Dimer algebras on surfaces and nonnoetherian geometry

Algebraic geometry is the study of certain geometric spaces using the functions that `live` on them. An example of such a geometric space is the parabola, given by the equation y = x^2. The parabola is the set of all points (x,y) in the plane that when plugged into the equation f(x,y) = y - x^2, give zero. The functions that live on the parabola are therefore all polynomials g(x,y) in the variables x and y, such that any two such polynomials are defined to be equal if they differ by some multiple of f(x,y) = y - x^2. So, for example, the two functions 5xy^4 + 8 and 5xy^4 + 8 + y - x^2 are equal on the parabola, because they give equal values at all points along the parabola. The set of all such functions is called the ` ring of functions` of the geometric space (the word `ring` means you can add and multiply the functions together). However, there are some unusual rings of functions that people had long believed could not be the ring of functions on any geometric space. In previous work I found that, contrary to this belief, these unusual rings do live on geometric spaces, but the spaces are quite strange: you could have, say, a surface containing curves that are single points. That is, such a ring may give rise to a geometric space that contains curves (such as lines) which are not made up of a continuum of points -- they are not made up of anything smaller -- but are themselves 1-dimensional points. Such curves may be thought of as `smeared-out` points. In my project, I will study this bizarre geometry, called `nonnoetherian geometry`. I will apply nonnoetherian geometry in two different settings: - to the study certain structures that arise from special arrangements of arrows on surfaces, such as donuts with many holes; and - to the problem of unifying gravity and quantum theory. An old view of time, held by Aristotle, Leibniz, and others, is that time passes if and only if something changes. I have found that this notion of time can be incorporated into Einstein`s theory of gravity using nonnoetherian geometry. Indeed, consider a collection of fundamental particles. If one of the particles is not interacting with the others, then it would not detect change, and thus it would not experience the passage of time. Consequently, time would not advance along the particle`s trajectory through spacetime, called it`s `worldline`. In other words, the particle`s worldline would be a single 1-dimensional point, which is exactly the type of geometry that occurs with our unusual rings. In my project, I aim to show how this modification to spacetime gives a concrete explanation to one of the most mysterious features of quantum mechanics: quantum nonlocality. Quantum nonlocality arises, for example, when two particles become entan gled, thus giving them the ability to influence each other instantaneously, no matter how far apart they are. Finally, I aim to use this new spacetime geometry to explain certain structures that appear in particle physics.