 Dr. Rudi Weikard Professor, UAB Department of Mathematics

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### Current Research Interests

My research interests are currently in Inverse Problems. I am also interested in the (not unrelated) subjects of differential equations in the complex domain and in abelian functions.

### Inverse Problems

My main interest is to understand which of the time honored methods in potential recovery for Schrödinger equations work also for complex potentials. At the same time I work on extensions of such methods to discrete problems and problems on trees. I am also interested in practical implementations of inverse problem algorithms.
Athens Talk
Atlanta Talk
München I
München II
Mexico City Talk
Santiago Talk
Lisbon Talk

### Differential Equations in the Complex Domain

Suppose the coefficient q in the differential equation y''+qy=zy is a meromorphic function of the independent variable. The requirement that, for every complex number z, every solution of the equation is also meromorphic is a very strong one. I am interested in understanding the implications of this requirement. There are connections to algebraic geometry, to spectral theory and inverse problems, and to the Korteweg-de Vries hierarchy and integrable Hamiltonian systems. For instance, if q is an elliptic function the above requirement forces q to be a solution of some stationary equation in the Korteweg-de Vries (KdV) hierachy. Stationary KdV equations may be represented by the equation [P,L]=0 where P and L are a Lax pair of ordinary linear differential expressions which are of order 2g+1 and 2, respectively and satisfy the algebraic relation P^2=R(L) where R is a polynomial of degree 2g+1.

### Abelian Functions

Closely related to the above topic is my interest in abelian functions, in particular hyperelliptic functions. Abelian functions are inverses of abelian integrals, i.e., integrals of rational functions of x and an algebraic function of x. If this algebraic function is the square root of a polynomial one is looking (in general) at hyperelliptic integrals and hyperelliptic functions. If the degree of the polynomial is 3 or 4 one has elliptic integrals and elliptic functions while for degree 1 or 2 the integrals may be solved in terms of inverses of trigonometric functions. Naturally the study of abelian functions involves the algebraic curve (or Riemann surface) associated with the algebraic function under consideration and this relationship leads back to KdV equations and generalizations thereof.

### Teaching

Notes for the seminar on Jacobi matrices (2003/2004)
Notes for my Algebra course (2010)
Notes for my Advanced Calculus course (2016)
Notes for my Complex Analysis course (2019)
Notes for my Real Analysis course (2019)

### Curriculum Vitae

I was born in 1958 and received my doctoral degree (Dr.rer.nat.) in 1987 from the Technische Universität Braunschweig in Germany. I worked as an assistant there from 1987 through 1990. In September 1990 I came to UAB as an Assistant Professor. I became an Associate Professor in 1994 and a Professor in 1999.