About Me

I have been a PhD student at Harvard since September 2018. Previously, I did my undergraduate and Part III at Cambridge (2014–2018).

Contact Me

You can email me at dexter@math.harvard.edu. My office is at 428a.

Adams Spectral Sequence for $tmf$

I have documented the calculation of the Adams spectral sequence of $tmf$ at the prime $2$ here (warning: this downloads a 12MB file behind the scenes).

This calculation was made with computer assistance. There is an online version of the software, and the source can be found on GitHub. See the README on GitHub for technical details.

The software computes the $E_2$ page and products, and propagates differentials via the Leibniz rule. The program is capable of resolving an arbitrary (finite dimensional or finitely generated) Steenrod module and assisting the computation of the associated Adams spectral sequence. It is also designed with the intention to be able to aid other spectral sequence calculations (as long as the modules are over $\mathbb{F}_p$), but no such applications have been coded at the moment. It was initially developed by Hood Chatham and I later joined the development.

The save file for the calculation can be found here, which can be imported into the resolver to reproduce the calculation. (However, doing it on the online version above is unwise)

Steenrod operations

Using an algorithm of Christian Nassau, I calculated some Steenrod operations in Ext of the Steenrod algebra and $A(2)$. The results are documented here, and they go beyond what is in Bruner's tables.

I believe the current implementation has huge room for improvement. Once such improvements have been made (or I have decided such improvements cannot be made), I will start systematically documenting all Steenrod operations within reach. If anyone is interested in any particular Steenrod operation not documented above, I can (attempt to) calculate it on demand.

The two main bottlenecks of the procedure is as follows:

• The first step of the algorithm involves finding a Yoneda representative of the relevant class. The size of the Yoneda representative directly and significantly affects the runtime of the rest of the algorithm, so it is crucial to find one that is as small as possible.

  Any minimal Yoneda representative would be a quotient of the minimal resolution of $\mathbb{F}_2$. Currently, the algorithm starts with the minimal resolution and iteratively quotients out elements that are not needed based on some heuristics. Improving these heuristics can potentially lead to much smaller representatives. At the moment the classes at $(t - s, s) = (69, 4)$ and $(41, 10)$ are examples where the algorithm performs extermely poorly.

• Currently, applying Steenrod operations on tensor product of modules is extremely slow, as the coproduct in the Steenrod algebra has a lot of terms. One possible way around this is to pre-compute the action of the $Sq^{2^n}$ using the coproduct, and then all other actions can be computed by decomposing the elements as a product of indecomposables. It is not clear to me whether this is actually beneficial, but it might be worth trying.

Expository Writings

Some miscellaneous expository writings. The word "expository" refers to the lack of originality, as opposed to any claim of comprehensibility or correctedness.

Clicking the title below will lead to a web version of the note, which is an experimental feature — let me know if anything seems broken. Click "pdf" for a downloadable pdf version.

• Chern–Weil forms and abstract homotopy theory (pdf)
• Introduction to Motivic Homotopy Theory (pdf)
• The Barratt–Priddy–Quillen–Segal theorem (pdf)
• Goodwillie filtration and factorization homology (pdf)
• Bott Periodicity (pdf)
• The Heat Kernel (pdf) ← this is my minor thesis
• Geometric Topology (pdf) ← this is my Part III essay
• Construction of v₁ and v₂ self-maps (pdf)
• Ring Structures on S/p (pdf)
• Global Analysis – Hodge Decomposition Theorem (pdf)
• Clifford Algebra and Bott Periodicity (pdf)
• The Étale Fundamental Group (pdf)
• The Lemniscate Sine (pdf)
• Period 3 orbits (pdf)

Cambridge Course Notes

When I was in Cambridge, I typed up my lecture notes for the courses I attended. They can be found here.

Computer Drawings

As a means of procrastination learning how to use HTML5 canvas, I produced the following drawings/simulations:

• Brownian motion
• Conformal maps

Miscellaneous

• Some Adem relations
• Some LaTeX tricks
• Some formulas about BP

Other Involvements

I co-organized the Bott periodicity seminar during Spring 2019.

I have been on the committee of the Cambridge student societies The Archimedeans, CUPS, the SRCF and the RCSA. I am also a sysadmin of the SRCF.

Privacy Statement

I have a Privacy Statement as required by law (maybe).