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.

My project focused on applying a new kind of geometry I previously introduced, where points can be 'smeared-out', to both spacetime geometry and to certain algebraic structures that come from surfaces. In this geometry, there can be curves, such as lines or circles, or surfaces, such as spheres, that are themselves single points. These smeared-out points are not made of smaller (0-dimensional) points, and this leads to surprising and unusual geometric features. On the physics side, I incorporated this new geometry into general relativity by taking time to be stationary---or frozen---along the trajectories of fundamental particles through spacetime. Doing so turns their trajectories into 1-dimensional points. The aim of this modification is to describe quantum phenomena using only spacetime geometry. I was successful in accomplishing this with both spin and polarization. In particular, measurements of spin and polarization are not well-defined in quantum theory (the so-called 'measurement problem'), and I showed that in my model such measurements become well-defined. I then used this geometry, together with tools from general relativity, to derive the standard model particles, their interactions, and the approximate masses of the first generation particles, namely, electrons, electron neutrinos, up quarks, and down quarks. I was also able to derive certain properties of light, such as what happens when light passes through two polarizers, as well as what happens when two photons or electrons become quantum entangled. On the mathematics side, I applied this geometry to certain algebraic structures that come from surfaces that look like donuts but with any number of holes in them, which are covered by polygons such as triangles. Variables are then assigned to the edges of these polygons. Taken together, these variables produce polynomial functions that define a geometric space (similar to how y = x^2 defines a parabola). In my project, I showed that these spaces necessarily contain smeared-out points. I also showed that the dimension of such a space is determined by the number of holes in the donut-like surface. In addition, I found that its smeared-out points are closely related to the topology of the surface (that is, properties of the surface that are independent of distances). My project therefore established a new bridge between different areas of mathematics that had not been known before (specifically, between algebra, representation theory, algebraic topology, and nonnoetherian algebraic geometry), with smeared-out points playing an essential role.