I guess this is it. This blog has been a persistent idea of mine for quite some time, and it’s interesting to see it materializing now.

Hi, everyone. I’m Patrick, currently a fourth-year undergraduate in pure mathematics at UCLA (formerly of UCSD, for two years). By way of background, I was born in Ann Arbor, Michigan, where I spent the first half of my life; my family moved to the Bay Area when I was around 11, and I spent my adolescence in Fremont, California.

I’ve begun this as a means of memorializing (and, optimistically, spreading) some things that I have written, which I hope others might find interesting or useful.

As the title of this blog should (might?) make clear, my main area of focus is mathematical analysis, and I mostly study harmonic analysis, with a mix of other things that catch my interest. Most of my work has to do with objects on Euclidean space, and their fine properties: obtaining results for regularity, oscillation, singularities, and decay from weaker forms of control: functionals and norms that can be interpreted as suitable variations on generalized notions of energy or mass.

I’m fond of results that rely on (or consist of) geometric arguments: covering lemmas, optimal decompositions and configurations, and close analyses of sets in physical space or frequency space. To be more specific: some areas I enjoy include

1. Euclidean harmonic analysis: singular integrals, and their associated maximal operators. This is the main avenue where I study: results that require careful tracking of cubes and balls in space; stopping-time arguments, identification of exceptional sets, and estimation according to forms of decay or singular behavior. I am interested in the work of establishing \(L^p\) bounds for operators with rough kernels, beyond the standard (Hölder or Dini) smooth-kernel theory, behavior near the endpoint \(p = 1\), and what can be said about pointwise properties. I am also interested in \(T(1)\) and \(T(b)\)–type theorems, and near-optimal conditions for boundedness of singular integral operators. (The early work of A. Seeger, for instance, is a good example of where my interests in this area lie, as well as that of F. Nazarov, S. Treil, E. T. Sawyer, and X. Tolsa.)
2. Dyadic harmonic analysis: in a similar vein, it is interesting to see how much of the Euclidean theory can be transplanted to an abstract setting, of metric measure spaces and other homogeneous settings. The questions of identifying proper generalizations of Hardy spaces, Calderón-Zygmund operators, and maximal function theory are all interesting; perhaps equally important are the correct notions of the dyadic grid, balls and cubes, and the Haar system beyond \(\mathbb{R}^d\). The recent philosophy in working with dyadic model operators, to pass to results about general Calderón-Zygmund operators like sharp weighted estimates, is also an intriguing development. In the words of Lerner and Nazarov, the work of dyadic methods in real-variable settings often proceeds via “a game of cubes”; it is our task now to identify how this is played in other terrains. (The work of M. C. Pereyra and L. A. Ward, and T. Hytönen and S. Petermichl, may collectively be illustrative of what I mean.)
3. Geometric measure theory: in a specific sense, that of the study of fine properties of mappings defined on Euclidean space. Estimation of pointwise or average behavior for weakly regular (i.e., Sobolev) functions, and other properties; this is related to the work of establishing good inequalities between function spaces, and careful analysis of concrete pointwise representations. Fourier methods also play a role, with Littlewood-Paley methods and other characterizations of function spaces being quite valuable in this setting. In my mind, I see these projects as encompassing many tools from classical real analysis, potential theory, and harmonic analysis. The generalization of these results to abstract settings (e.g., analysis on metric spaces), also interests me. (The flavor I am thinking of can be seen in the work of Ziemer and Federer and W. Fleming, the work of M. Torres and P. Hajłasz, and especially the work of D. Spector and P. Mironescu.)
4. Mathematical physics: I am interested in the work of Elliott Lieb and Michael Loss, on the stability of matter and quantum many-body systems. The derivation of physical principles for macroscopic configurations of objects, from the fundamental results in classical and quantum theory, is an especially-daunting task that requires a tour de force of hard analysis and spectral theory to pull off (and, when successful, serves as extraordinary backing for our physical intuitions). The classical work on the stability of matter, later extended to relativistic and highly-magnetized settings, stands in my mind as a landmark achievement of mathematics in the twentieth century. Much modern progress in functional inequalities and progress toward sharp(er) constants, as well as the theory of Schrödinger operators, was a notable benefit of this field’s development. (Some other researchers in this area whose contributions I have in mind include J. P. Solovej, R. Seiringer, R. Frank, and P. T. Nam.)
5. Rearrangement methods: finally, broadly speaking, I am interested in the dependence of nonlinear functionals on the geometry of functions, in a general sense. The general sense of what I mean is the behavior of normed or seminormed quantities, or measures of “energy,” under highly geometric operations like symmetric decreasing and two-point rearrangement, heat-flow evolution, and mass transport for functions on Euclidean space. The action of two-point, symmetric, and Steiner symmetrization on sets, and the corresponding behavior of various functionals, may also come into play. I am especially interested in how these methods can help us achieve sharp constants, like in Sobolev or HLS-type inequalities. Related is the question of identifying extremizers for these sharp inequalities, and associated forms of stability. My understanding of this area is far less firm than the other fields listed above, but some researchers whose work I find intriguing include Loss and A. Burchard, M. J. Esteban, A. Figalli.
6. Some other things I’m interested in: many of my other interests derive from the field of PDE. My long-standing interest is in the field of dispersive PDE, and nonlinear equations modelling water wave behavior, especially the KdV and its associated hierarchy, and variants of the NLS. I’m interested in the use of harmonic analysis methods to show well-posedness, at low regularities. Other precise behaviors for well-behaved equations (integrable systems which allow for inverse scattering results, and asymptotics for solutions, with an eye toward soliton resolution) are also intriguing. Kato-Ponce inequalities and fractional Leibniz rules, and aspects of paraproducts and multilinear operators, are also of interest. Also, I like functional analysis generally, with particular attention to classical Banach space theory, spectral theory, operator theory and trace ideals, and functional calculi. The theory of \(C^*\) and von Neumann algebras, noncommutative topology and integration, are among topics I’d like to study in the future.

In this blog, I certainly have no hopes of saying anything definitive on any of these subjects, but I do hope to provide a few small signposts and road maps that might be useful, for anyone else who’s similarly lost in this terrain we’re passing through. Some personal musings, on the spirit of mathematics in general and analysis in particular, might crop up from time to time.

For a variety of reasons, this is certainly a strange time for me to be starting a blog; I’m in the middle of my graduate school applications for the upcoming year (either a Ph.D. or a masters, at home or abroad, also in pure mathematics), and my course schedule this fall is in flux (I’m not even sure how many mathematics courses I’ll be taking, if any, and which ones; this is due to some comically-bad planning on my part: namely, spending nearly all of my three prior years in undergrad taking mathematics courses, which has now necessitated back-loading most of my university’s irritatingly-numerous GE requirements at the very end, in the span of just three quarters). All of this is just to say: it may be a while between posts.

Nonetheless, I’ll write when and if I have the time. To kick things off, I’ve prepared a few items to start strong, mainly uploading some writings and mathematical works I spent time creating this summer. (Not to be overly self-congratulatory, but these really did take a fair amount of time to plan, work-out, and fill-in.) This should give some flavor of what I’m doing, and what types of mathematics I enjoy.

If anyone out there is reading this and finds the materials on this blog useful, I’ll be glad. If the notes help anybody, this will have been worth it. Hope to see and hear from y’all.