### Global Injectivity, Manifold Estimation and Universality of Neural Networks

September 27, 4 p.m. - 5 p.m. , UP 117

Michael Puthawala, South Dakota State University

In recent years machine learning, and in particular deep learning has emerged as a powerful and robust tool for solving problems in fields ranging from robotics, to medicine, materials science, cosmology and beyond. As work on applications has advanced, so too has theory advanced to guide, explain, interpret deep learning. In this talk I will provide an overview on some of that theory in three parts. In the first, I will present a connection between injectivity of ReLU layers and vector geometry, which yields a simple criterion for a ReLU network to be end-to-end injective. Second, I will introduce the concept of universality in the context of neural networks and reveal a surprising connection to knot theory, which sheds light on what kinds of manifold-supported functions can be learned by a neural network. Finally, I will discuss a work that exploits the connections between topological covering spaces and locally bilipschitz maps to develop a recipe for constructing neural networks that can learn to approximate `topologically interesting’ maps between manifolds.

### Random variables for which the number operator algebra and Weyl algebra intersect nontrivially

October 2, 4 p.m. - 5 p.m. , UP 117

Aurel Stan, Ohio State University

We introduce first the quantum operators: creation, preservation, and annihilation, and the number operator generated by a random variable X having finite moments of all orders. Every linear operator T, from the space of polynomial random variables in X to itself, can be uniquely written as an infinite sum of terms of the form A_n(X)D^n, where A_n(X) is a polynomial in X, viewed as a multiplication operator and interpreted as a position operator, and D is the classic differentiation operator, interpreted as the momentum operator. We call this sum, the position-momentum decomposition of T. The Weyl algebra is the space of all linear operators T having a finite position-momentum decomposition. We show first that if a continuous random variable has the property that a non-constant polynomial function of its number operator belongs to the Weyl algebra, then its density function satisfies a first order linear equation, which we call the generalized Pearson equation. We apply then this equation to the case when the number operator is quadratic in D, showing that the random variable must be Gaussian or Gamma distributed.

Finally, we discuss the random variables for which the number operator satisfies a quadratic equation in D, proving that these random variables are beta distributed.

### Predicting Time to Failure, Even with Incomplete Data: Lifetime Data Analysis via Probability Functions

October 19, 6:30-8 pm, Farber Hall - 71st Annual Harrington Lecture

Yuhlong Lio (University of South Dakota)

### Building Thinking Classrooms

October 25, 4 p.m. - 5 p.m. , UP 117

Dan Van Peursem, University of South Dakota

Peter Liljedahl wrote a book “Building Thinking Classrooms in Mathematics” and it has gained a lot of traction in many high schools in South Dakota. This seminar will provide a hands on example of some of the components of this new teaching method, and we will discuss the rationale behind the structure. No need to bring anything, just an inquisitive mind.