We compute the descent spectral sequence (DSS) for Tmf, which validates all other computations of the homotopy groups of Tmf and tmf that we are aware of in the literature. This has been a problem for many years, that the previous literature on the homotopy groups of tmf is circular, but this concern only seems to appear in print in Meier's review here. There is a lot that one can do with topological modular forms without knowing its homotopy groups, but most of its applications to date use some kind of computation in this direction. Our solution to this problem is to compute the homotopy groups of Tmf, hence also those of tmf, using the spectral sequence associated to the synthetic spectrum Smf of (9) below. One advantage of this approach is that this spectral sequence is equivalent to the DSS for Tmf (proven in (9)) and also receives a natural map from the Adams--Novikov spectral sequence for the sphere. This is a key ingredient used in Bauer's computations of the homotopy groups of tmf, except he uses the translation between these two spectral sequences without proof or reason (a proof of which is provided by Mathew in Homology of tmf, which in turn relies on knowing some homotopy groups of Tmf...). Another key ingredient is to use all of the power of synthetic spectra. We prove synthetic versions of Moss' theorem and the Leibniz rule (which may appear in Burklund's Synthetic Cookware) which also allow us to answer questions of Isaksen about the DSS for Tmf/ANSS for tmf. As a result of our techniques too, we obtain not only the homotopy groups of Tmf and tmf, but also the ANSS's for TMF and tmf as a localisation and a retract, respectively, of the DSS for Tmf. There are lots of pictures and tables too!

Christian Carrick and I deduce various new nonvanishing statements about elements in the divided beta-family, a collection of 144-periodic classes in the stable homotopy groups of spheres defined using height 2 chromatic homotopy theory, at the prime 3. Similar to how we studied the divided alpha-family in (8) below, we use a ring spectrum E, show that E detects the desired divided beta-family elements (that's the hard bit), and then the multiplicative structure of E tells us that products of these classes cannot vanish in the sphere. This last step is sometimes obvious. For example, if x and y are detected by E, and xy is nonzero in E, then xy also must be nonzero in the sphere. Sometime this is a little less obvious though. For example, if we know that if a class z is not in the image of the unit map, but can be written as a Toda bracket <a,b,c> where a,b,c all lie in the image of the unit map, then either ab or bc cannot vanish in the sphere. The E used in this article is the height 2 image-of-J spectrum j² from (3), which is perfectly balanced to both have simple homotopy groups to compute with but also detects many products amongst the divided beta-family (for the experts: for synthetic filtration reasons, j² detects as many beta-family products as Behrens' spectrum Q(2)). To calculate the image of the unit map in the homotopy groups of j² we use a modified Adams--Novikov spectral sequence constructed using synthetic spectra as done in (8).

We discuss the interactions of synthetic spectra with spectral algebraic geometry. In particular, we take spectral stacks X coming from chromatic homotopy theory called even-periodic refinements and show that the global sections of the associated synthetic spectral stack implements the descent spectral sequence (DSS) for the original global sections. Doing this all in synthetic spectra allows us to discuss DSSs in the same way we would discuss Adams--Novikov spectral sequences (ANSSs)---in future work, we will use this idea to produce a computation of the homotopy groups of Tmf independent on the current literature (which is a little circular---we will get back to this in a follow-up paper). In this paper though, we also come up with a cohomological criterion on the underlying stack of X such that the DSS agrees with the ANSS for its global sections in a highly structured manner (with respect to naturality and multiplicative structures). As an application, we define SMF and Smf, synthetic versions of the usual periodic and projective versions of topological modular forms, which give (counter-) examples to this criterion above. (For the experts: we also give some counter-examples to a potential synthetic affineness statement along the lines of Mathew--Meier.)

A simple proof that the unit map from the sphere to the connective image-of-J spectrum j is surjective on homotopy groups is given. This is a known fact, although the usual proof technique seen in the literature is quite complicated, as one needs to know a lot about the homology of j and compute its Adams spectral sequence. Our simplification is to use synthetic spectra to construct modified Adams and Adams--Novikov spectral sequence for j which are very easy to calculate---we would say of comparable difficulty to calculating the homotopy groups of j; hurray! We then use these modified spectral sequences to show that the unit map is surjective using v_1 self-maps. We also give two refinements of this classical statement: at odd primes, the unit map in BP-synthetic spectra induces a split surjection on bigraded homotopy groups, and at the prime 2, the unit map from the sphere to j induces a split surjection of filtered abelian groups, where the filtration on the homotopy groups of j comes from this modified Adams--Novikov spectral sequence.

The equivariant cohomology theories of Lurie and Gepner--Meier, called tempered cohomology and equivariant elliptic cohomology, are shown to be equivalent, in some sense. In more detail, we first extend Lurie's definition of tempered cohomology from E∞-rings to spectral Deligne--Mumford stacks, and restrict Gepner--Meier's construction of equivariant elliptic cohomology from compact Lie groups to finite groups. These two functors are then shown to be naturally equivalent, the comparison map coming from the inclusion of the P-divisible group of an abelian variety into itself. We also emphasise that this comparison, as well as the functors involved in the comparison, are natural with respect to the (preoriented) abelian variety and the stack these theories are based upon. As an application, we import some theorems in tempered cohomology to their equivariant elliptic counterparts.

The classical Goerss--Hopkins obstruction theory based on p-complete complex K-theory is refined to an obstruction theory for real E∞-rings. As an application of this obstruction theory, we show that the topological q-expansion map is unique up to 3-homotopy, and this further refines to uniqueness up to (2p-3)-homotopy after taking p-completions and some homotopy fixed points. As an application of this uniqueness result, we show that the Adams operations on tmf (from (3) below) compose in a multiplicative fashion up to 2-homotopy, and up to (2p-4)-homotopy after completion and fixed points. Further applications of these uniqueness statements will be used in future work to construct connective models for Behrens' Q(N)-spectra.

Inspired by operators and endomorphisms on the classical groups of modular forms over the complex numbers, we build machinery to construct stable operations on the cohomology theory TMF of periodic topological modular forms which lift these classical morphisms. In particular, we construct Adams operations, Hecke operators, and Atkin--Lehner involutions on TMF. Our setup allows for simple proofs of all the basic properties of these operations, for example, showing that these families mutually commute up to higher homotopies. We also provide some applications in number theory of these operations on the torsion classes in TMF such as rederiving some congruences of Ramanujan and providing some extra evidence for Maeda's conjecture.

Folklore tells us that the sheaf defining Tmf is unique up to homotopy, and in this article we provide a proof of this result. This a posteriori reconciles all previous (and future) constructions of Tmf. (Note: the K(1)-local argument in the arXiv version has an inaccuracy which is fixed in the published version.)

In this article, the cohomology theory Tmf of dualisable topological modular forms is endowed with stable Adams operations. This is done using Hill--Lawson's description of Tmf using TMF and Tate K-theory, the fact that these latter two theories have natural Adams operations by (2) below, and gluing these operations together on Tmf using Goerss--Hopkins obstruction theory---this construction is rather delicate. We also calculate the effect these Adams operations have on homotopy groups. A conjecture on the effect of dual endomorphisms of Anderson self-dual spectra is also formulated, motivated by our calculations. As an application, we construct connective height 2 analogues of Adams summands and image of J spectra, denoted by u and j^2, respectively, and show that the p-completion of tmf splits into sums of shifts of u if and only if p-1 divides 12, and that j^2 detects all of the height 1 image of J as well as the Hurewicz image of tmf. 

Lurie's theorem states that upon a suitable site over the moduli stacks of p-divisible groups of height n, there is a sheaf of E∞-rings which when evaluated on affines looks like the associated Landweber exact theory. In this article, we provide the first publicly available proof of Lurie's theorem, relying heavily upon Lurie's work on spectral algebraic geometry and elliptic cohomology. The titular theorem is then applied to constructions of topological K-theory, Lubin--Tate theory, and topological modular and automorphic forms, as well as a general study of stable Adams operations on such theories. (Have a look at previous arXiv versions or my thesis above for some more details, although the writing can be quite rough.)

In this article, we study realisation problems in global homotopy theory. In particular, given an ultra-commutative ring spectra R, we show that projective modules (and in general projective dimension 2 modules with a flatness assumption) over its non-equivariant graded homotopy groups can be realised by global R-modules and that various algebras over these graded rings can also be realised by homotopy commutative, E∞-, and even G∞-objects in global R-modules. The key concept in maintaining control over these realisations is that of global flatness which allows for computation. As applications, we see study localisations of E∞-global ring spectra and lifts of stable homotopy types from chromatic homotopy theory to globally flat global homotopy types over MU.

Let me clarify some of the constructions of various operations in, on, and around topological modular forms, as the lack of a single source can potentially lead to confusion.

Adams operations on TMF: Constructed and act exactly as one would like up to all higher homotopies; see (5) above. Calculations follow from (3) above. One also has these operations on TMF with all kinds of level structures and equivariant TMF too by unwinding what's happening in (5).

Adams operations on Tmf and tmf: Constructed but their structural properties are difficult to pin down. They do however commute with those on TMF, KO[[q]], and KO by construction. In (3) they are constructed, and in (6) it is shown that away from the prime 2, they multiply and commute up to 2-homotopy. If one wants Adams operations on Tmf with level structure, one can copy the arguments found in (3) or use the uniquenss results of (6); I have something written that summarises this, which will hopefully soon see the light of day.

Hecke operators on TMF: Constructed and act exactly as one would like up to all higher homotopies, up to the caveat that the general Hecke decomposition formula only holds up to noncanonical homotopy and some other integers must be inverted (if the two Hecke operators are coprime, then they commute up to all higher homotopies); see (5) above. For computations one can use (3) above, but there are limitations as Hecke operators are not multiplicative, so be careful!. Some specific computations are performed in (5).

Hecke operators on Tmf and tmf: Their construction seems posisble, but their structural properties are difficult to pin down. They will, however, commute with those on TMF, KO[[q]], and KO by construction. On could try to work this out from (3) and (6).

Hecke operators on height 2 image of J spectra: Constructed, but again we have to separate the periodic from the dualisable/connective case. In the former case, everything works up to higher homotopy (at least, as well as Hecke operators behave on TMF); see (2). In the latter case, we only obtain such Hecke operators working away from the prime 2; this is not available, but it should be a simplification of the  connective Q(N) work below.

Connective/dualisable models of Behrens' Q(N) spectra: Constructed but only away from the prime 2, meaning each q(N) has been constructed and admits an E∞-map to tmf[1/2N]; this is typed up, but is waiting for further calculations and applications.

Stable operations on Behrens' Q(N) spectra: For the periodic Q(N) over TMF[1/N], all operations exist with the level of coherence dictated by those on TMF; this is implicit in (2). For the connective/dualisable q(N) over tmf[1/2N] all operation exist but there is a priori no coherence or structural properties between them; this is typed up, but is waiting for further calculations and applications.

Stable operations and topological modular forms (February 2022, Instituut Mittag-Leffler) (video link)

Adams operations on elliptic cohomology (June 2021, Warwick algebraic topology seminar 20/21) 

Adams operations on elliptic cohomology (November 2020, Bochum Oberseminar topologie)

TMF, or: how I learned to stop worried and love spectral algebraic geometry (July, 2020, GROOT seminar) (video link)

Barsotti--Tate theory and reality (from mid-2021)

Illusie-style deformation theory for spectrally ringed ∞-topoi, (from mid-2019)

A quick note on a graded Lazard's theorem, (from early-2019)

Globally coherent RO(G)-gradings, (from around 2018-19) [super rough, needs fixing]

Multiplicative localisations in global homotopy theory, (from around 2018-19)

I do like writing notes for various seminars I have given or courses I have taken. Find my email on my homepage and send me any typos, grammar mistakes, mathematical errors, or general feedback. I also removed a bunch of older notes from my masters/PhD studies, but let me know if you want something you once saw but has now disappeared.

Algebraic geometry I (Winter 2016-17, Bonn Universität)

Algebraic geometry II (Summer 2017, Bonn Universität)

Handwritten notes from the Oberwolfach Arbeitsgemeinschaft on Lurie's elliptic cohomology (2019)

I helped edit the notes from the European Talbot Workshop on algebraic K-theory (2019)

I helped edit notes for Danny Shi's talk at the Viva Talbot (2021)

I contributed an article to the Homotopy Theory Oberwolfach report (2023)

I typed my notes for a minicourse on stable homotopy theory given at the Galois theoretic aspects of stable homotopy theory workshop (2024)

I have typed notes from my lecture with Liz Tatum on ``Stable and chromatic homotopy theory'' (2024)