Abstract: In their periodicity theorem Hopkins and Smith constructed a v_n^N self map on each finite type n spectra where N is a constant that depends on the chosen type n spectrum. I discuss my work on showing that there exist finite spectra for which N=1. More specifically, I have proven that there exists an X for each height and prime for which the gcd of the powers of the self-maps it admits is 1. After telescopic localization these spectra admit v_n^1 self-maps and are convenient in examining the telescope conjecture.

Abstract: Secondary stability is a recently discovered stability pattern for the homology of a sequence of spaces exhibiting homological stability in a range where homological stability does not hold. We prove secondary stability for unordered configuration spaces of manifolds. The main difficulty is the compact case (the non-compact case was previously known by some experts). In the compact case, there are no obvious stabilization maps and the homology does not stabilize but is periodic.  We resolve this issue by constructing a chain-level stabilization map for configuration spaces of compact manifolds.

Abstract: My research is primarily about v_1-periodicity and v_1-selfmaps. In my first project, we establish a v_1-periodicity theorem over the complex-motivic Steenrod algebra. In joint work with Prasit Bhattacharya and Bertrand Guillou, we show that one of the 1-periodic v_1−self-maps of Y, whose cofiber is a realization of the real-motivic Steenrod subalgebra A(1), can be lifted to a C_2 equivariant self-map as well as real-motivic. We show that A(1) can be given 128 different module structures, and all of which can be realized.

Abstract: The A^1-degree is the motivic homotopical analog of the Brouwer degree. In this video, I discuss my work on computing the A^1-degree in two ways: by lifting and transferring, and in terms of commutative algebraic formulas. I also discuss a few applications of the A^1-degree, including computing trace forms of number fields and studying enumerative geometry over general base fields.

Abstract: Hi! I’m David Mehrle. I'm a PhD student at Cornell University working with Inna Zakharevich. I work in equivariant homotopy theory. Specifically, my thesis is on commutative and homological algebra in the category of Mackey functors. Although computations in equivariant homotopy theory are done with Mackey functors, many aspects of their algebra have not yet been developed. In this video, I  share results from two different projects of mine in the algebra of Mackey functors. 

Free incomplete Tambara functors are almost never flat: https://arxiv.org/abs/2105.11513

David's website: math.cornell.edu/~dmehrle

Abstract: My research focuses on computations in equivariant homotopy theory. In my thesis work, I investigate multiplicative structures in order to compute the RO(C_2 -graded homology of C_2-equivariant Eilenberg-MacLane spaces. This approach extends a non-equivariant Hopf ring argument of Ravenel-Wilson computing the mod p homology of non-equivariant Eilenberg-Maclane spaces to the RO(C_2)-graded setting. An important tool that arises in this equivariant context is the twisted bar spectral sequence which in general is quite complicated, lacking an explicit E_2 page and having arbitrarily long equivariant degree shifting differentials. I avoid working directly with these differentials and instead use a computational lemma of Behrens-Wilson to complete the computation. In ongoing and future work, I apply these methods more broadly, aiming to produce additional equivariant analogues of classical nonequivariant computations.

If you would like to get in touch, or read more about me and my projects, you can find more information on my webpage: https://sites.google.com/view/sarahpetersen/home

Abstract: My name is Emma Phillips, and I am a Ph.D. candidate at the University of New Hampshire. Here, I discuss three research projects: 1. I study a localization of operads, and work to prove that this localization plays the same role as the Dwyer-Kan hammock localization; 2. in a collaboration, we describe algebraically the bicategory of 2-group bundles over a manifold with a contractible universal cover and use this to prove several results about principal 2-group bundles; 3. in another collaboration, we aim to use a double categorical model of bicategories to consider applications of category theory in higher dimensions.

Abstract: In the 1980s, Mahowald and Kane used Brown-Gitler spectra to construct splittings of bo ⋀ bo and l ⋀ l. These splittings helped make it feasible to do computations using the bo- and l-based Adams spectral sequences. I will discuss progress towards an analogous splitting for BP<2> ⋀ BP<2> at odd primes.

Abstract: My primary research area is homotopy theory, a subfield of algebraic topology.  In past and ongoing work I focus on topologically motivated module theory and homological algebra, with connections to category theory and representation theory.  

Recently, I have also been expanding my area of focus to include problems applicable to data science. In this video statement, I focus on this most recent work in topological data analysis.

Abstract: Rational homotopy theory has deep and powerful results about the growth of the free part of the unstable homotopy groups of a finite space, but we know much less about how the torsion behaves. I'll tell you about work in this direction, following a program of Huang and Wu. The questions are easy to pose, but difficult to resolve in any generality, and I'm interested in how they connect to other parts of homotopy theory, especially Lie models and Goodwillie calculus.

Abstract: In this video, I give an overview of the 2-Segal spaces of Dyckerhoff-Kapranov and Gálvez-Kock-Tonks and provide the motivation for my thesis project, which is to find a robust 2-Segal generalization of quasi-categories.

Abstract: In this talk I give an overview of my research. My focus is on invariants of dg-categories and structures on them. In particular, it is conjectured that periodic cyclic homology of smooth proper dg-categories over complex numbers come with Hodge decomposition and a rational structure. I mention my results in these directions based on topological techniques and name several further questions. 

Abstract: In classical algebraic topology, the Hopf invariant allows stable homotopy theory to be applied to problems in topology, geometry, and algebra. In this video, I summarize work (joint with Balderrama and Culver) on the motivic Hopf invariant and its applications in algebraic geometry. The video also contains brief descriptions of my other research projects. 

Abstract: Let C3 denote the cyclic group of order 3. For spaces with an action of C3, we have a cohomology theory graded on RO(C3) called Bredon cohomology. In my thesis project, I classify all surfaces with an action of C3 using equivariant surgery methods. I then use this classification to compute the RO(C3)-graded Bredon cohomology of all C3-surfaces with coefficients in the constant Z/3-Mackey functor.

Abstract: The slice filtration is an equivariant filtration of genuine G-spectra. Due to results from Hill, Hopkins, Ravenel, and Ullman, in order to characterize all slices over a finite group G, it is only necessary to characterize n-slices where n is between 0 and |G|-1. For the Klein Four group we are only missing a characterization of the (4k+2)-slices. Today, we will characterize 2-slices over Klein Four. We will also determine the slices of integral (de)suspensions of the Eilenberg-Mac Lane spectrum HZ and look at some resulting (co)towers and the relationship between them.