### Applied Mathematics Seminar

**Location:**Minard 308**Time**: Monday, 2:00pm-2:50pm**Organizer:**Indranil SenGupta (Spring) and Artem Novozhilov (Fall)

### Fall 2021 Schedule

##### November 1, 2021

**Nikita Barabanov**** **(Math, NDSU): Clifford algebra and Mobius addition

**Abstract:** We introduce the geometric algebra over vector spaces with quadratic forms, and corresponding Clifford algebra. Examples of this algebra include the algebras of complex numbers and quaternions. We consider the standard operations, including the twisted adjoint action, and their basic properties. Using these properties, it will be shown that the standard Mobius transformation of the unit ball represents a binary operation called Mobius addition that generates the Mobius gyrogroup. In particular, the Mobius gyrations will be represented in terms of twisted adjoint action.

##### October 11, 2021

**Nicholas Salmon**** **(Math, NDSU): Rough Paths and Rough Differential Equations

**Abstract:** Ever since Newton and Leibniz first described differential equations, they have been crucial to our understanding of the natural world and a source of great math research. A variety of ways to generalize the notion of solution to a differential equation to include non-differentiable functions have been studied. Recently a new technique, the theory of rough paths, has emerged and provided much insight into the nature of integration and differential equations. In this talk we aim to describe the framework of rough paths, see what new intuition can be gained from rough paths, apply rough paths to problems in stochastic analysis, and finally discuss some ways that rough paths have impacted real world problems.

##### September 27, 2021

**Artem Novozhilov**** **(Math, NDSU): On a special class of mixed functional differential equations and on one specific example from this class (Part II)

**Abstract:** My presentation will consist of two parts. In the first part I will discuss a special class of functional differential equations that got relatively little attention in the existing mathematical literature. I am fortunate enough to be familiar with Anatoly Myshkis, who systematically studied this class in the 90th of the previous century, so my talk will also contain some historical details. In the second part of my talk, I will present a recent mathematical model that is very close in its form to the equations discussed in the first part. This model was formulated by my former PhD advisor, Alexander Bratus and his recent PhD student, and some of the analysis of the model was also performed by Ivan Yegorov, and (to a lesser extend) by myself. The prerequisites for the talk will be kept minimal (it is useful to have some general ideas about dynamical systems, their equilibria, and stability), and I especially invite all the graduate students, since quite a few basic questions with respect to this class of differential equations are yet to be answered.

##### September 13, 2021

**Artem Novozhilov**** **(Math, NDSU): On a special class of mixed functional differential equations and on one specific example from this class (Part I)

**Abstract:** My presentation will consist of two parts. In the first part I will discuss a special class of functional differential equations that got relatively little attention in the existing mathematical literature. I am fortunate enough to be familiar with Anatoly Myshkis, who systematically studied this class in the 90th of the previous century, so my talk will also contain some historical details. In the second part of my talk, I will present a recent mathematical model that is very close in its form to the equations discussed in the first part. This model was formulated by my former PhD advisor, Alexander Bratus and his recent PhD student, and some of the analysis of the model was also performed by Ivan Yegorov, and (to a lesser extend) by myself. The prerequisites for the talk will be kept minimal (it is useful to have some general ideas about dynamical systems, their equilibria, and stability), and I especially invite all the graduate students, since quite a few basic questions with respect to this class of differential equations are yet to be answered.