Bernd Schröder

Research Interests

 

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Bernd S. W. Schröder - Research Interests
 
 

I am interested in pretty much all subjects that relate to mathematics.

My central area of research undoubtedly is the area of ordered sets. I have been very interested in the fixed point property (an ordered set has the fixed point property iff every order-preserving self-map has a fixed point). This beautiful property is related to many aspects of the theory of ordered sets as outlined in my survey article.

As a consequence of work on algorithms for fixed points, I have become very interested in decision problems and complexity. Currently I am working on algorithms for constraint satisfaction problems in general, with main focus on analysis/prediction and on preprocessing techniques. My main tool is the steadily expanding CSP solver.
 
The pictorial aspects of work on the fixed point property have led me to working on the reconstruction problem for ordered sets (Is every ordered set determined up to isomorphism by its set of one-point-deleted unlabeled subsets?). Recent work by Kratsch and Rampon has shed much light on this problem. Many order-theoretical parameters turn out to be reconstructible and there seems to be hope that the problem is solvable soon after all.

All the above topics are reflected in my textbook on the theory of ordered sets.

I am also interested in possible applications of order in analysis. But my main analytical focus was on harmonizable stochastic processes: Completeness of the spectral domain, positivity of the angle between past and future for multivariate stationary process, etc.

I recently completed a text on analysis. This text has certainly revived my interest in the area. No plans for specialized projects as of now. It seems I'll be enthusiastically expanding my background in analysis and maybe I'll write a book that connects analysis with physics.

 

In the area of education I am interested in curriculum integration. Research has shown that knowledge is retained better if it can be made meaningful to the learner. Thus the explicit connection of topics to each other should allow us to produce better students. My main focus so far has been the integrated engineering curriculum at Louisiana Tech University and all its ramifications. Click here for a power point presentation about the implementation of the integrated engineering curriculum. Click here for the videos on differential equations. Click here for the animations on various calculus and differential equations topics.