Math

A non-exhaustive list of things I am thinking about (I don’t claim that any of these ideas are either good or original):

• Crossed simplicial groups: there are some categorical perspectives on crossed simplicial groups that I haven’t seen explored much; for instance, a crossed simplicial group is a distributive law between certain monads. Do these perspectives lend any insight to the classical classification? Can we use them to help with the extension problem (for instance can we define cohomology of crossed simplicial groups), or to build a theory of higher crossed simplicial groups?
• Parameterized cohomology: it is a folklore observation that $\text{RO}(G)$-graded cohomology might be better thought of as $\text{Pic}(\text{Sp}_G)$-graded, so that we don’t have to pick canonical representatives of each representation. (See, for instance, the discussion at the start of section 6 of these notes of Adams.) This might motivate us to try to consider $\text{Pic}(\text{Sp}_T)$-graded cohomology, where $\text{Sp}_T$ is the category of $T$-spectra coming from parameterized higher category theory. Is there a Brown representability theorem for such things, for instance? What do they look like in the global setting?
• Stable discrete Morse theory: discrete Morse theory is a powerful toolbox for proving homology equivalences between CW complexes. Much of the theory seems to me to depend only on the combinatorics of cells; and for instance, it has been generalized to the equivariant setting by Freij (although the extension seems far from formal to me). Can we build an analogous theory for cellular spectra? More generally, can we identify some categorical description of the settings in which such theories exist?

I would love to talk about these things or anything else—send me an email at [firstinitial][lastname][at]sas[dot]upenn[dot]edu!

Papers

I’m young and there is not much here yet… but hopefully there will be more soon!

• Enumerating submonoids of finite commutative monoids, with Caoilainn Kirkpatrick, Amelie el Mahmoud, Kyle Ormsby, Angélica M. Osorno, Dale Schandelmeier-Lynch, Lixing Yi, Avery Young, and Saron Zhu — arXiv:2508.20786 (submitted) Show abstract of Enumerating submonoids of finite commutative monoids

  Given a finite commutative monoid $M$, we show that submonoids of $M\times [n]$ — where $[n] = \{0,1,\ldots,n\}$ is equipped with the max operation $\vee$ — may be enumerated via the transfer matrix method. When $M$ is also idempotent, we show that there are finitely many integers $\lambda$ and rational numbers $b_\lambda$ (only depending on $M$) such that the number of submonoids of $M\times [n]$ is $\sum_\lambda b_\lambda\lambda^n$. This answers a question of Knuth regarding ternary (and higher order) max-closed relations, and has applications to the enumeration of saturated transfer systems in equivariant infinite loop space theory.

Talks

• Parameterized higher category theory—Penn Oral Qualifying Exam, April 2026 show abstract of Parameterized higher category theory

  We motivate and introduce the theory of parameterized higher categories, with an eye towards Nardin’s universal property of the parameterized category $\underline{Sp}_G$ of $G$-spectra and the corresponding comparison of spectral Mackey functors with orthgonal $G$-spectra.

• Categorical Fraïssé theory—Penn Model Theory Seminar, April 2026 show abstract of Categorical Fraïssé theory

  We introduce the 2007 paper of Kubiś on Fraïssé sequences. We compare this approach to the traditional model-theoretic approach, and discuss subtelties involved in transfinite Fraïssé sequences.

• A survey of directed topology—Penn Graduate Geometry-Topology Seminar, September 2025 show abstract of A survey of directed topology

  Sometimes spaces seem to come with a natural sense of direction—consider a manifold equipped with a vector field, or a spacetime manifold, or a configuration space in concurrency theory, or just the ordinary interval. Directed topological spaces are models for this kind of geometry. The primary goal of this talk is to introduce the two most popular models—d-spaces and streams—and discuss the sense in which they are equivalent. We will then introduce the core ideas of directed homotopy theory.

• The sequence 1, 11, 21, 1211, ... (its name is a spoiler!)—Penn Pizza Seminar, September 2025 show abstract of The sequence 1, 11, 21, 1211, ... (its name is a spoiler!)

  This talk is about a very playful sequence, originally studied by John Conway, whose first terms are displayed in the title. We’ll define the sequence, prove some of its basic properties, and then discuss Conway’s “cosmological theorem,” which determines the limiting behavior of the sequence. A surprising role is played by a particular endomorphism of the free monoid on 92 generators.

• What if equality but geometry is just vibes?—Mathcamp Colloquium, July 2025 show abstract of What if equality but geometry is just vibes?

  In math we often encounter operations—like the multiplication of real numbers—with nice properties, like the associative property (xy)z = x(yz). We then define algebraic gadgets via listing certain desir- able properties, and algebra is in some sense the study of these abstract algebraic gadgets. But what if these equalities held not literally ”on the nose,” but only up to some weaker notion of equivalence? And then what if that notion of equivalence, itself, has some properties of its own, each of which also only hold up to some second notion of equivalence? And then what if this process repeats forever? This seemingly far-fetched situation in fact occurs in many places in modern mathematics. The study of situations like this—sometimes called homotopy theory or higher algebra—is an extremely active area of research. This talk will give a zero-prerequisite geometric example of such a situation, and discuss some of the abstract tools we use to get a handle on it.

• The K-theory of Lawvere theories—Penn Kan Seminar on Algebraic K-Theory, April 2025 show abstract of The K-theory of Lawvere theories

  We introduce Lawvere theories and then discuss Bohmann-Szymik’s work on their K-theory. From the K-theoretic perspective, Lawvere theories are a convenient generalization of rings in which Quillen’s +=Q theorem still holds, enabling a more computationally tractable K-theory than more general settings.

• Deloopings of categories—Penn Kan Seminar on Algebraic K-Theory, April 2025 show abstract of Deloopings of categories

  We give a high-level intuition for why spaces with highly-coherent homotopy-commutative and associative multiplications should model infinite loop spaces. We discuss two machines for identifying such space spaces: May’s operads and Segal’s gamma spaces. We conclude with a discussion of categories of operators and the May-Thomason uniqueness theorem for infinite loop space machines.

• A connectivity toolbox—AMTRaK IV, April 2025 show abstract of A connectivity toolbox

  This talk will elaborate on the connectivity arguments that were swept under the rug during pre-talk 1. In particular, we will focus on tools and techniques coming from discrete Morse theory. After introducing the basics of discrete Morse theory, we will use it to prove the Solomon-Tits Theorem.

• Classifying spaces of categories—Penn Kan Seminar on Algebraic K-Theory, March 2025 show abstract of Classifying spaces of categories

  We motivate and define the classifying space of a category and discuss Quillen’s Theorems A and B, which give powerful tools for computing homotopical information about a functor.

• Artin gluing in logic—Penn Graduate Logic Seminar, February 2025 show abstract of Artin gluing in logic

  A standard approach to proving meta-theoretic properties of a proof system or type theory is to first define a stronger dependent predicate R—often called a logical relation—by induction on syntax, and prove the desired property by showing that every well-formed term satisfies this stronger property. In categorical semantics, this technique looks like an Artin gluing of the syntactic category along a hom-functor. This talk will introduce this technique, at least in the special case of so-called “sconing,” and utilize it for a categorical proof of the existence property of intuitionistic logic.

• The logical interpretation of topology—Penn Graduate Geometry-Topology Seminar, February 2025 show abstract of The logical interpretation of topology

  Our standard intuition for abstract topology is geometric: the notion of open set axiomatized the properties of open balls in metric spaces. Maybe surprisingly, there is an equally applicable, and in some sense formally dual, logical interpretation of topological spaces. In this interpretation, open sets behave like “propositions subject to finite proof.” This talk—which assumes no knowledge of mathematical logic—will explore this perspective, in particular as it relates to point-free topology and sheaves.

• Categorical semantics, via vibes—Penn Pizza Seminar, January 2025 show abstract of Categorical semantics, via vibes

  You might have heard of “categorical semantics” or “internal logics,” or of the relationship between topoi and intuitionistic logic, perhaps in Jin and Carmine’s pizza seminar talks from last semester. You also might not have! Regardless, this talk aims to explain the big picture of what these terms mean, how they relate, and why non-logicians might care.

• Girard's paradox for set theorists, via Hurkens—Penn Graduate Logic Seminar, November 2024 show abstract of Girard's paradox for set theorists, via Hurkens

  Girard’s paradox is an analogue of Russell’s paradox which demonstrates the inconsistency of a sufficiently rich type theory with a “type of all types.” In this talk, I will present a simplified version of the paradox due to Hurkens (1995), which in particular has a nice analogue in naive set theory. After treating this classical intuition and the type-theoretic argument, if we have time we will implement the paradox in Lean.

• Saturated transfer systems on cylinders—Penn Graduate Geometry-Topology Seminar, October 2024 show abstract of Saturated transfer systems on cylinders

  Transfer systems are relatively simple combinatorial objects with deep connections to the theory of $N_\infty$ operads, an equivariant analogue of $E_\infty$ operads. In this talk, we informally present this homotopy-theoretic story so as to motivate the combinatorial study of saturated transfer systems, which in good cases classify linear isometries operads. We then give a novel method for enumerating saturated transfer systems on cylindrical modular lattices, for instance the subgroup lattices of finite cyclic groups, and tightly characterize the asymptotic behavior in the height of the cylinder. This talk presents joint work with several Reed College students along with Angélica Osorno and Kyle Ormsby.

• Towards linear topoi—Penn Graduate Logic Seminar, September 2024 show abstract of Towards linear topoi

  Topos theory plays a central role in geometry and categorical logic. For instance, topoi yield a natural semantic theory of intuitionistic dependent type theories. One important step in this theory is the observation that topoi are Heyting categories, i.e. that propositions in the internal logic of a topos can be refined along intuitionistic predicates. In this talk, I will attempt to give an accessible outline of this classical story and motivate the study of “linear topoi”, categories which exhibit topos-like behavior with respect to a linear internal logic. I will conclude with a discussion of some progress towards an axiomatic presentation of these categories.