Welcome to the Department of Mathematical Sciences
As our world becomes increasingly connected by the internet, math is more important than ever. It provides the basis of every technology we have and helps us plan the technology of the future.
At USD, our internationally recognized faculty represent a wide variety of research and teaching interests, including both pure and applied mathematics. As a mathematical sciences student, you'll benefit from small class sizes and hands-on research projects. You'll also have the opportunity to intern across the country or across the world, preparing you for a career in accounting, business, teaching and many more.
Join our community of researchers, future planners and problem solvers in the USD Department of Mathematical Sciences.
Meet the Department
Two students in the University of South Dakota Department of Mathematical Sciences, Oleksandra “Sasha” Lukina and Olivia Roberts, were accepted to attend the BRING MATH (Bridges for the Next Generation: Mathematical Science Research and Our Future) Workshop at Argonne National Laboratory in Lemont, Illinois, Oct. 5-6.
For 2023 National Teacher of the Year and University of South Dakota alumna Rebecka Peterson ‘11, being an educator is about more than teaching – it’s about sharing stories.
The University of South Dakota Department of Computer Science is set to host its third annual Artificial Intelligence (AI) Symposium on March 28 from 9 a.m. – 3:10 p.m. in person in Farber Hall, located inside Old Main, and livestreamed online.
Known For Excellence
Our professors are internationally recognized for their research, making them ideal mentors for students interested in ground breaking research.
Get a personalized education with small class sizes.
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.
Departments & Facilities
Merten Hasse Mathematics Contest
The Department of Mathematical Sciences annually hosts the Merten Hasse Math Competition for area middle school and high school students. Students can qualify for several prizes, including a scholarship to USD.