David E. V. Rose
I am the Busemann Assistant Professor in the
USC Mathematics Department.
Contact and professional info:
I am interested in categorification and its interaction with representation theory and low-dimensional topology.
In particular, my recent work explores the relationship between link homology and categorified quantum groups:
The sl_n foam 2-category:
a combinatorial formulation of Khovanov-Rozansky homology via categorical skew-Howe duality: In this paper
(joint with Hoel Queffelec), we construct the 2-category of “enhanced” sl_n foams, the setting for a completely combinatorial
description of (colored, integral) Khovanov-Rozansky homology. We prove that our foam relations suffice to evaluate any
(enhanced) closed foam, answering a question of Mackaay-Stosic-Vaz . Along the way, we give a “topological ” description of
(the direct limit of) categorified q-Schur algebras, and connect Khovanov-Rozansky homology to categorified quantum sl_m
using categorical skew Howe duality.
Khovanov homology is a skew Howe 2-representation of
categorified quantum sl(m): In this paper (joint with Aaron Lauda and Hoel Queffelec), we show that
sl_2 and sl_3 Khovanov homology give 2-representations of categorified quantum sl_m via categorical skew
Howe duality. This resolves a conjecture of Cautis and allows
for the application of his methods to foam-based constructions of link homology. In particular, this gives
a uniform construction of categorified highest weight projectors in this setting and provides a method
for comparing link homology theories.
In the past, I have worked on the following projects:
Categorification of highest weight quantum sl_3 projectors
(to appear in Quantum Topology):
In this paper, I show that the complexes assigned to k-twist torus braids (suitably shifted) in
Morrison and Nieh's formulation of sl_3 link homology stabilize as
k goes to infinity. This stable limit gives a categorified quantum sl_3 projector; I use these projectors to give a
categorification of the sl_3 Reshetikhin-Turaev invariant of tangles (with components colored by irreducible representations).
This work extends Rozansky's results to the sl_3 case.
Grothendieck groups of additive categories:
In this short note, I show that the split Grothendieck group of an additive category A is isomorphic to the triangulated
Grothendieck group of the homotopy category of (bounded) complexes in A. In more down to earth terms, this is equivalent to
showing that the "Euler characteristic" of a complex, viewed as an element in the split Grothendieck group of the underlying
additive category, is invariant under homotopy equivalence. This result allows relations which hold up to homotopy to descend to
equations in the Grothendieck group of A and as such has implications for categorification.
As an undergraduate at The College of William and Mary, I studied matrix theory with Ilya Spitkovsky.
We analyzed properties of the Aluthge transform, in particular the sequence of iterated Aluthge transforms and the behavior of the
numerical range under the Aluthge transform:
On the stabilization of the Aluthge sequence
On the numerical range behavior under the generalized Aluthge transform.
I also wrote two honors theses:
Results concerning the Aluthge transform
Minimal length uncertainty and the quantum mechanics of non-commutative space-time
although the former contains little more than the content of the above papers.
In Spring 2014, I taught Math 435 (curves and surfaces) and Math 226 (multivariable calculus),
and in Fall 2013, I taught Math 440 (topology).
In Spring 2013, I taught Math 225 (linear algebra and linear differential equations), and in Fall 2012, I taught two sections of Math 226 (multivariable calculus).
In Spring 2012, I taught an undergraduate topics course on
Algebraic Methods in Knot Theory (through the Duke University
Bass Fellowship program). This course served as an introduction to knot theory, the Jones polynomial, and Khovanov homology.
If you are interested in any of my course materials or about the feasibility of teaching such a class at the undergraduate level,
feel free to shoot me an email.
While at Duke, I also taught first and second semester calculus (Math 25L and Math 32) and served as a TA for multivariable
calculus for economics majors (Math 102).
In August of 2010 and 2011, I led the linear algebra portion of the Duke mathematics department pre-qualifying preparatory program.
This program is an intensive week-long review of the linear algebra covered on the
written qualifying exam.
When I'm not thinking about mathematics (and sometimes, while I am), I am typically doing one of the following:
- running near the Silverlake reservoir,
- hiking in the Santa Monica or San Gabriel Mountains, or
- listening to my vinyl collection.