新基石研究员项目系列研讨会19：

题目：Whitney Extension Problems:  Whitney Extension, Glaeser Refinement, and Finite Criteria

报告人：Kevin G. Luli （University of California, Davis）

报告时间及地点：

          6月22日（周一）上午9:00-12:00――文北楼102  第一讲，第二讲

          6月23日（周二）上午9:00-12:00――文北楼102 第三讲，第四讲

          6月24日（周三）上午9:00-12:00――文北楼102  第五讲，第六讲

第八讲至第十二讲时间地点待定（课堂上安排）

摘要：A fundamental problem throughout analysis and geometry is to determine when local information can be realized by a globally smooth object. Given data on a closed set of $\mathbb{R}^n$ or more generally a collection of algebraic constraints, one would like to know whether there exists a smooth function $F$ on $\mathbb{R}^n$ satisfying those constraints and, if so, how can one detects solvability efficiently.

Typical examples include the following:

1. Prescribing Taylor polynomials: Given $E \subset \mathbb{R}^n$, at each $x \in E$, a polynomial of degree at most $m$ $P^x$, determine if there exists a $F\in C^m(\mathbb{R}^n)$ such that \[J_x^mF=P^x\]

where $J_x^mF$ is the m-th degree Taylor polynomial at $x$.

2. Solving systems of linear equations: Given $A_{ij}, f_i: E \subset \mathbb{R}^n \to \mathbb{R}^M$, find solutions $F=(F_1, \cdots, F_N)$ with $F_j \in C^m(\mathbb{R}^n)$ such that \[\sum_{j=1}^N A_{ij}(x)F_j(x)=f_i(x), \text{ on } E\]

3. Solving systems of linear inequalities: Given $A_{ij}, f_i: E \subset \mathbb{R}^n \to \mathbb{R}^M$, find solutions $F=(F_1, \cdots, F_N)$ with $F_j \in C^m(\mathbb{R}^n)$ such that \[\sum_{j=1}^N A_{ij}(x)F_j(x)\leq f_i(x), \text{ on } E\]

The central question we address in this course is the following:

Can the existence of a solution be determined by checking finitely many local conditions on the data?  

In many important cases, the answer is yes.

This question lies at the heart of Whitney extension theory. Whitney's classical theorem (1934) provides a complete characterization of when prescribed jets arise from a $C^m$ function. In the 1960s, Glaeser introduced a geometric procedure for detecting local obstructions to solvability. Building on these ideas, beginning in the early 2000s, Fefferman and collaborators developed a powerful framework that led to finite criteria for broad classes of extension and interpolation problems, including systems of linear equations with semialgebraic coefficients.

This course introduces the main ideas behind these developments and culminates in a recent result showing that, although finite criteria fail for systems of linear inequalities in dimensions $n\ge 2$, they do exist in dimension one.

Lecture 1,2. Overview and Whitney's Extension Theorem

The course begins with a classical question: when can local data be assembled into a global function? The main question of this lecture is:

When does prescribed local differential data arise from a global $C^m$ function?

We begin with the classical Whitney Extension Theorem, which may be viewed as a converse to Taylor's theorem with remainder. Whitney's theorem gives necessary and sufficient conditions for a family of polynomials $\{P^x\}_{x\in E}$

to be realized as the Taylor jets $J_x^m F$ of a single function $F\in C^m(\mathbb R^n)$.

In this lecture, we discuss the main ideas in the proof.

Lecture 3,4. Glaeser Refinements

Whitney's theorem treats the case in which a single jet is prescribed at each point. A more flexible problem allows many possible jets at each point. The question then becomes: how can one systematically identify and discard local data that cannot occur in any global solution?

Glaeser's refinement procedure gives a natural answer. One repeatedly removes local candidates that are incompatible with the existence of a global $C^m$ function. The surviving candidates contain increasingly refined information about solvability.

Using concrete one-dimensional examples and divided differences, we explain why this elimination process works and why it eventually stabilizes in the settings relevant to extension theory. This introduces one of the central ideas behind modern extension theory, including Fefferman's solution of the Whitney extension problem: given $f:E\subset \mathbb R^n \to \mathbb R$, how can one decide whether there exists$F\in C^m(\mathbb R^n)$ such that $F|_E=f$?

Lecture 5,6. Finiteness of the Glaeser Refinements

Can an infinite extension problem be reduced to finitely many local tests? For $C^m$ extension problems, one often obtains a finite stabilization phenomenon: after finitely many Glaeser refinements, the bundle stabilizes, meaning that further refinements no longer remove any jets.

This viewpoint leads to finiteness principles: global solvability can often be determined by checking only finitely many local configurations.

Lecture 7,8. Finite Criteria for Linear Equations

We now apply the preceding ideas. Once the relevant fibers have stabilized, the stabilized bundle encodes the obstruction to solving the corresponding $C^m$ extension problem.

Consider a system of linear equations with variable coefficients, $\sum_{j=1}^{N} A_{ij}(x) F_j(x)=f_i(x),

\qquad i=1,\dots,M$. Although this problem may at first seem unrelated to Whitney extension theory, it can be reformulated in terms of bundles of allowable jets. Joint work with Fefferman shows that, for such systems with semialgebraic coefficients, solvability can be characterized by finitely many differential conditions on the data.

In this lecture, we explain the meaning of this finite criterion and the geometric ideas behind it.

Lecture 9,10. Why Inequalities Are Harder

We next replace equations by inequalities:$\sum_{j=1}^{N} A_{ij}(x) F_j(x)\le f_i(x), \qquad i=1,\dots,M.$

At first sight, this appears to be a minor change. In fact, it fundamentally changes the nature of the problem. Inequalities introduce convexity, one-sided constraints, and boundary phenomena that do not arise in the purely linear-equation setting.

We discuss counterexamples showing that finite differential criteria fail in dimensions $n\ge 2$.

Lecture 11, 12. The One-Dimensional Theory

The final lecture presents a positive result in the remaining one-dimensional case. Let $x\in \mathbb R$, and consider the system \[\sum_{j=1}^{N} A_{ij}(x)F_j(x)\le f_i(x), \qquad i=1,\dots,M,\]

with unknown functions$F_j\in C^m(\mathbb R).$

In one dimension, solvability of this system can be characterized by finitely many ordinary differential equations and inequalities involving the data. This lecture gives a complete proof of this finite one-dimensional criterion.