Many of the issues in modern physics depend on computations in homotopy theory. In this talk I will survey some of these situations and state some open problems arising from this interaction with physics.

In this talk, I will introduce the elements h₀h_n in the E₂-term of the classical Adams spectral sequence and of the Adams-Novikov spectral sequence. I will also introduce the "method of infinite descent," by which we proved that h₀h₃ is a permanent cycle. At last I will introduce our further consideration on the elements h₀h_n.

I will introduce chromatic homotopy theory which uses Bousfield localization with respect to Morava K-theories K(n) to filter the category of spectra. This filtration by height allows us to simplify calculations of stable homotopy groups of spheres by working one prime and one chromatic height at a time. I will introduce the main tools from number theory that help with these computations.
Then I will talk specifically about current work at chromatic height 2 and describe how the sphere at height 2 can be decomposed in terms of spectra related to the spectrum of topological modular forms TMF. I will talk about computing the Spanier-Whitehead duals of TMF and related spectra and describe how this can be useful for understanding the K(2)-local sphere.

Let G be a 1-connected Lie group with a maximal torus T. Combining the Schubert presentation of the integral cohomology of the flag manifold G/T obtained by Duan and Zhao (2015) with the Leray-Serre spectral sequence of the fibration G → G/T , we construct the integral cohomology ring H^∗(G; ℤ) uniformly for all G.

The rational homotopy theory of Quillen and Sullivan identifies homotopy types of topological spaces with differential graded commutative (co)algebras, and with differential graded Lie algebras, after inverting primes. Given any non-negative integer n, we can instead invert certain "v_n-self maps" and seek algebraic models for the resulting unstable "v_n-periodic" homotopy types. I'll explain why this is a natural and useful generalization of the classical story, and how a version of it has been achieved through Goodwillie calculus in recent work of Behrens and Rezk. I'll then explain my work on its applications to calculating unstable homotopy types in the case of n = 2. A key tool is power operations in Morava E-theory. Time permitting, I'll report further joint work in progress with Guozhen Wang.

Chromatic localization can be seen as a way to calculate a particular infinite piece of the homotopy of a spectrum. For example, the (infinite) chromatic localization of a p-local sphere is its rationalization, and recovers only the zero stem in π_∗𝕊. Palmieri has developed an analogue of chromatic homotopy theory in an algebraic category Stable(A), which gives information about Adams E₂-pages. Unlike in stable homotopy, in this world there are many non-nilpotent operators acting on the unit object, and unlike localization at p, localizing at some of these operators can provide nontrivial information. We discuss one of these localizations, which recovers partial information about the height 2 part of the Adams E₂-page for the sphere.

This talk concerns a twenty-thousand-year old mistake: The natural numbers record only the result of counting and not the process of counting. As algebra is rooted in the natural numbers, the higher algebra of Joyal and Lurie is rooted in a more basic notion of number which also records the process of counting. Long advocated by Waldhausen, the arithmetic of these more basic numbers should eliminate denominators. Notable manifestations of this vision include the Bökstedt-Hsiang-Madsen topological cyclic homology, which receives a denominator-free Chern character, and the related Bhatt-Morrow-Scholze integral p-adic Hodge theory, which makes it possible to exploit torsion cohomology classes in arithmetic geometry. Moreover, for schemes smooth and proper over a finite field, the analogue of de Rham cohomology in this setting naturally gives rise to a cohomological interpretation of the Hasse-Weil zeta function by regularized determinants, as envisioned by Deninger.

I will talk about irreducible components of the Thom spectrum MString associated to the 7-connected cover of BO. I will develop a Milnor-Moore theorem for non-graded Hopf algebras in order to show that in the K(2)-local category the smash product of MString and the "small" spectrum T₂ splits into copies of Morava E-theories. Here, T₂ is related to the Thom spectrum of the canonical bundle over ΩSU(4).

Motivic homotopy theory is the homotopy theory for smooth schemes. The application of motivic homotopy theory to algebraic geometry is a success, solving important problems such as the Milnor conjecture and the Bloch-Kato conjecture.
On the other hand, the Betti realization functor relates motivic homotopy theory to classical homotopy theory. This produces powerful methods for studying classical homotopy theory, such as Isaksen's computations of stable homotopy groups of spheres up to stem 59.
In this talk, I will present joint work with Gheorghe, Isaksen and Xu on the latter direction. We will show that the category of cellular Cτ-modules over the complex numbers is equivalent to the derived category of BP_∗BP-comodules as infinity categories. This implies that the algebraic Novikov spectral sequence is isomorphic to the motivic Adams spectral sequence for Cτ, providing a systematic way to generate nontrivial Adams differentials using algebraic computations. Using this method, we can do the computations of stable homotopy groups of spheres up to the 90-stem, and many new phenomena in stable homotopy groups are discovered in this new range.

The Teichmuller space of negatively curved metrics on a hyperbolic manifold M is the quotient of the space of all negatively curved Riemannian metrics on M by the action of the group of all self-diffeomorphisms that are homotopic to the identity. F. T. Farrell and P. Ontaneda have proved that the Teichmuller space of negatively curved metrics on a real hyperbolic manifold is, in general, not contractable. In this talk, I will present joint work with M. Bustamante and F. T. Farrell on showing that the Teichmuller spaces of negatively curved metrics on some real hyperbolic manifolds have nontrivial higher rational homotopy groups.

The characters of categorical representations behave much like Hopkins-Kuhn-Ravenel characters at chromatic level 2. In particular, there are formulas for the Strickland inner product, and these have a meaning, describing the dimensions of categorical fixed points. The application I will revisit in this talk is the Orbifold Hochschild-Kostant-Rosenberg isomorphism, where the formalism of categorical representation theory gives a very clear picture of multiplicativity.

Thanks to work of Ando, Hopkins, Strickland, Rezk, Zhu, Stapleton, Barthel, and others, we now know a great deal about the operations that act on the homotopy of K(n)-local 𝔼_∞ E-algebras. I will discuss work, joint with Brantner and Knudsen, that seeks to understand operations on 𝔼_n algebras by calculating the Morava E-theory of loop spaces of spheres. The key input is the use of Koszul duality and spectral Lie algebras to reduce the problem to the 𝔼_∞ case.

This talk will be an introduction to the use of Quillen model categories in stable homotopy theory. There is more than one way to define a model structure in the category of spectra, and they all involve Bousfield localization, which I will also introduce. I will also talk about how to define the category in such a way that it has a closed symmetric monoidal structure given by smash product.

At a fixed height, chromatic homotopy theory simplifies as the prime p grows. In this talk we will explore the asymptotic behavior of chromatic homotopy theory by constructing versions of the E(n)-local category and K(n)-local category at non-principal ultrafilters on the set of prime numbers. Further, we will show that the resulting categories are purely algebraic in nature. This is all joint work with Tobias Barthel and Tomer Schlank.

I will talk about the slice spectral sequence of a C₄-equivariant spectrum. This spectrum is a variant of the detection spectrum of Hill-Hopkins-Ravenel and is very closely related to the height 4 Lubin-Tate theory. This is joint work with Mike Hill, Guozhen Wang, and Zhouli Xu.

(Joint with Fedor Pavutnitskiy)
The project was carried out in the PhD thesis of Fedor Pavutnitskiy. We use combinatorial group theory methods to extend the definition of a classical James-Hopf invariant to a simplicial group setting. This allows us to realize certain coalgebra idempotents at simplicial set level and obtain a functorial decomposition of the spectral sequence, associated with the lower p-central series filtration on the free simplicial group.
The talk will aim to general audience, starting from the introduction of basic notions and techniques on the topic.

It is a classical result that if G is an appropriate Lie group, then its loop space ΩG is homotopy equivalent to a certain infinite-dimensional manifold (homogeneous space) called the Grassmannian model. In (for example) geometric representation theory, there is a well-studied algebro-geometric analogue of the Grassmannian model, the so-called affine Grassmannian Gr_G of an algebraic group G. This is an ind-variety. We show that there is a canonical 𝔸¹-equivalence between Gr_G and the space of 𝔾_m-loops Ω_𝔾mG on G, in the sense of 𝔸¹-homotopy theory.

Elliptic cohomology is a natural big brother to ordinary cohomology and K-theory. However, in contrast to the geometric objects that provide representatives for ordinary cohomology and K-theory classes, there is no such geometric description of elliptic cohomology. This talk will explain some recent progress, namely a geometric model for equivariant elliptic cohomology over the complex numbers. The construction is motivated by techniques in supersymmetric gauge theory. This is joint work with Arnav Tripathy.

Spectra can be considered as the universal presentable stable ∞-category, similarly Sets can be considered as the universal presentable 1-category and pointed spaces can be considered as the universal pointed presentable ∞-category. More generally some properties of presentable ∞-categories can be classified as equivalent to being a module over a universal symmetric monoidal ∞-category. We call such universal symmetric monoidal ∞-category "modes." We describe certain facts about the general theory of modes and present how we can generate new ones from old ones.

The space of rational maps, as defined by Gaitsgory, is the fiber of an approximation map from the factorizable version of the affine Grassmannian to the space of G-bundles on a curve. Gaitsgory and Gaitsgory-Lurie proved that the space of rational maps from a curve to a quasi-affine variety has acyclic etale homology. I will explain a proof of this statement that enhances the acyclicity on homology to contractibility in the sense of motivic homotopy theory. The proof uses a moving lemma of Suslin.