**[Photo]** taken June 9th, 2018, Shenzhen, China

**Note:** All talks take place in Teaching Building 1, Room 301, except that the **afternoon** talks on Thu. June 7
are in Room 107.

The lecture rooms are classrooms and there will be rings at
class times.

### Wed. June 6

8:30 am, Coffee & Refreshments

9:00–9:45, **Michael Hopkins**,
*Some problems in homotopy theory inspired by questions in physics*

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.

**[Slides]** 10:05–10:50, **Xiangjun Wang**,
*On the homotopy elements h*_{0}h_{n}

In this talk, I will introduce the elements h_{0}h_{n} in the E_{2}-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 h_{0}h_{3} is a permanent cycle. At last I will introduce our further
consideration on the elements h_{0}h_{n}.

11:10–11:55, **Irina Bobkova**,
*Spanier-Whitehead duals in the K(2)-local category*

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.

12:00 pm, Lunch Break

**[Slides]** 2:30–3:15, **Haibao Duan**,
*Schubert calculus and cohomology of Lie groups*

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.

3:35–4:20, **Yifei Zhu**,
*Toward calculating unstable higher-periodic homotopy types*

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.

4:40–5:25, **Eva Belmont**,
*Chromatic localization in an algebraic category*

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 E_{2}-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 E_{2}-page for the sphere.

6:00 pm, Banquet

### Thu. June 7

8:30 am, Coffee & Refreshments

**[Slides]** 9:00–9:45, **Lars Hesselholt**,
*Higher algebra and arithmetic*

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.

10:05–10:50, **Gerd Laures**,
*Chromatic atoms of Thom spectra*

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 T_{2} splits into copies of Morava E-theories. Here, T_{2}
is related to the Thom spectrum of the canonical bundle over ΩSU(4).

**[Slides]** 11:10–11:55, **Guozhen Wang**,
*Motivic Cτ modules*

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.

12:00 pm, Lunch Break, **Afternoon Talks in Room 107**

2:30–3:15, **Yi Jiang**,
*Teichmuller spaces of negatively curved metrics on hyperbolic manifolds*

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.

3:35–4:20, **Nora Ganter**,
*Categorical traces and orbifold invariants*

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.

4:40–5:25, **Jeremy Hahn**,
*Dyer-Lashof operations in the Morava E-theory of n-fold loop spaces*

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.

### Fri. June 8

8:30 am, Coffee & Refreshments

**[Slides]** 9:00–9:45, **Douglas Ravenel**,
*Model categories and stable homotopy theory*

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.

10:05–10:50, **Nathaniel Stapleton**,
*Chromatic homotopy theory is asymptotically algebraic*

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.

**[Slides]** 11:10–11:55, **XiaoLin Danny Shi**,
*The slice spectral sequence of a height 4 theory*

I will talk about the slice spectral sequence of a C_{4}-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.

12:00 pm, Free Afternoon

### Sat. June 9

8:30 am, Coffee & Refreshments

**[Slides]** 9:00–9:45, **Jie Wu**,
*Simplicial James-Hopf map and decompositions of the unstable Adams spectral
sequence for suspensions*

(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.

10:05–10:50, **Tom Bachmann**,
*Loop groups in 𝔸*^{1}-homotopy theory

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
𝔸^{1}-equivalence between Gr_{G} and the space of 𝔾_{m}-loops Ω_{𝔾m}G on G,
in the sense of 𝔸^{1}-homotopy theory.

11:10–11:55, **Daniel Berwick-Evans**,
*A geometric model for complex analytic equivariant elliptic cohomology*

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.

12:00 pm, Lunch Break

2:30–3:15, **Tomer Schlank**,
*Anabelian Bousfield lattice and presentable modes*

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.

3:35–4:20, **Elden Elmanto**,
*Motivic contractibility of the space of rational maps*

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.