Higher Categorical Structures with Applications to Orbifolds and Computational Semantics

My research program studies the algebra of transformations. Here,`transformation' may stand for a procedure in a computer program or a function in calculus or a symmetry of an object, such as a rotation or reflection. Transformations are often only partially defined; for instance, part of an object may have rotational symmetry, but this symmetry may not be applicable to the whole object. Another example is that of the square root function. If one wants a real number as outcome, then one should not apply it to negative numbers. The current program studies two (related) aspects: the construction and study of models for the algebra of objects with partial/local symmetry and the construction and study of models of the effects of small changes in the input and/or output parameters of a transformation. Some of these models allow one to describe and compute certain characteristics of objects with local symmetry that have proven useful in mathematical physics for instance. One such characteristic is obtained by considering all distinct paths one can construct on the object, up to deformation, where one is allowed to take jumps using the symmetry of the object (such as a reflection or a rotation) as part of the path. My collaborators and I continue to develop and describe new features of these objects in order to enable new applications to areas such as robotics. With my students I am developing new models that take particular features such as smoothness into account. When describing the effect of small changes in an input parameter on the outcome of a computer program, one normally uses differential calculus. This is important in optimization for complex models. The derivative measures how a small change in a given parameter will affect the output. In the literature, there is an algebraic model to describe the role of the derivative in programming semantics. However, in practice one often wants to use something called "reverse differentiation" instead of ordinary forward differentiation. This is for instance the case in back-propagation for neural network training. Here the goal is for each small change in the output to trace back the contribution that each small change in the input variables made. It turns out that this can be done more efficiently and more accurately using the reverse derivative than by calculating the usual forward derivative. I have developed an algebraic model that can be used to describe and study the semantics of computer programs involving backward derivatives. This model also shows how the backward derivative and the forward derivative are related. The backward derivative is more powerful and my works shows exactly what one needs to add to a forward derivative to obtain a backward one. In the current program, I want to combine this formalism with the algebra for partially defined procedures and symmetries as well as the algebra for parallel computation and combine it with the axiomatization of the state of the program.