David E. V. Rose
I am a graduate student in the
Duke University
Mathematics Department
studying categorification and link homology under the supervision of
Lenny Ng. In Fall 2012, I will be joining the
mathematics department
at USC as a
Busemann Assistant Professor (NTT).
Contact info:
Research
I am interested in categorification and its interaction with low dimensional topology and representation theory.
In particular, I am currently thinking about various approaches to Khovanov homology and the construction of categorified
highest weight projectors in these approaches.
More broadly, I enjoy any mathematics which involves a mix of algebra, topology, geometry, and (higher) category theory. For more
information, see my CV and research statement.
Recently, I have been working on the following projects:
Categorification of highest weight quantum sl_3 projectors:
In this project, 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.
Teaching
In Spring 2012, I will be teaching an undergraduate topics course on
Algebraic Methods in Knot Theory (through the Duke University
Bass Fellowship program). This course will serve as an introduction to knot theory, the Jones polynomial, and Khovanov homology.
In past semesters, I have taught first and second semester calculus (Math 25L and Math 32) and served as a TA for multivariable
calculus for economics majors (Math 102).
During the past two summers, I have 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.
Personal
When I'm not thinking about mathematics (and sometimes, while I am), I am typically doing one of the following: