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: TBD
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Monday October 12th: TBD
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Monday October 19th: Keita Allen (MIT)
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