## Spring 2019

### Organizer

If you would like to give a talk or propose an activity, please email me.

### Meeting Times

Tuesday, 4:00–4:50pm, in Pearce 227.

### Schedule

 Date Speaker Title (click Title for Abstract or scroll down to find it) 1/8/2019 C-Y. Jean Chan (Organizer) Organizational Meeting at PE 227 1/15/2019 Xiaoming Zheng (CMU) A Mathematical Model of Cell Fate 1/22/2019 No Seminar No Seminar 1/29/2019 No Seminar No Seminar 2/5/2019 Meera Mainkar (CMU) The Matrix Exponential 2/12/2019 Jordan Watts (CMU) (Rescheduled on April 16) Introduction to Smootheology for Students 2/19/2019 David Anderson (Ohio State University) Department Colloquium 2/26/2019 Debraj Chakrabarti (CMU) Several Complex Variables Are Better Than Just One 3/52019 Spring Recess (No Seminar) Spring Recess (No Seminar) 3/12/2019 Adam Moreno (University of Notre Dame) The Boundary Conjecture for Leaf Spaces 3/19/2019 Pin Hung (Chris) Kao (Flagler College) (Canceled) Almost-Prime Values Of Reducible Polynomials At Prime Argument 3/26/2019 Adam Mock (School of Engineering & Technology, CMU) Electromagnetic Eigenmodes of Parity-time Symmetric Periodic Materials Characterized Using Heesh-Shubnikov Group Theory 4/2/2019 Arkabrata Ghosh (CMU) Bezout's Theorem And Applications 4/9/2019 Kati Moug (CMU) Power Domination and Vertex Sum Graphs 4/16/2019 Jordan Watts (CMU) Introduction to Smootheology For Students 4/16/2019 Nancy Matar (CMU) Canceled 4/23/2019 Krishna Graikipati (University of Michigan) Department Colloquium (Special Colloquium in Mathematics Area)

### Abstracts

Speaker: C-Y. Jean Chan (January 8)
Title: Organizational Meeting
Abstract: Students who register MTH 693 for the current semester, and those who are planning or interested in doing so, should come to PE 227 for the course policy and essential guidelines for the assignments.

Speaker: Xiaoming Zheng (January 15)
Title: A Mathematical Model of Cell Fate
Abstract: This talk presents how mathematical analysis is useful in analyzing the cell fate. In many developing biological tissues, an alternating salt-and-pepper pattern of cells is formed from an initially uniform cell type. Two proteins, Delta and Notch, plays a critical role in generating this structure by switching neighbor cells into opposite cell fates. A very simple model from Sprinzak et al, Nature 2010, was developed to study this mechanism. We will present steady state and sensitivity analysis of this model to investigate under what conditions the cell can develop a switch of cell fate. This is a joint work with Matt Moore (a CMU undergraduate student) and Yican Zhang (a high school student).

Speaker: Meera Mainkar (February 5)
Title: The Matrix Exponential
Abstract: The exponential of a matrix plays a crucial role in the theory of Lie groups. In this talk, we will see some basic properties of the matrix exponential and the matrix logarithm.

Speaker: David Anderson (February 19)
Title: Department Colloquium

Speaker: Debraj Chakrabarti (February 26)
Title: Several Complex Variables Are Better Than Just One
Abstract: We examine a few ways in which the theory of holomorphic functions of several complex variables is different from the theory in one variable studied in 636 and 637.

Title: The Boundary Conjecture for Leaf Spaces
Abstract: The boundary conjecture asks "Is the boundary of an Alexandrov space itself an Alexandrov space?" Attacking this problem is messy in general. However, quotients of singular Riemannian foliations (with closed leaves), called leaf spaces, are a particularly nice type of Alexandrov space with a more approachable geometry. In this talk, we will use this geometry to prove the boundary conjecture for this special case. While the tools used are a bit specialized, the story and flow of the proof is very intuitive and should be accessible to graduate students.

Speaker: Pin Hung (Chris) Kao (March 19)
Title: Almost-Prime Values Of Reducible Polynomials At Prime Argument (This event was canceled.)
Abstract: We adopt A. J. Irving's double-sieve method to study the almost-prime values produced by irreducible polynomials and products of irreducible polynomials evaluated at prime arguments. For the first part of the talk, we will provide a historical background on this classical problem in additive number theory. For the second part of the talk, we will discuss the main idea behind the double-sieve method and the improved results from their classical counterparts.

Title: Electromagnetic eigenmodes of parity-time symmetric periodic materials characterized using Heesh-Shubnikov group theory
Abstract: In the past twenty years, it has been recognized that physical systems possessing balanced regions of wave absorption and amplification possess interesting mathematical and physical properties. This presentation will discuss these properties within the context of electromagnetics. I will begin with a brief overview of electricity and magnetism. A mathematical description of absorption and amplification of electromagnetic waves will follow. The concept of balanced absorption and amplification will then be discussed in the context of Parity-Time symmetry, and it will be shown how group theory techniques that incorporate both spatial and color symmetry can be used to understand and predict the behavior of complicated Parity-Time symmetric electromagnetic systems.

Speaker: Arkabrata Ghosh (April 2)
Title: Bezout's Theorem And Applications
Abstract: Algebraic curves plays a central role in Algebraic Geometry that concerns the zero set of polynomials. A general question asks whether there exists a method or algorithm to determine the number of intersection points (counted with multiplicities). If we consider the spaces over $\mathbb{R}$ or $\mathbb{C}$, there exist examples such that, in these spaces, one can have two curves where they have less number of intersection points than the product of the degree of the corresponding polynomials defining the curves. Then Bezout's theorem plays a significant role in solving the possible inconsistency. Bezout's Theorem says that if you have a homogeneous polynomial $f$ with degree $m$ and a homogeneous polynomial $g$ of degree $n$ in the complex projective plane $\mathbb{CP}^{2}$, then we have exactly $mn$ points of intersection counted according to the multiplicity. I will prove the theorem and will compute some examples of this theorem.

Speaker: Kati Moug (April 9)
Title: Power Domination and Vertex Sum Graphs
Abstract: An electrical power grid can be represented as a set of vertices and edges, where electrical substations are vertices, and the power transmission lines that connect them are edges. Within the electrical grid, it is important to monitor changes in variables such as voltage and current. The problem of placing observational devices, called Phase Measurement Units, throughout a graph, so that all electrical substations (vertices) are observed, is called the power domination problem. One of the main goals in this area is to find a minimum cardinality power dominating set. In this talk, I will discuss vertex sums, a particular kind of graph product that allows us to break large graphs into smaller components. By finding power dominating sets for the subgraphs, under specific conditions, we will be able to find power dominating sets for the vertex sum supergraphs, as well.

Speaker: Jordan Watts (April 16)
Title: Introduction to Smootheology for Students
Abstract: You have been doing calculus for a long time now. Why? What is this thing called a derivative and what business does it have occupying so much of your life?
In this talk, I will give an answer to this question, which will help make a connection between calculus and linear algebra that you probably have seen hints of in past classes, but never quite knew why. But then, after building up some intuition as to what derivatives are, I will smash this intuition with some strange examples. These strange examples, though, show up naturally in various areas of math, and mathematicians over the last century have invented various methods of dealing with them, and getting their derivatives back. I'll discuss some of these, and time-permitting how we are getting glimpses of something deeper hiding behind many of these methods.

Speaker: Krishna Graikipati (April 23)
Title: Special Department Colloquium in Mathematics