Margaret A. M. Murray

Regularity properties of commutators and layer potentials associated to the heat equation

In recent years there has been renewed interest in the solution of parabolic boundary value problems by the method of layer potentials. In this paper we consider graph domains D = {(x, t): x > f(t)} in 9j2, where the boundary function f is in I112(BMO), This class of domains would appear to be the minimal smoothness class for the solvability of the Dirichlet problem for the heat equation by the method of layer potentials. We show that, for 1 < p < oo, the boundary single-layer potential operator for D maps LP into the homogeneous Sobolev space I1I2(Lp) . This regularity result is obtained by studying the regularity properties of a related family of commutators. Along the way, we prove LP estimates for a class of singular integral operators to which the T 1 Theorem of David and Journe does not apply. The necessary estimates are obtained by a variety of real-variable methods.

Uniform analyticity of orthogonal projections

Let X denote the circle T or the interval [-1, 1], and let di denote a nonnegative, absolutely continuous measure on X. Under what conditions does the Gram-Schmidt procedure in the weighted space L2(X, 02 dj) depend analytically on the logarithm of the weight function co? In this paper, we show that, in numerous examples of interest, log o E BMO is a sufficient (often necessary!) condition for analyticity of the Gram-Schmidt procedure. These results are then applied to establish the local analyticity of certain infinitedimensional Toda flows.

Absolute Continuity of Parabolic Measure

Let ℝ be the real numbers and if E \( \subseteq \) ℝ, let Ē,∂E, |E|, denote the closure, boundary, and outer Lebesgue measure of E, respectively.

Toward a Documentary History of American Women Mathematics PhDs: The Doctoral Classes of 1940–1959

In 1993, I started to compile a database of biographical information on the roughly 200 women who earned PhDs in mathematics from US colleges and universities during the 1940s and 1950s. At the time, my primary motivation was to locate and interview as many of those still living as I reasonably could. My book, Women Becoming Mathematicians (MIT Press 2000)—while providing an overview of the entire group—was devoted in the main to what I had learned by conducting oral history interviews with 36 of the women. In recent years, I have returned to the task of completing the database and publishing it online. In this essay, I describe the personal motivations that led me to this project, surprises that emerged in the course of my research, and my ongoing efforts to complete and publish the database. In light of the trailblazing work of Green and LaDuke on the pre-1940 doctorates, and the project of Leggett and Case on PhDs of the 1960s and 1970s, I see this work as key to a larger program of compiling a documentary history of the first century of American women mathematics PhDs.

Pioneering Women in American Mathematics: The Pre-1940 PhD's

In the thick of studying for my mathematics Ph.D. qualifying exams at Yale in 1980, a headline in the New York Times taunted me with the question, Are boys better at math? [2] According to Johns Hopkins University researchers Camilla Benbow and Julian Stanley, SAT math scores showed that answer was yes [3]. In the 30 years since, I’ve probably read at least a hundred articles purporting to give the answer, invariably based upon the results of mathematics tests of one kind or another (see [1], [6], [7] for a non-representative recent sampling). Given our culture’s endless fascination with numerical measures of women’s mathematical aptitude, we have largely failed to attend to women’s actual mathematical achievements. And when we have attended to those achievements, we’ve done so haphazardly, invoking the names of the same few women, over and over again. But way back in 1978—two years before the Benbow and Stanley bombshell—Judy Green and Jeanne LaDuke set out to chronicle the lives and careers of the first American women in mathematics—to trace each woman’s trajectory “from birth to death,” collecting the details of “family background, education, work history, special contributions, extracurricular interests, and professional recognition,” and compiling a complete list of “publications and professional presentations” (p. xv). Three decades later, Pioneering Women in American Mathematics is the spectacular result. And it turns out that women—hundreds of women!—were present at the creation of the American mathematical community. Prior to 1940, exactly 228 American women earned Ph.D.’s in mathematics and most went on to active careers in mathematical research and teaching. Thanks to the tireless efforts of Green and LaDuke, we now know more about this first cohort of American women mathematicians than we know about any cohort of mathematicians, male or female. Nowadays, a Ph.D. in mathematics is essentially mandatory for anyone pursuing a career of college-level research and teaching in the United States. But the modern Ph.D. degree, with its research emphasis, is a German invention barely 200 years old [10, p. 2]. In 1874, Russian-born Sofya Kovalevskaya was the first woman to earn a Ph.D. in mathematics; this was at Göttingen, where—with her friend, the chemist Yulia Lermontova—she became one of the first women to earn a modern Ph.D. in any field [9]. In America, for almost as long as women have been admitted to institutions of higher education, they have been earning Ph.D.’s [12, p. 133], including Ph.D.’s in mathematics. In 1862, Yale University awarded the first American Ph.D. to anyone,