Department of Mathematics

Research Interests

My research is in complex analysis and geometry. As to my teaching, during my career in academia I have taught classes on various levels, from freshman calculus to advanced graduate courses and to the occasional summer schools for postdocs. What we teach in calculus is mostly 17th and 18th century mathematics; it is excellent mathematics, widely applicable---even indispensable---now and in the foreseeable future, but it is over two hundred years old. It always amazes me that in spite of this how little it takes to go from the material of a calculus course to the problems I research in the 21st century.

Freshman calculus teaches students how to understand functions of one independent variable and how to work with them; also how such functions can be used to study plane (and space) curves, and vice versa, how curves can be used to study functions---the stuff of analytic geometry. Sophomore calculus is about functions of two or three variables, and higher dimensional figures they describe: surfaces and solid bodies. The objects of my research are essentially the same: functions and the geometric figures they describe, but with a twist. The variables of my functions are complex numbers, unlike the real number variables of calculus. And there are infinitely many of them. So, in a nutshell, I study problems concerning functions of infinitely many complex variables, and the related infinite dimensional geometric figures.

As with most theoretical investigations, a lot of my research is driven by curiosity, by the desire to understand what hitherto nobody has understood, but perhaps only conjectured based on some analogy or other sort of incomplete evidence. The direction of the research is determined by what one often calls inner logic of the subject. Yet it is legitimate to ask whether these functions of infinitely many complex variables have anything to do with the real world at all. In general, mathematics communicates with the real world through the sciences, so the really legitimate question would be whether those functions have anything to do with other sciences. Yes, they do, something that is perhaps surprising at first sight.

Real numbers, as results of measurements such as length, area, weight, etc., have been known at least naively since the dawn of human civilization. They occur whenever we try to describe our world quantitatively. By contrast, complex numbers were first discovered (or invented, depending on your philosophical disposition) in the 16th century and did not become fully accepted in mathematics until the early 19th century. A complex number has a real and an imaginary part and can be represented by a point in a plane, much like real numbers are represented on the number line. It has been a major discovery of the 19th century that even if quantities in the real world are measured by real numbers, many real world problems are easier to treat if complex numbers are introduced in them. For example, voltage in an alternating current at time t may be given by v cos t, where v is voltage at moment t=0; all quantities involved are real at this point. Yet it simplifies computations with alternating currents if we think of this real voltage quantity as the real part of a carefully constructed "complex voltage". (If i denotes the imaginary unit, then the complex voltage at time t should be defined as v cos t + iv sin t.) The 20th century produced many more physical phenomena that can be understood only with the help of complex numbers. Not only electromagnetism, but quantum physics, the theory of weak and strong forces in atomic nuclei would be unimaginable without complex numbers.

The discussion above explains the relevance of functions of complex variables. On the other hand, the number of variables of functions enters for example when one wants to describe systems with a certain number of degrees of freedom. A very simple system, a point mass in space has three degrees of freedom: once we know the three coordinates of its location, we fully know the location. Two point masses, joined by a rigid rod have five degrees of freedom, because among the coordinates of the locations, six real numbers, there is one relation expressing that the distance of the two masses is fixed, and this brings down the freedom to five degrees. Many quantities associated with a system of n degrees of freedom can be expressed through functions of n variables: as an example, the potential energy of the point mass above is determined by the location, hence it is a function of three variables, namely of the three coordinates of the location. Logically, if one wants to study systems with infinitely many degrees of freedom, one needs functions of infinitely many variables. But does nature know systems with infinitely many degrees of freedom? Some would immediately say yes, fields, such as an electric field, can be described only by infinitely many numbers: at each of the infinitely many points in space one has to prescribe the electric force, a three dimensional vector. Others might argue that the space with infinitely many points is but an idealization, and it does not really exist in nature. Yet it matters little which side of this debate we support. Even if systems with infinitely many degrees of freedom should not really exist in nature, systems with gigantically many degrees of freedom do exist. For instance, at zero temperature and one atmosphere pressure, in 22 liters of gas the number of molecules is about N , Avogadro's number (about 600,000,000…, with twenty--three zeros after the initial 6). Therefore the gas in question has about 3N degrees of freedom. Properties of such systems can only be understood if one approximates them by systems with infinitely many degrees of freedom; and to study those, one needs functions of infinitely many variables.

Even though there is a connection with the real world, the problems I work on are not motivated by immediate applications in the real world, or in other sciences. Most of my problems are "cohomological" in nature. Cohomology has been one of the great inventions of 20th century mathematics. Together with her slightly older sister, homology, originally devised to distinguish between geometric shapes by their very crude ("topological") properties, they have by now permeated many branches of mathematics. In analytic and geometric questions cohomologies arise when some global construction needs to be performed, but initially it is only known that it can be locally performed, in some small neighborhood of an arbitrary point. To pass from the local constructions to a global construction one must show that a so called cohomology class is zero. We will finish this little write up by formulating a concrete problem of cohomological nature from the theory of functions of infinitely many complex variables, a problem that has been largely, though not completely solved.

At this point we must get more rigorous about what functions we are considering, especially what their domain is to be. We start with a complex Banach space, and we will consider complex valued functions defined on this Banach space as models of functions of infinitely many variables. When the Banach space is a sequence space such as l^p, so that its points are given by (certain) infinite sequences of complex numbers, then indeed, any function on the space is just a function of infinitely many complex variables. We will only consider functions that are complex differentiable (or "holomorphic"); the importance of this class, in the simpler setting of functions of a single complex variable, is amply demonstrated in introductory courses to the subject, here at Purdue in MA 530. Suppose now we are given a sequence of points in our Banach space, without an accumulation point. The interpolation problem asks if one can find a complex differentiable function on the space that takes arbitrarily preassigned values at each point of the given sequence.

After the finite dimensional positive results of H. Cartan, K. Oka, and J.-P. Serre in the 1940s and 50s, the first infinite dimensional result on this problem is due to S. Dineen. Dineen showed in 1971 that in the Banach space consisting of all bounded complex sequences the interpolation problem is in general not solvable. By contrast, in 2007 with my former student I. Patyi we proved that in many Banach spaces, including for example the separable Hilbert space, the interpolation problem can be solved. Thanks to Patyi's subsequent work in 2011 the interpolation problem is now known to be solvable in all Banach spaces that have a Schauder basis---as are many other cohomological problems. A conspicuous open problem is whether interpolation problems can be solved in all separable Banach spaces.

Laszlo Lempert
Distinguished Professor of Mathematics
lempert@purdue.edu

Dept of Mathematics
Purdue University
150 N Univ St
West Lafayette, IN 47907
ph. (765) 496-1908
Fax (765) 494-0548


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