic effective filtration and equivariant Betti realization. Using it, we compute the homotopy groups of C_2-equivariant connective real K-theory.

2. (Joint with Bachmann, Wang, Xu) We define the Chow t-structure on the motivic stable homotopy category over general base fields. Its heart turns out to be purely algebraic. One application is to use it to determine the Adams differentials.

Website: http://math.uchicago.edu/~hanajk/

Abstract: We consider a nontrivial action of C_2 on the type 1 spectrum Y:=S/2/\S/\eta. This can also be viewed as the complex points of a finite real-motivic spectrum. One of the v_1−self-maps of Y can be lifted to a C_2 equivariant self-map as well as as well as a real-motivic self-map. Further, the cofiber of the self-map of the R-motivic lift of Y is a realization of the real-motivic Steenrod subalgebra A(1).

Abstract: My name is Jonathan Rubin, and I am a postdoc at UCLA. My research is in equivariant homotopy theory. More specifically, I have done work on Elmendorf--Piacenza type theorems, N-infinity theory, and equivariant category theory. In this video, I discuss some of my research and plans for the future.

Abstract: I am a postdoc interested in classical, equivariant, and motivic homotopy theory. In this video I talk about a project, joint with Dan Isaksen, to compute R-motivic stable homotopy groups using the R-motivic Adams spectral sequence. This has applications to the Mahowald invariant. I also briefly mention some of my other research interests, including the 3-primary classical Adams spectral sequence.

Abstract: Deninger proposed using a Weil cohomology theory in Arakelov geometry to prove the Riemann Hypothesis. I describe my work on enriched higher category theory and how it can be used to produce cohomology theories in Arakelov geometry, including K-theory and THH. K-theory is not a Weil cohomology theory but still has rich connections to the Riemann zeta function.

Abstract: In the 70s, J.P. May and F. Cohen developed a theory of operations on the mod p homology of E_n-algebras. My research aims to develop such a theory for homology with twisted coefficients, and to apply this theory to examples that cannot be analyzed using the classical theory. Here I describe the key differences between the classical and twisted theories and discuss the application of this theory to the homology of special linear groups.

Abstract: In this talk we will demonstrate how letting the cyclic group of order four act on the real Grassmannians can show the Atiyah-Hirzebruch spectral sequence calculating their nth Morava K-theory collapses. This uses chromatic fixed point theory coming from the classification of the equivariant Balmer spectrum of the cyclic groups. This work is joint with Nicholas Kuhn.

Abstract: My name is Apurva Nakade. I'm currently a postdoc at UWO. I'm interested in applying to liberal arts colleges and teaching track positions. My research is in homotopy theory. I'm currently working on two projects on 2-groups and in formal verification using computers. In the first 2-group project, we show that 2-groups naturally show up when studying finite Chern-Simons. In the second 2-group project, we set up the foundations of 2-groups in Homotopy Type Theory. In the formal verification project, I'm working on adding math to the existing math library in the Lean.

Abstract: My research focuses on problems dealing with completion, localization, and functor calculus in the setting of operadic algebras in spectra -- that is, spectra with extra algebraic "flavor" described by an operad. In studying completion of O-algebras with respect to TQ-homology, I have identified a new class of fibration sequences which are preserved by TQ-completion. My current and planned future work lies in the direction of functor calculus. I have proven a preliminary result in this area, which identifies sufficient conditions for the Taylor of the identity in O-algebras to recover TQ-completion.