What Are You Working On? June 20, 2022

My current research consists of higher dimensional complex analysis stirred with complex geometry and a sprinkle of algebraic geometry.

I will give a talk about the fractal dimensions, more explicitly the Hausdorff Dimension.
Started trigonometry last week, I’m a little disappointed that it’s just triangles and solving for angles etc.
Going through McMullen's lectures on geometry and ergodic theory.
Reading Hartshorne's algebraic geometry with some support from Eisenbud's commutative algebra.
Trying to understand the proof of runge's approximation theorem. I think the proof in the book I have is incorrect, it applies the cauchy integral formula to a function which isn't defined on all of the interior of the contour

Two of the recent complex analysis quals at my university have had a problem where you need to prove a special case of this theorem (that any holomorphic function on a compact disk minus two smaller disks is uniformly approximable by rational functions), so I want to be really certain about the details
I ran into a hiccup the other week with my work on finding a good definition of schemas. After mulling over the problem a bit, I realized a part of why I even had it to begin with is that even though I knew what I ultimately wanted schemas to let me do, I didn't have a clear idea of what a schema should be other than "a generalization of a set". I more or less scrapped what I had been doing up to that point and got to task with figuring out a clear answer to this question, since I didn't really feel I could properly move forward without having a solid idea of the end-goal, so to speak.

Since my original motivation for this whole thing came from certain issues with dealing with the hyperreals (and, more generally, objects built out of the ultrapower construction) I decided that would be a good place to start. If you don't know about the ultrapower construction you don't really need to know a lot except this (which I'll frame specifically in terms of the hyperreals): the construction introduces a distinction between *internal* subsets and *external* subsets of \*R. An important fact of the hyperreals is that any infinite internal subset of \*R has infinitesimal and/or infinite elements. As a result, even though N, Z, Q, and R can all be embedded in the reals in a natural way, for the purposes of reasoning about the internal properties of the hyperreals we should use the hyperextensions of each of these. E.g., the hyperextension of N is \*N, called the *hypernaturals*, which contains elements that are infinite in the sense that they are larger than any standard natural.

The internal subsets of \*R are given by \*P(R), the ultrapower of the powerset of the reals. In turn, we can talk about internal *collections* of internal subsets of \*R—this corresponds to \*P²(R), the ultrapower of the powerset of P(R). This continues ad infinitum.

How I've taken to interpret this is that \*P(R) is the "correct setting" for second-order logic, not P(\*R). Likewise, \*P²(R) is the "correct setting" for third-order logic on \*R rather than P³(\*R), and so forth. To extend this to schemas, we should think of them as a setting for a restricted form of higher-order logic on some set. This idea isn't entirely novel (there's something called Henken semantics for HOL which seems to be similar), but I believe the way I need to wrap it up in an object to achieve my eventual goal is a new idea.

This has led me to my definition of a *concrete schema complex* on a set, which is I believe how any "good" definition of a schema should be realized (at least in some cases) when we're considering a way to put this structure on an actual set. A concrete schema complex on a set X is a sequence of sets ∆ = (∆(0) = X, ∆(1), ∆(2), ∆(3), ...) along with embeddings ∆(k) → ∆(k+1) and embeddings ∆(k) → P^(k)(X). Moreover, when we include the natural embeddings P^(k)(X) → P^(k+1)(X) where P^(0)(X) = X, there is an obvious (infinite) diagram and we impose that this diagram must commute. Lastly, with respect to the language {\exists, \forall, \vee, \wedge, \neg, \sup, \inf} where \sup and \inf are the *binary* operators on the lattice structure of these sets, we require that ∆(k) and P^(k)(X) are elementarily equivalent (which I think means that the map ∆(k) → P^(k)(X) is an elementary embedding?) Intuitively, this says that a schema complex on X is a collection of substructures of the (iterates) powersets of X which behave just like powersets as long as we restrict ourselves to first-order sentences in the appropriate language.

I've already begun working to abstract this definition, and I believe I'm making good progress though I'm far from finished as of yet. I'll probably end up dropping "complex" from the name and just refer to the whole collection of data as a schema. Originally I had planned on a schema as being a much smaller collection of data, but some concerns led me to the larger collection in using now and that's when "schema complex" got started. Now that I've zeroed in on this "restricted HOL" idea it feels like the larger collection of data is the right choice and now having "schema" and "schema complex" be distinct things is a bit pointless.
Currently reviewing my Calculus by going through the entire CLP textbook series online, and I'm loving it. Literally the best calc books I have read yet.
Putting together a short course on logarithms for weak undergraduate students.
As soon as I go up to my dad's for the summer, I have to start my revision. I'm going to start with group theory; have to finish up matrix groups.
A few months ago one of my mentors suggested that I look into the representation theory of quivers. I've finally gotten around to reading some of it this past week and it's pretty interesting. I've just characterized simple representations of a quiver without oriented cycles and I'm currently verifying that the standard resolution is actually a projective resolution.

I remember reading somewhere that every finite-dimensional algebra over an algebraically closed field is Morita equivalent to a quotient of the path algebra of a quiver. Knowing absolutely nothing about how to prove this, I thought about it a little bit in the case of group algebras and came up with an idea. Since a group is nothing more than a groupoid with one object, we can turn this into a quiver. Namely, there's one vertex and an arrow for each element of the group. Then we consider the path algebra and quotient out by the ideal generated by whatever relations hold in the group. I have no idea if this actually works but if it does that would be neat.

Edit: Upon some reconsideration, you might want to look to a finite presentation of the group instead.

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