David E. V. Rose



I am the Busemann Assistant Professor in the USC Mathematics Department and currently
organize the geometry/topology seminar; the schedule can be found here.

Contact and professional info:

Research

I am interested in categorification and its interaction with representation theory and low-dimensional topology. In particular, I am currently thinking about the relationship between Khovanov (and Khovanov-Rozansky) homology and categorified quantum groups:

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.

Khovanov-Rozansky homology is a skew Howe 2-representation of categorified quantum sl(m): In work in progress (joint with Aaron Lauda and Hoel Queffelec), we extend our results above to sl_n link homology for n>3.

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.

Teaching

In Fall 2013, I am teaching Math 440 (topology) at USC.

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.

Personal

When I'm not thinking about mathematics (and sometimes, while I am), I am typically doing one of the following: