Higher Categorical Structures with Applications to Orbifolds and Computational Semantics

我的研究项目是研究变换的代数。在这里，“变换”可以代表计算机程序中的一个过程，也可以代表微积分中的一个函数，也可以代表物体的对称性，比如旋转或反射。转换通常只被部分定义；例如，一个物体的一部分可能具有旋转对称性，但这种对称性可能不适用于整个物体。另一个例子是平方根函数。如果你想要一个实数作为结果，那么你就不应该把它应用到负数上。目前的程序研究两个（相关）方面：部分/局部对称对象的代数模型的构建和研究，以及转换的输入和/或输出参数的微小变化的影响模型的构建和研究。其中一些模型允许人们描述和计算具有局部对称性的物体的某些特征，这些特征在数学物理中已经被证明是有用的。一个这样的特征是通过考虑可以在物体上构建的所有不同路径来获得的，直到变形，其中允许使用物体的对称性（例如反射或旋转）作为路径的一部分进行跳跃。我和我的合作者继续开发和描述这些物体的新特性，以便在机器人等领域实现新的应用。我和我的学生们正在开发新的模型，这些模型考虑了平滑等特殊特征。当描述输入参数的微小变化对计算机程序结果的影响时，人们通常使用微分学。这对于复杂模型的优化非常重要。导数测量给定参数的微小变化对输出的影响。在文献中，有一个代数模型来描述导数在规划语义中的作用。然而，在实践中，人们经常想使用一种叫做“反向微分”的东西，而不是普通的正微分。这就是神经网络训练中反向传播的例子。这里的目标是让输出中的每一个小变化都追溯到输入变量中的每一个小变化所产生的贡献。结果表明用逆导数比用正导数更有效更精确。我开发了一个代数模型，可以用来描述和研究涉及后向导数的计算机程序的语义。该模型还显示了后向导数和正向导数之间的关系。后向导数更强大，我的作品正好说明了在正导数上加什么才能得到后向导数。在当前的程序中，我想将这种形式主义与部分定义过程和对称的代数以及并行计算的代数结合起来，并将其与程序状态的公理化结合起来。

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.