In this talk, we will construct a stratification for the equivariant slice spectral sequence. This stratification is achieved through the localized slice spectral sequences, which compute the geometric fixed points equipped with residual quotient group actions.  As an application, we will utilize this stratification to investigate norms of Real bordism theories and their quotients. These quotients hold significant importance in Hill—Hopkins—Ravenel’s resolution of the Kervaire invariant one problem, as well as in the study of fixed points of Lubin—Tate theories by finite subgroups of the Morava stabilizer group. For these theories, the stratification exhibits a transchromatic phenomenon: the slice spectral sequence of a higher height theory is stratified into distinct regions, each isomorphic to the slice spectral sequences of the lower height theories. This provides an inductive approach and various structural insights when computing the fixed points of Lubin—Tate theories.   This is joint work with Lennart Meier and Mingcong Zeng.

Calculating the coefficients of equivariant generalized cohomology theories has been a fundamental question for equivariant homotopy theory. In this talk, I will talk about some of my joint work with Po Hu and Igor Kriz in the nonabelian cases. Examples include RO(G)-graded Eilenberg-MacLane cohomology of a point with constant coefficient when G is a dihedral group of order 2p or the quaternion group Q_8, and coefficient ring of Σ₃-equivariant complex cobordism. If time permits, I will also talk about some possible calculations for compact Lie groups.

Given an odd prime p, I will explain how the mod p Chern classes of the conjugation representation of PU(p^r) are related to the Dickson invariants, which, by definition, are certain elements in the polynomial ring F_p[x_1,··· ,x_n] invariant under a GL_n(F_p)-action.

The Gröbner basis is a powerful tool in commutative algebra. We can use it to do many calculations such as computing the presentations of the kernel and cokernel of a map between finitely presented modules over a commutative algebra. However, many important algebras including the Steenrod algebra in algebraic topology are not commutative. We make a noncommutative generalization of the Gröbner basis which can be applied to the Steenrod algebra A. This leads to highly efficient calculations in the category of A-modules including the computation of E2 pages of Adams spectral sequences.

A moduli space is a space of parameters that label a certain family of structured objects we are interested in.  I'll report on using methods of algebraic topology to understand aspects of a diverse set of moduli problems: (1, joint with Guozhen Wang et al.) in connection with p-adic arithmetic geometry, a filtered equivariant quasi-syntomic sheaf of Koszul complexes for computing unstable chromatic homotopy of spheres, over moduli spaces that parametrize deformations of a formal group with level structures; (2, joint with Hongwei Jia et al.) in connection with condensed-matter physics and materials science, monodromy of stratified vector bundles as moduli for gapless quantum mechanical systems, which arise from non-Hermitian symmetries; and (3, joint with Pingyao Feng et al.) in connection with data science, topological distribution spaces for image and speech signals, as revealed from persistent homology, and applied to the design of convolutional layers for deep learning.  For each, I will introduce the context of study and describe the mathematical objects in question, with all technical terms above explained.

The classification of manifolds in various categories is a classical problem in topology. It has been widely investigated by applying techniques in geometric topology in the last century. However, the known results tell very few information about the homotopy of manifolds. In the last ten years, there are attempts to study the homotopy properties of manifolds by using techniques in unstable homotopy theory. In this talk, we will discuss the loop decomposition method in this topic and review the known results and our recent work.

Ravenel introduced a filtration of MU by Thom spectra X(n), which played a key role in the Devinatz-Hopkins-Smith proof of the nilpotence theorem. We say a ring spectrum E has chromatic number <= n if E becomes complex orientable after tensoring with X(n). We discuss general properties of spectra with finite chromatic number and compute this number for the equivariant versions of Johnson-Wilson theories introduced by Hu-Kriz and Beaudry-Hill-Shi-Zeng.

In the 1980's, Kane and Mahowald used Brown-Gitler spectra to construct spectrum-level splittings of the bo and BP<1>-cooperations algebras. These splittings helped make it feasible to do computations using the bo and BP<1>-Adams spectral sequences. In this talk, we will present an analogous splitting for the BP<2>-cooperations algebra.

(joint with Asok-Elmanto-Hopkins)
The real étale topology on schemes arose out of the study of semialgebraic geometry and semialgebraic topology, that is, the study of varieties (and their cohomology) in the presence of a notion of "inequality". The prime example is, of course, working over the real numbers with their usual ordering.
Some years ago I showed that a piece of stable motivic homotopy theory is closely related to semialgebraic topology: essentially, stable semialgebraic (=real étale) homotopy theory arises from motivic homotopy theory by inverting a certain map called rho. In this talk I will report on an unstable upgrade of this theory. In other words, I will speak about the real étale localization and rho-periodization of motivic spaces.

We construct explicit cocycle representatives of twisted equivariant Thom classes in real and complex K-theory. Basic properties of Clifford algebras can then be used to verify that power operations are compatible with the spin and spin^c orientations. This is an ongoing work joint with Daniel Berwick-Evans.

Classical Steenrod operations is one of the most fundamental and formidable tools in stable homotopy theory. It led to calculation of homotopy groups of spheres, calculation of cobordism rings, characteristic classes, and many other celebrated applications of homotopy theory to geometry. However, equivariant analogs of Steenrod operations are not known beyond the group of order 2. 
In this talk, I will demonstrate a geometric method that constructs G-equivariant Steenrod operations for any finite group G. This method recovers known C_2-equivariant Steenrod operations and produces new families of equivariant Steenrod operations for all G. Time permitting, we will discuss a few open problems and potential applications to equivariant geometry.

Chromatic homotopy theory is a powerful tool to study periodic phenomena in stable homotopy groups of spheres. Under this framework, the homotopy groups of spheres can be built from the fixed points of Lubin--Tate theories. The homotopy groups of these fixed points are periodic and computed via homotopy fixed points spectral sequences. In this talk, we give an upper bound of the complexity of these computations. In particular, at the prime 2, for any given height, and a finite subgroup of the Morava stabilizer group, we find a number N such that the homotopy fixed point spectral sequence of collapses after page N and admits a horizontal vanishing line of a certain filtration N.
Our proof uses new equivariant techniques developed by Hill--Hopkins--Ravenel in their solution of the Kervaire invariant one problem and has applications to computations. This is joint work with Zhipeng Duan and XiaoLin Danny Shi.

The stable homotopy groups of spheres are one of the central objects of study in algebraic topology. In this talk, I will discuss some motivation for studying these groups and review some of what is known about them. I will then explain the detection of some new 192-periodic families of elements in the 2-primary stable homotopy groups of spheres using recent progress on non-nilpotent self-maps on certain finite cell complexes and topological modular forms. This is joint work with Prasit Bhattacharya and Irina Bobkova. 

Synthetic homotopy theory is a general framework for constructing interesting contexts for doing homotopy theory: using the data of a spectral sequence in some category C, one can construct another category which can be viewed as a deformation of C. The motivating example is the fact, due to Gheorghe-Wang-Xu, that (p-complete, cellular) ℂ-motivic homotopy theory can be described as a deformation of the ordinary stable homotopy category, simply using the data of the Adams-Novikov spectral sequence. Burklund, Hahn, and Senger used this framework to study ℝ-motivic homotopy theory as a deformation of C_2-equivariant homotopy theory. In joint work with Gabe Angelini-Knoll, Mark Behrens, and Hana Jia Kong, we give (up to completion) a different synthetic description of this deformation, which generalizes to give a deformation of (Borel-complete) G-equivariant homotopy theory for other groups G.

Non-equivariantly, the dual Steenrod algebra spectrum is a wedge of suspensions of HZ/p. I will talk about the computation of the equivariant dual Steenrod algebra for G=C_p, the cyclic group of order p. It turns out that when p is odd, the dual Steenrod algebra spectrum is a wedge of suspensions of HZ_p and another spectrum, which we call HT. I will talk about how to obtain the generators of these summands. This is joint work with Po Hu, Igor Kriz, and Petr Somberg.


I will describe how the G-equivariant J-homomorphism can be "desuspended" in a way that gives rise to nontrivial RO(G)-graded periodicities in equivariant stable homotopy theory, such as in the G-equivariant stable stems. When G = C_2, these are essentially versions of James periodicity, and I will explain how this recovers and unifies theorems of Bredon, Araki and Iriye, and Behrens and Shah.

Inspired by similar such constructions for topological K-theory, we present an algebra-geometric perspective on stable cohomology operations in elliptic cohomology. The advantage of this point of view, is that the families of operations it produces are highly structured, leading to families of operations on certain universal elliptic cohomology theories. We would like to highlight some of the immediate applications of these operations, some of their structural properties, as well as some future directions.