# Eva Belmont

 UC San Diego Department of Mathematics Office: AP&M 6432 Email:   ebelmont at ucsd dot edu

I am an S. E. Warschawski Assistant Professor at UC San Diego. My main research interests are in stable homotopy theory. Previously, I was a Boas Assistant Professor at Northwestern University. I completed my doctoral work at MIT in 2018 under the supervision of Haynes Miller.

## Research

My current research focuses on computational aspects of classical, motivic, and equivariant stable homotopy theory.
• My Ph.D. thesis, Localization at $b_{10}$ in the stable category of comodules over the Steenrod reduced powers, represents progress towards computing the $b_{10}$-periodic part of the Adams $E_2$ page for the sphere at $p = 3$. This is published as two papers:
• Localizing the $E_2$ page of the Adams spectral sequence (published in Algebraic and Geometric Topology). This paper is about a spectral sequence converging to an infinite piece of the Adams $E_2$ term for the sphere at $p = 3$. We compute up to the $E_9$ page of an Adams spectral sequence in the category $\mathrm{Stable}(P)$ converging to $b_{10}^{-1}\mathrm{Ext}_P(\mathbb{F}_3, \mathbb{F}_3)$, and conjecture that the spectral sequence collapses at $E_9$.
• A Cartan-Eilenberg spectral sequence for a non-normal extension (published in Journal of Pure and Applied Algebra)
This paper compares three ways to construct a Cartan-Eilenberg type spectral sequence associated to a map $\Phi \to \Gamma$ where $\Gamma$ is a Hopf algebra and $\Phi$ is a $\Gamma$-comodule algebra. This is the main tool required for the above paper.
• I have worked with Dan Isaksen to compute the $\mathbb{R}$-motivic Adams spectral sequence for the sphere in a range of dimensions.
• (with Dan Isaksen) $\mathbb{R}$-motivic stable stems, a computation of the (2-complete) $\mathbb{R}$-motivic homotopy groups of spheres using the Adams spectral sequence, up to co-weight 11. We define a variant of the Mahowald invariant, which can be computed using the $\mathbb{R}$-motivic stable stems, which agrees with the Mahowald invariant in many interesting cases.
• (with Dan Isaksen) Charts for the above paper can be found here.
• (with Bert Guillou and Dan Isaksen) $C_2$-equivariant and $\mathbb{R}$-motivic stable stems, II (accepted in Proceedings of the AMS) expands the known range of dimensions in which the map from the (2-complete) $\mathbb{R}$-motivic homotopy groups of the sphere to the $C_2$-equivariant homotopy groups is an isomorphism.
• I am part of a Women in Topology III collaborative project, with Natalia Castellana, Jelena Grbic, Kathryn Lesh, and Michelle Strumila. We used the theory of fusion systems to give a normalizer decomposition for compact Lie groups, and computed this decomposition for some examples.
• Our first paper, A new approach to mod 2 decompositions of $BSU(2)$ and $BSO(3)$ uses the theory of fusion systems associated to $p$-local compact groups to recover the 2-complete homotopy colimit decomposition of $BSU(2)$ and $BSO(3)$ originally due to Dwyer, Miller, and Wilkerson.
• Our second paper, A normalizer decomposition for $p$-local compact groups (in preparation), generalizes the normalizer decomposition due to Dwyer (in the case of finite groups) and Libman (in the setting of saturated fusion systems over finite $p$-groups) to $p$-local compact groups, a generalization of compact Lie groups. In particular, we obtain normalizer decompositions for compact Lie groups. We compute this explicitly for $p$-completed $U(p)$ and $SU(p)$, and for the Aguad\'e-Zabrodsky $p$-compact groups.

## Research talks

• $\mathbb{R}$-motivic homotopy theory and the Mahowald invariant, my 2020 Midwest Topology Seminar talk about my work with Dan Isaksen about the $\mathbb{R}$-motivic Adams spectral sequence and its relationship to the Mahowald invariant.
• Slides for my talk in ECHT in April 2019 about the same project.
• Chromatic localization in an algebraic category, a talk I gave at the conference "Chromatic homotopy: Journey to the frontier" held in Boulder, Colorado in May 2018. I give an introduction to Palmieri's work on the homotopy theory of modules over the Steenrod algebra, with emphasis on the analogues of the nilpotence and periodicity theorems, and discuss how this relates to my thesis work about the $E_2$ page of the Adams spectral sequence at $p=3$.

## Teaching

I am currently teaching Math 142A, Introduction to Analysis I in Fall 2021 at UCSD. Click here for the current course website.

At Northwestern, I have been an instructor for the following courses.
• Winter 2021: Math 220-2 (single variable integral calculus) and Math 334 (linear algebra for math majors)
• Fall 2020: Math 228-1, Multivariable differential calculus for engineers
• Spring 2020: Math 240, Linear algebra
• Winter 2020: Math 300, Foundations of higher mathematics (intro to proofs class)
• Fall 2019: Math 230-1, Differential calculus of multivariable functions.
• Spring 2019: Math 224, Integral calculus of one-variable functions.
• Winter 2019: Math 230, Differential calculus of multivariable functions.
• Fall 2018: Math 224, Integral calculus of one-variable functions.

## Expository Writing

• Complex Cobordism and Formal Group Laws, Part III essay about Quillen's theorem that $MU$ has the universal formal group law, including the construction of the Adams spectral sequence and basics about formal group laws. (If you want to look at this, email me to ask for a copy.)
• Stable Homotopy and the J-Homomorphism, undergraduate senior thesis about Adams' work on the splitting of the stable homotopy groups of spheres.

## Class Notes

As a student, I live-TeXed many classes and seminars. Some of the notes can be found here.