Hi, I'm Howard! I'm a senior at MIT, majoring in Pure Mathematics Research interests: stable homotopy theory - primarily chromatic and equivariant homotopy theory and algebraic K-theory Contact information: Howard Beck, (firstinitial)(lastname)[at]mit[dot]edu (a futile attempt at spam protection) CV - updated Mar 14, 25 (or the short version - updated Feb 5, 25), Resume - updated Nov 11, 24

Preprints

In progress: Chromatic blueshift conjecture: the simple case and an algebraic analogue, joint with Kyle Roke.
Supervised by Tristan Yang and Professor Jeremy Hahn

Using power operations, we show that the Chromatic Blueshift Conjecture of Burkland, Schlank, and Yuan holds for certain E_$\infty$-rings, at prime cyclic groups $C$_$p$ with the trivial family and at all chromatic heights.
> draft available upon request

An Elementary Introduction to Stable Homotopy Theory (show BibTeX - for my own reference, you should cite something more reputable)

@unpublished{beck2024stable,
        title = {An Elementary Introduction to Stable Homotopy Theory},
        author = {Beck, Howard},
        year = {2024},
        month = nov,
        url = {https://www.mit.edu/~hbeck/papers/elem_stable.pdf}
}

This is an expository paper introducing stable homotopy theory. The ideal readership has taken an introduction to topology course, and is familiar with the language of groups, and ideally (no pun intended) also rings, and modules. It was submitted to a certain unnamed-for-now journal - we'll see what happens!

Some of my writings that may be useful or of interest:

Presentation notes from the Kan Seminar at MIT, spring 2025:

La cohomologie modulo 2 de certains espaces homogènes (Feb 7th, 2025 - first talk of the Kan Seminar)

Notes from my talk at Zygotop on chromatic redshift and blueshift will appear here at some point...

Live notes from Babytop Spring 2025 can be found here

The Slice, Reduction, and Gap Theorems of Hill-Hopkins-Ravenel

Given on December 4th, 2024 at the graduate-student run joint MIT/Harvard Babytop seminar, organized this semester by Natalie Stewart. My talk covered sections 6-8 of the Hill-Hopkins-Ravenel paper about the Kervaire invariant one problem.
Note: These notes are undergoing historical revisionism. They have some small errors and places deserving more clarification. Also, two sections are missing since they were handwritten... that'll be fixed.

Notes on the Chromatic Blueshift Conjecture

This is intended to get someone with only basic knowledge of (chromatic, stable) homotopy theory up to speed to understand Conjecture 9.9 of Burkland, Schlank, and Yuan (see: The chromatic nullstellensatz). In its current state, this document is mostly complete but will be revised. The first section covers the basics of Bousfield localization, introduces Morava $K$($n$)-theories and the telescope spectra $T$($n$), and of course briefly mentions the (recently disproved) Telescope Conjecture of Ravenel. The second section covers the basics of equivariant spaces, and then develops stable equivariant homotopy theory. By the end, I introduce the geometric fixed point functors on genuine $G$-spectra which puts the reader (hopefully) in a position to understand the statement of the Chromatic Blueshift Conjecture. If you have comments on this document, feel free to email me!
Note: Claim 1.4.5. is correct but the proof I presented is horribly wrong - I will fix this soon.

Presentation notes on classifying spaces, simplicial sets, and Čech categories
Presentation notes on characteristic classes

These are notes from presentations I gave for the 18.906 (Algebraic Topology II) reading group in Fall 2024 at MIT. The class is unusually not being offered this year, so the reading group is "replacing" the class. It's based on Professor Haynes Miller's lecture notes, available on his website. I would like to thank Professor Miller for his contributions to these notes, by pointing out mistakes and giving insightful comments during the presentation that were included in the notes afterwards. My other presentations used written notes that are... somewhere.

Deep Learning Imitation of Particle Filter for Autonomous Vertical Optical Lunar Lander (show BibTeX)

@unpublished{beck2021deep,
        author = {Beck, Howard and Furfaro, Roberto and Gaudet, Brian},
        title = {Deep Learning Imitation of Particle Filter for Autonomous Vertical Optical Lunar Lander},
        year = 2021,
        month = aug,
        url = {https://www.mit.edu/~hbeck/papers/lunar_learning.pdf}
}

This project was done in the second half of my senior year of high school for my AP Research Class. The idea was that modern spacecraft may be able to perform descent using cameras, and there's a desire to outfit their computer systems with hardware capable of calling deep learning models. A statistically robust way to combine image data into the pose estimation algorithm is very computationally expensive, and this paper offers a way to replace this with a recurrent neural network. Frankly, the method I employed is still terribly inefficient and more classical methods that may be less Bayes-optimal are better-suited. However, I'm still begrudgingly making this document publicly accessible for the sake of open research. For a brief while back then and even at the start of my undergrad at MIT, I was considering being some flavor of engineer - here is the evidence of such a time. The code can theoretically be found here, but beware I have absolutely no idea where any of the neural network code is. I fear it is a mystery lost to time.

Other shenanigans

A homotopy theorist's LaTeX refinements

For some reason this tool made for scientists and mathematicians of all fields and walks of life wasn't optimized for homotopy theorists - how outrageous! Here's some commands I use to redefine some badly named macros. Before I get angry emails from higher algebraists: sometimes I too use the tensor product notation for smash products of spectra! This is just a template (and is buggy).

Various resources to block unwanted generative AI features

Companies like to prioritze search results from their generative AI models and often can't assess which types of questions are not suited for such results. Here I'm documenting various resources to disable these.

If I were a Springer-Verlag Graduate Text in Mathematics, I would be Frank Warner's Foundations of Differentiable Manifolds and Lie Groups. I give a clear, detailed, and careful development of the basic facts on manifold theory and Lie Groups. I include differentiable manifolds, tensors and differentiable forms. Lie groups and homogenous spaces, integration on manifolds, and in addition provide a proof of the de Rham theorem via sheaf cohomology theory, and develop the local theory of elliptic operators culminating in a proof of the Hodge theorem. Those interested in any of the diverse areas of mathematics requiring the notion of a differentiable manifold will find me extremely useful. Which Springer GTM would you be? The Springer GTM Test

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