Baby RAT Seminar

RAT stands for “Researchers Are Talking”. 🐀

Participants may give an hour talk about their research (invited seminar style) or present a paper of their choice (as an alternative to running a seminar for a whole semester). This seminar aims to give grad students a space to practice giving research talks and learn about what their peers are doing. The target audience will primarily consist of topologists (algebraic topologists/ homotopy theorists).

There might be pizza! 🍕

We will meet in room 528 on Mondays from 11:30 am to 12:30 pm.

Please find below the schedule for Fall 2026:

Monday September 14th: Tan Su
Classification of bundle with connection
Differential cohomology theories refine ordinary cohomology by keeping track not only of topological information, but also of geometric data such as connections and curvature. A classical example is the work of Cheeger–Simons and Sullivan–Simons, which gives numerical invariants describing complex vector bundles with unitary connection and leads to a concrete model for differential K-theory.

In this talk I will discuss a real analogue of this story for differential KO-theory. Starting from a real vector bundle with orthogonal connection, we construct a collection of (R mod Z)-valued eta invariants, together with (Z2)-valued index invariants, which completely classify the bundle and connection up to “chern-simons” equivalence thus determining its differential KO-class. We call this collection the differential KO-character.

Monday September 21st: Alex Scheffelin
The Second Vanishing Theorem in Ramified Mixed Characteristic
Abstract: To a triple of a ring $R$, an $R$-module $M$, and an ideal $I$ we can associate the local cohomology modules $H^n_I(M)$. One classical problem dating back to a question of Grothendieck is to identify when these vanish for all $n > i$ and all modules $M$, the smallest such $i$ we denote the cohomological dimension of $R$ with respect to $I$. Grothendieck showed that this is bounded by $d = \dim R$, while a very simple condition on $\widehat{R}$ and $I\widehat{R}$ controls the vanishing at $d$. A more subtle topological condition controls the vanishing at $d-1$ for regular local rings as posed by Hartshorne in the late 60s, and was proven in equicharacteristic in the early 70s. After 50 years it was shown to be true in unramified mixed characteristic, and the main result of this talk is the ramified mixed characteristic case.

Monday September 28th: Serena An
Faithful specializations of the Burau representation
Abstract: This talk is about an expository paper which appeared in the Columbia Journal of Undergraduate Mathematics Vol. 3 No. 2 (2026).

The Burau representation, first introduced by Werner Burau in 1935, is a well-studied matrix representation of the braid group $B_n$ with connections to the Alexander polynomial. The Burau representation of $B_n$ is faithful for $n\le 4$ and unfaithful for $n\ge 5$, with the longstanding case of $n = 4$ resolved by Bharathram, Birman, and Brendle in 2026.

For $n = 3$ case, we consider whether the Burau representation $\psi_3 : B_3\to GL_{2}(\mathbb{Z}[t, t^{-1}])$ still faithful when $t$ is replaced with a real number. Following Scherich, we prove that it is faithful for all $t < 0$ with $t\neq -1$, while taking a surprising detour into hyperbolic geometry!

Monday October 5th: No Talk

Monday October 12th: Azélie Picot
From Geometry to Homotopy Theory
A celebrated theorem of Thom states that classifying manifolds up to cobordism instead of diffeomorphism is fairly understandable. In this talk, I will describe the case for homotopy-theoretic analogues of manifolds, aka Poincaré duality spaces.

Monday October 19th: Keita Allen (MIT)
Evenness and equivariant motivic homotopy theory
Work of Pstrągowski and Gheorghe-Isaksen-Krause-Ricka provide a description of (cellular, p-complete) motivic homotopy theory over (\mathbb{C}) which does not reference algebraic geometry, in terms of what have come to be known as synthetic spectra. This category can be thought as a categorification of the Adams-Novikov spectral sequence, which work of Hahn-Raksit-Wilson and Pstrągowski show is intimately connected to the notion of evenness of (ring) spectra. I will discuss joint work with Lucas Piessevaux which shows that many aspects of this story can be generalized in the presence of the action of a finite abelian group, hopefully highlighting phenomena which are not as visible in the non-equivariant setting.

Monday October 26th: Sangmin Ko
Efimov K-theory and the chromatic tower
Algebraic K-theory plays a central role in algebra, geometry and topology. Classically, it is only useful for small categories. For large categories it breaks down: by the Eilenberg swindle, any category with infinite direct sums has trivial K-theory. Recently, Efimov introduced continuous K-theory (also known as Efimov K-theory), which gives meaningful answers for a large class of big categories, namely dualizable categories.

In this talk, I will discuss the ideas behind his construction through examples, such as sheaves on $\mathbb{R}^n$ and the $p$-adic integers $\mathbb{Z}_p$. I’ll also explain how Efimov K-theory commutes with inverse limits of categories in nice situations. At the end, I’ll briefly describe my current project, which applies these ideas to the chromatic tower in stable homotopy theory.

Monday November 2nd: No talk (University Holiday)

Monday November 9th: Carlos Andrés Alvarado Álvarez
p-adic Adams Operations on Algebraic K-theory We talk about some progress on constructing p-adic Adams operations on Algebraic K-theory.