David Richter's Research and other Projects

This page summarizes my scholarly work. The intended audience is anyone in the academic community who has significant background in mathematics. My mathematical interests are far-reaching and my projects are eclectic. I enjoy working on anything related to classical geometry. I am also interested in finding new ways to visualize sophisticated mathematical ideas. Much of my work is available on this website. However, reprints and preprints can also be made available upon request.

Algebra and Mathematical Physics

Throughout graduate school and as a post-doctoral fellow (1994-2000), I studied Lie algebras and related objects. In this regard, I have primarily been interested in explicit constructions of these objects using differential operators. It is an outstanding problem, for example, to determine all realizations of a given Lie superalgebra as a space of matrix differential operators. The motivation for realizing these objects in this way comes from certain problems in mathematical physics. In the course of studying Lie superalgebras, I discovered an interesting connection to differential geometry; to every manifold with constant sectional curvature, one may naturally associate a particular Lie superalgebra constructed using its Riemannian curvature tensor.

My interest in representations of semisimple Lie algebras has led me to invent a far-reaching generalization of Lie superalgebras. The key properties of a Lie superalgebra which I have chosen to generalize are (a) the even subalgebra is usually a semisimple Lie algebra, (b) the odd subspace is a module for the even subalgebra, and (c) there is a multiplicative composition rule which respects the grading by the 2-element group. Essentially, I want to replace the group appearing in (c) by any finite abelian group and replace the module in (b) by a collection of modules for the semisimple Lie algebra appearing in (a). I call this generalization a "hybrid algebra". Obviously a Lie superalgebra is a special type of hybrid algebra. Moreover, many simple Lie algebras can be realized in a non-trivial way as a hybrid algebra. However, the interesting structures which one may construct as hybrid algebras are not known in general.

Concrete Visualization of Mathematical Ideas

I have long been an avid builder of mathematical models. I believe that many ideas in geometry are beautiful because they can be visualized in an elegant way. Moreover, I have found many ways to put this belief into practice. One of my most prized inventions, for example, is the keyring model of the Hopf fibration. The Hopf fibration is widely recognized by geometers and topologists as a beautiful and important object, but it is difficult for the non-specialist to visualize. However, it is easy to imagine assembling a large collection of key rings so that every pair is linked, thus providing a physical model of one of the critical facts about the Hopf fibration. Another example is my Zome model of the E(8) root system. Again, many mathematicians hold E(8) in high regard for the beauty of its intricate structure. The Lie group underlying E(8) is extremely complicated, but it is relatively straightforward to build a model of its root system using Zome pieces.

It is an ongoing project to document these models using some media, probably a book or a website. I have already assembled many photos and notes on my website. Ideally, individual documentation for each model should include (a) a quality photograph, drawing, or some other type of visualization, (b) instructions or advice for building the model, and (c) comprehensive information on the mathematical aspects of the model. These should be done in a uniform way so that one can see the breadth of the concept and efficiently grasp any desired model. Maintaining this on a website has an advantage over a book because one may continually refine the work.

Ghost Symmetry

I have long been interested in visualizing regular polytopes and related objects using linear projections, i.e. "shadows". Recently this has led me to discover a phenomenon which I have called "ghost symmetry" or "non-inherited symmetry". The basic idea is that while a subset of a vector space may not have an interesting symmetry group, it may have several shadows with non-trivial symmetry. A good example is furnished by the regular 4-dimensional 24-cell. Let S be any 3-dimensional shadow of the 24-cell. Then S always has several 2-dimensional shadows with 4-fold and 6-fold rotational symmetry. This happens in spite of the fact that the symmetry group of S generally does not contain more than 2 elements. Thus, even though S generally does not have this symmetry, a "ghost" of the symmetry of the 24-cell can be revealed by subsequently projecting.

Once one recognizes that an object may have ghost symmetry, it is not hard to create examples among higher dimensional objects or in closely related contexts such as coding theory and design theory. In spite of the elementary ideas used to discuss it and its consequent ubiquity, this phenomenon has been completely missed by the mathematical community until recently. The main reason for this is that there is a general feeling that projecting an object inevitably means a loss of information; this feeling is often justified by associating the rank of a linear transformation with information. In fact, as the example above shows, loss of information is not inevitable when one considers symmetries of several particular shadows.

The consideration of ghost symmetries has led to a new theorem about combinatorics of constellations (configurations of points) in the plane. Studying ghost symmetries of planar constellations is relatively straightforward because the only shadows which might have non-trivial symmetry result from projecting to a 1-dimensional subspace. Moreover, if a 1-dimensional shadow of a planar constellation has non-trivial symmetry, then it must be bilateral symmetry. My theorem concerns the circumstances when one may prescribe that a constellation have precisely three 1-dimensional shadows with bilateral symmetry. To summarize the main result, one may one may associate a 3-connected, 3-edge-colored cubic graph to each such constellation and vice-versa.

This result is stated and proved in a manner strikingly similar to a theorem which characterizes edge skeleta of 3-dimensional convex polytopes due to Steinitz. A natural question therefore arises about prescribability of 4 shadows with bilateral symmetry. Richter-Gebert's universality theorem states that the realization space of 4-dimensional convex polytopes may be homotopy equivalent to any given semialgebraic variety. By analogy, therefore, it is believed that realization spaces of constellations with 4 symmetrical shadows should also be universal. One of my major projects is to prove this assertion. Given the combinatorics associated with this problem, I believe that a resolution of this shall shed light on Richter-Gebert's corresponding conjecture on realization spaces of simplicial 4-dimensional polytopes.

The combinatorics associated with a planar constellation with three symmetrical shadows are quite beautiful, again by analogy to the combinatorics of 3-dimensional convex polytopes. If the constellation has 2n points, then one may also associate an arrangement of hyperplanes in dimension n+1. It is conjectured that this arrangement is always free. The complement of this arrangement (the realization space of the constellation) is a disjoint union of open cells whose number is unknown. It is speculated that the resolution of these problems may be facilitated through the use of an invariant of 3-edge-colored graphs which has not yet been invented. It is thought that such an invariant may be defined in a manner similar to the Tutte polynomial in matroid theory.

Another major research goal is to obtain a Galois connection theorem for ghost symmetry. Evidence for such a theorem arises from the structure of the ideas and terminology which have proved useful in discussing ghost symmetry. For example, if an object already has some symmetries, then the symmetries which arise in their shadows must be conjugate via symmetries of the original object. If S is a finite constellation of points in some vector space, then the poset of shadows of S is an extremely complicated object. However, it is speculated that the poset of shadows with certain symmetry properties may possess some structure reflected in the subgroup structure of several of the shadows. In this regard, it may be of interest to study constellations S for which the symmetry group is transitive but every shadow of S has symmetry inherited from S.

This provides only a taste of the prominent problems in ghost symmetry, but there are many others. One of the features of ghost symmetry is that, due to the elementary nature of some of the ideas (linear algebra, symmetry, group theory) an advanced undergraduate can understand and appreciate some of its problems. However, as one can see, one can also glimpse connections to higher ideas in geometry and combinatorics.

February 2011