Geometric analysis bridges the areas of differential geometry and differential equations, bringing to bear the most powerful tools of each to tackle seemingly-intractable problems in the other. On the differential equations side, for instance, the specific geometry of a space can provide enough constraints to make it reasonable to even ask for a solution to a partial differential equation. In the realm of geometry, by studying certain partial differential equations associated to a space, we can gain deep topological insights. In my research I follow this later perspective, employing geometric analysis techniques to develop generalizations of symplectic geometry, such as b-symplectic (log-symplectic) structures. Often times my results link many a priori unrelated areas of mathematics - such as contact and Poisson geometry - in new and exciting ways.
A User-Friendly Look at My Research
Symplectic geometry tackles fundamental questions for conservation laws and the generalized "motions" they permit. Joseph-Louis Lagrange introduced this geometry in 1808 to understand slow variations of the orbits of the planets in the solar system. He showed that symplectic geometry is the fundamental ingredient in a mathematical formulation of any scenario in mechanics, scenarios such as a swinging pendulum or a spinning top. Unfortunately, this geometry can not model scenarios with external forces that evolve through time, for instance a swinging pendulum in shifting winds or a spinning top on a moving train. In my research, I build a more general version of symplectic geometry that is adapted to tackle these scenarios.
A symplectic structure can perform a wide variety of geometric tasks, such as computing the volume of an object. For the purpose of developing intuition, we will limit our scope to thinking of a symplectic structure as a sensor on a space. It is an operation that inputs a function and outputs the direction where the function does not change.
For example, if we input the function that tells us the temperature at every point in the United States, a symplectic structure will output the heat map that you typically see in your daily newspaper. A symplectic structure conveys where a quantity, temperature in the case above, is conserved, i.e., stays the same. Thus symplectic structures are especially vital in physics because they relay where energy is conserved in a system such as the swinging pendulum.
However, there are times we come across a structure that is mostly symplectic, but has degeneracy or places where the sensor is out. We can think of a symplectic structure on a space as an apple with a shiny bright, unblemished red facade. A bruise or dent on the skin of a fruit represents places where the structure is not symplectic, that is the sensor is out. These more general structures are called Poisson structures - named after Simeon Poisson, not fish - and they provide a mathematical formulation for the situation of a pendulum swinging in the wind or a top spinning on a train.
In my thesis research I focus on minimally degenerate Poisson structures (a mildly bruised fruit).
Goal: Build symplectic machinery in minimally degenerate Poisson settings.
The machinery I develop permits a powerful shift of Poisson problems to the symplectic setting where we can use symplectic tools:
Poisson cohomology. I have spent much of time computing an invariant of a Poisson manifold called Poisson cohomology. Each degree n cohomology group that we associate to a Poisson manifold M has an interpretation analogous to the way we say "the nth singular cohomology group counts the number of n-dimensional holes in a manifold". We interpret the second poisson cohomology as the quotient of the space of all possible infinitesimal deformations of our Poisson structure by the space of trivial deformations. In other words, this cohomology is supposed to count the number of Poisson structures nearby that are actually different from our original structure. Accordingly, this invariant has the potential to tell us a lot about our Poisson structure, particularly about local normal forms.
Contact structures. Further, I can use my machinery to understand subtle nuances in the geometry that other mathematicians were not able to see before. One class of Poisson manifold I study are called scattering-symplectic. At the degeneracy locus of the Poisson structure, these spaces inherit what is called a contact structure. Analogous to a symplectic structure's key role in modeling situations, a contact structure has broad applications. For example, in physics it is used to model and understand how rays of light bend.
symmetry, simplicity, and generality,
and an indefinable sense
of the fitness of things,
creative mathematicians now,
as in the past,
are inspired by
the art of mathematics
rather than by any prospect
of ultimate usefulness.