Abstract: Topological Azumaya algebras arise as a generalization of Central Simple Algebras over a field. Tensor product is a well-defined operation on topological Azumaya algebras. Hence given a topological Azumaya algebra A of degree mn, where m and n are positive integers, it is a natural question to ask whether A can be decomposed according to this factorization of mn. In this talk, I briefly explain the definition of a topological Azumaya algebra over a topological space X, and present a result about what conditions should m, n, and X satisfy so that A can be decomposed.

Abstract: My current research provides a new homotopy theoretic perspective on problems from manifold theory, particularly relating to those involving connected sums. In this video presentation, I discuss some of my results, obtained throughout the course of my PhD. I also briefly touch on some speculative future work.

Abstract: My research interests lie in stable and chromatic homotopy theory, with overlapping interests in modular representation theory and tensor triangular geometry. In particular, my thesis uses tools and techniques from these fields to study modules over the Steenrod algebra. In this video, I discuss the main result of my thesis, which is a classification of the thick subcategories of modules over the Steenrod algebra.

Abstract: My research focuses on computations in equivariant homotopy theory. In my thesis, we introduce a special computational method, which we call the splitting method. When the order of the acting group G contains multiple primes, this method enables us to compute the homotopy of G-spectra in an inductive way. In this video, we will explain the main procedure and key theorems for the splitting method, and some computations on equivariant Eilenberg-Maclane spectra as examples. I will try to apply the splitting method on more types of equivariant spectra and study its interactions with other theories in the future.

Abstract: In the Batalin-Vilkovisky formalism, one finds that the classical observables of a field theory on a manifold N possess a compatible Poisson bracket of cohomological degree +1. Furthermore, the observables form what is known as a factorization algebra on N. Given a "sufficiently nice" factorization algebra with Poisson bracket on N, one may associate to it a factorization algebra on NxR_{greater than 0}. The aim of the research statement is to explain the sense in which the latter factorization algebra "knows all the classical data" of the former. This is the bulk-boundary correspondence of the title.

Abstract: Algebraic topology has grown from its early roots, studying the fundamental group, to an extensive mathematical branch which includes ideas ranging from derived geometry to equivariant homotopy theory. However, this advancement has primarily relied on classical mathematical foundations, and particularly classical set theory. As a result it is largely unknown how classical results in algebraic topology will adapt to general foundations. In this talk I will provide some motivation why we should address this issue and then explain a possible approach via higher topos theory.

Abstract: I prove a result computing the factorization homology of a smooth map of $E_\infty$ algebras. It is a generalization of the Hochschild Kostant Rosenberg Theorem.

Our theorem, restricted to THH, is a brainchild of McCarthy-Minasian (https://arxiv.org/abs/math/0306243). We fix the errors in their proof using ideas from a subsequent paper (https://arxiv.org/abs/math/0401346).

We hope to use this result to find an interpretation of the Hopkins Kuhn Ravenel character map, and the Bismut Chern character as a Dennis trace map. Our theorem led us to guesses involving $A$-theory and thom spectra. For more information see my website https://hariraumurthy.github.io/.

My main result: 2:09
Special case of my result, proven by McCarthy and Minasian: 3:42
Errors in their result: 4:10
How I fix these errors: 5:08
The motivation for my main result: 6:40

Abstract: My research is focused on using the techniques of category theory to describe complicated structures such as the homotopy theory of spaces, network infrastructures for machine learning, or the K-theories of finite sets and algebraic varieties. This typically involves using various types of higher categories, so I am especially interested in systematically developing the theory of different notions of higher category and their interactions.

Abstract: Classifying topological spaces up to various equivalence relations is an important part of algebraic topology. One effective way to do this is to study localisations of topological spaces. Fixing a prime number p, the v_h-periodic localisation of topological spaces for every natural number h provides a decomposition of the p-local homotopy types of topological spaces into torsion-free and "v_h-periodic" torsion parts, which are the building blocks of unstable chromatic homotopy theory. I will give an introduction of these localisations and present two projects from my PhD thesis on this subject.

Abstract: A major theme in my research is the connections between algebraic K-theory and L-functions. In this video, I will first give a brief review on L-functions and algebraic K-theory, as well as some of the known connections between them. Then I will summarize my work in generalizing those connections to Dirichlet and Artin L-functions.

Abstract: I study very general notions of trace and their compatibility with various kinds of structure. In particular, I have formulated a notion of bicategorical cotrace, dual to Ponto's bicategorical trace, and applied it to Lipman's comparison of traces and cotraces and to Ganter and Kapranov's induction formula for 2-characters.

Abstract: Classical enumerative geometry asks geometric questions of the form "how many?" and expects an integral answer. For example, how many circles can we draw tangent to a given three? How many lines lie on a cubic surface? The fact that these answers are well-defined integers, independent upon the initial parameters of the problem, is Schubert’s principle of conservation of number. In this talk we will outline a program of "equivariant enumerative geometry", which wields equivariant homotopy theory to explore enumerative questions in the presence of symmetry. Our main result is equivariant conservation of number, which states roughly that the sum of regular representations of the orbits of solutions to an equivariant enumerative problem are conserved.

Abstract: Hi, I am a postdoc in Freibrug, Germany, and I am working on positive scalar curvature, index theory, and stable homotopy theory. In this video I will talk about two projects of mine. The first is about my PhD thesis in which I constructed a positive scalar curvature analog of the space of block diffeomorphism and show that the index difference factors through this new space. The second project concerns parametrised unitary bordism spectra. There I show that twisted K-homology are not determined by twisted unitary bordism in contrast to the untwisted case.

More information about me or my projects can be found on my website: https://home.mathematik.uni-freiburg.de/geometrie/hertl/?l=en

Abstract: In this video I describe my research in equivariant homotopy theory which constructs new tools for Real topological Hochschild homology (THR) in the form of Bökstedt-style spectral sequences and uses those tools in homology computations of THR.

More about me and my work can be found on my website: https://www.chloelewis.net/

Abstract: My research focuses on new chromatic computations in higher heights and study corresponding algebraic structures rising in homotopy computations. The goal is to encode the homotopy computation in as clean an algebraic manner as possible. In this video, I briefly motivate chromatic homotopy theory and summarize work (joint with Zhipeng Duan and XiaoLin Danny Shi) of a vanishing line in the chromatic computation at prime 2.

Abstract: An important tool in stable homotopy theory for computing stable homotopy classes of maps is the Adams spectral sequence. In recent years, Pstragowski has formed a category of synthetic spectra which, in a sense, is a "categorification of the Adams spectral sequence." In this video, I talk briefly about my research interests, describe some details of synthetic spectra, and discuss my recent research in this area.

Abstract: By importing tools from p-adic Hodge Theory, arithmetic geometry, and moduli spaces of curves, my thesis builds a new perspective which, by overcoming a 40 year old computational stalemate that has so far kept us from understanding the Lubin-Tate action, enables us to do chromatic homotopy theory in ways that were not possible before. Specifically, we use ramified families of curves to partially compute the E2 page of the homotopy fixed point spectral sequence of the K(h)-local homotopy groups of spheres of height h=p^{k-1}(p-1), for all such h and p simultaneously.

Abstract: An N-infinity operad is an equivariant extension of E-Infinity operads developed by Blumberg and Hill. My Research is about extending this theory to equivariant extensions of Ek-operads. In this talk, I will give some background on N-Infinity-operads, the idea of “Nk-operads” and talk about some work I’ve done on proving a version of Dunn additivity for equivariant little disks, as well as some applications on quotients of norms of MU.