Arick Shao 邵崇哲


A big part of my work involves research - attempts to add just a bit more to what we currently know about mathematics.

Partial differential equations

A differential equation is a mathematical equation that also involves differentiation. A simple example is the equation \(f' = 2 f\). Unlike for an algebraic equation such as \(x^2 - 1 = 15\), in which the unknown is a number \(x\), the unknown in the differential equation is a function \(f\). In this case, the solutions to this equation are given by functions of the form \(f(x) = C e^{2x}\) for any constant \(C\).

By partial differential equation (abbreviated PDEs), one generally refers to a differential equation involving functions of two or more variables, which hence involve partial derivatives. One fundamental example is the Laplace equation in two dimensions, \( \partial_{x x} u + \partial_{y y} u = 0 \). This is in contrast to ordinary differential equations (abbreviated ODEs), which deal with functions of a single variable. Unlike for the ODE \( f' = 2 f \), the picture for a general ODE or PDE is far more complicated, and one usually cannot find solutions that can be so explicitly described.

We also use the term "PDEs" to describe the branch of mathematics that involve the study of partial differential equations. In this discipline, one is generally interested in several questions, with some fundamental examples including:

  • Do there exist solutions of a given differential equation?
  • If solutions do exist, then how many are there? Can one pose additional conditions that pick out only one solution?
  • How do solutions to a differential equation behave? What properties can we establish for these solutions?

One appeal of PDEs is that they are found in a wide variety of real-world settings. For instance, many theories in physics, describing aspects of the universe, are modeled by PDEs. The examples are at least as diverse as there are areas of physics:

Assuming these physical theories are accurate enough, then in order to understand the phenomena they describe, one must first understand the PDEs that model them. Moreover, one could use a refined, quantitative understanding of these PDEs to not only understand physical phenomena, but also to predict future or past behavior. This applicability of PDEs lies not only in physics, but also in other disciplines, including biology, chemistry, and economics. Some notable examples include the Lotka–Volterra (or predator–prey) equations and the Black–Scholes equation.

PDEs are useful not only outside of mathematics, as many fundamental concepts and problems in mathematics are also connected to PDEs. For example, in complex analysis, the notion of complex differentiability is closely tied to a system of PDE known as the Cauchy–Riemann equations. More recently, a geometric system of PDE called the Ricci flow was used to solve a major open problem in mathematics, the Poincaré conjecture.

Yet another appeal of PDEs is that it is a mathematically diverse area that envelops both pure and applied mathematics. For example, research in PDEs could vary from rigorously proving results for solutions of PDEs to numerically simulating these solutions. Moreover, different PDEs will have very distinct and rich theories associated with them, and there is very little general knowledge applicable to all, or most, PDEs. Thus, it is possible to do tremendous amounts of research on one class of PDEs but know very little about other classes.

A third appealing feature of PDEs as a field is that it utilizes and brings together many areas of mathematical analysis.

Thus, PDEs offers a fertile playground for mathematical analysis, where more abstract principles can be applied in clever ways to solve interesting problems. On the other hand, this feature could also be a disadvantage, since the diverse amounts of knowledge required means there can be a steep learning curve associated with doing cutting-edge research.

Because of its innovative use of mathematics within and its wide applicability to other fields, PDEs is a fascinating subject in which to conduct research. Given the ubiquity of PDEs in physics, economics, and other fields, and the sheer multitude of problems yet to be solved, the mathematical study of PDEs will remain important for a long time to come.

General relativity

Many modern theories in physics seek to model and describe the world using a multitude of advanced mathematical concepts. In fact, the underlying mathematics can be so rich that entire careers could be devoted to studying them! If a mathematical construction is supposed to "model the universe" in some sense, then we can pose the following sensible general questions: what can we mathematically prove about our model, and what could this imply about the universe? Furthermore, can we use this knowledge to make predictions about the future or to infer information about the past?

One revolutionary example of such a modern physical theory is general relativity, which has forced mankind to reinterpret our view of gravitation, as well as space and time altogether. In terms of mathematics, relativity has found rigorous footing and has produced many fascinating problems in geometry and differential equations. Although the theory was conceived over one hundred years ago, it is stunning in its sheer complexity, and even to this day, there are many fundamental questions that will likely remain unresolved in the near future.

Studies in mathematical relativity are also inextricably tied to differential geometry and partial differential equations. In particular, in general relativity, the "universe" is modeled as a four-dimensional "curved spacetime" (more specifically, a Lorentzian manifold), with gravity being represented by the curvature of this spacetime. Moreover, such spacetimes are postulated to satisfy the Einstein field equations, a system of geometric partial differential equations coupling the gravity to the matter fields. Thus, one of the main appeals in studying mathematical relativity is the ability to apply a diverse range of mathematical knowledge toward solving problems that may be physically relevant.

The following series of notes provide a brief and informal introduction to the mathematical theory behind relativity. These are based on a short informal talk first given at Imperial College London in fall 2014.

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