## Research

My research lies in (stable) homotopy theory, as well as the spectral algebraic geometry and chromatic homotopy theory. I am particularly interested in ``height two'' phenomena, so elliptic cohomology and topological modular forms. Recently, aside from my research into operations in elliptic cohomology and further consequences therefrom, and I have been thinking about a few different avenues of research:

Equivariant multiplicative structures on global topological modular forms (with William Balderrama and Sil Linksens).

Global (in an equivariant homotopy-theoretic sense) Tate K-theory (with Sil Linskens).

Synthetic methods for detecting elements in the stable homotopy groups of spheres (with Christian Carrick).

An approach to descent spectral sequences using synthetic spectra (with Christian Carrick and Sven van Nigtevecht).

Here are some links to my zbmath.org reviews, my arXiv.org papers, my mathscinet page, and my genealogy.

My PhD thesis is also available online here---although the statements contained within are improved upon (and sometimes corrected) in other articles, so use at your own risk.

Below my articles is a summary of some of the objects I have constructed, and at what stage they are at now.

### Articles (in reverse chronological order)

9. Descent spectral sequences through synthetic spectra [2024] (submitted | arXiv) Christian Carrick, Sven van Nigtevecht, and I 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). 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.)

8. A synthetic approach to detecting v_1-periodic families [2024] (submitted | arXiv) Christian Carrick and myself provide a simple proof that the unit map from the sphere to the connective image-of-J spectrum j is surjective on homotopy groups. 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.

7. Comparing tempered and equivariant elliptic cohomology [2023] (submitted | arXiv) 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.

6. Uniqueness of real ring spectra up to higher homotopy [2023] (Annals of K-theory | arXiv) 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 Constructing and calculating Adams operations on dualisable topological modular forms 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.

5. Hecke operators on topological modular forms [2022] (Advanced in Mathematics| arXiv) 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.

4. Elliptic cohomology is unique up to homotopy [2021] (Journal of the Australian Mathematical Society | arXiv ) 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.)

3. Constructing and calculating Adams operations on dualisable topological modular forms [2021] (submitted | arXiv) - 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 ``On Lurie's theorem and applications'' 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.

2. On Lurie's theorem and applications [2020] (in revision | arXiv) 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.)

1. Realising $\pi_\ast^e R$-algebras by global ring spectra [2019] (Algebraic & Geometric Topology | arXiv) 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.

### Status of constructions

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.

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.

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 additive. Some specific computations are performed in (5).

Hecke operators 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. This is typed up, but not yet uploaded to the arXiv.

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.

### Research seminar notes

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)

### Unfinished notes

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)

### Notes and reports from lectures, seminars, and workshops

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)