Kenya, Summer 2012 |
ResearchTeaching and seminarsBiography |
---|
I'm currently a math PhD student at UC Berkeley
Mathematical Interests: Geometry/Topology, Representation Theory, Physics
Email: danielhl@math.berkeley.edu
1037 Evans Hall, Berkeley, CA 94720
Spring 2013: I am running a seminar on moduli spaces in algebraic geometry. I am also teaching Math 191, an upper division undergraduate course on the geometry and topology of surfaces.
I have been a teaching assistant for Math 1A,1B,53, and 54, and I have taught math 1B over the summer 2009. My last semester as a GSI was Spring 2011, for Math 53 with Prof. Frenkel, worksheets and quizzes from the course are available here.
Spring 2010: I ran a student Gromov-Witten theory seminar with Constantin Teleman
PhD Thesis:
My thesis consists of three parts. The first and second are the papers on derived Kirwan surjectivity and derived autoequivalences below. The third develops a general notion of instability in algebraic geometry. It will eventually appear as its own paper.
Wall crossings and derived autoequivalences in geometric invariant theory:
In geometric invariant theory (see below), the GIT quotient of X/G depends on a choice from a continuous set of parameters. Nevertheless, the parameter space breaks down into "chambers" within which the GIT quotient does not vary, and these chambers are separated by "walls." When the parameters cross a wall, the GIT quotient is modified by a "birational transformation." I have been studying how the geometry and especially the derived geometry of the GIT quotient changes under such a wall crossing. For the special case of these wall crossings known as a "generalized flop," the derived geometry of the GIT quotient does not change at all. These cases are especially interesting -- they can reveal new symmetries of the derived category of the GIT quotient which do not arise in the classical geometry.
Derived Kirwan surjectivity:
If an algebraic group G acts on an algebraic variety X, like C* acting on affine space by dilation, is there a meaningful notion of a "space of orbits" for that action? Mumford's geometric invariant theory (GIT) answers this question by constructing a well-behaved orbit space for the action of G on an open subset of "semistable points" of X. Many of the algebraic varieties we know and love (partial flag varieties, toric varieties,...) can be presented as GIT quotients of affine spaces. Since the 1980's, many beautiful relationships between the geometry and topology of the GIT quotient and the "equivariant" geometry of X have been discovered. My research extends these relationships to the "derived" equivariant geometry of X and the derived geometry of the GIT quotient.
Topology of stacks:
Deligne-Mumford stacks are an abstract method that modern algebraic geometers use to handle spaces whose points have "internal symmetry." Such spaces arise naturally as parameter spaces of objects (such as pointed elliptic curves) which generically have no symmetry, but for which certain points parameterize an object with discrete symmetry. By enlarging your notion of "space," you can deal with these naturally occuring geometric objects, and with the proper formulation, you can even use concrete geometric reasoning such as Morse theory to show that classical theorems still apply in this setting.
I grew up in Cheltenham, a suburb of Philadelphia. Growing up I was into literary magazines and table tennis, in addition to the usual list of computer programming / physics nerd activities. In my undergraduate studies at Princeton, I focused on math and physics, and in my graduate work I'm still studying pure math inspired by and related to string theory and high energy physics. Despite the specialized nature of my research, I love learning about some of the awesome things going on in biology, physics, and machine learning / statistics. I'm no longer a ping pong champ, but I enjoy running, swimming, and whenever I can I love hiking in the bay area.
As an undergraduate I thought about algebraic approaches to information theory. Here is a primer on my work (Last update 4/30/08)
I also have something to say about nonnegativity of entropy, as well as on the theory of entropy on metric spaces, but I haven't written that up in a nice primer.