# Interdisciplinary Mathematics Program

Mathematics has become increasingly interdisciplinary. On one hand, mathematics continues making fundamental contribution to traditional and new fields in science and engineering; on the other hand, emerging problems, new discoveries, and innovative technology from other disciplines bring enormous developments to contemporary mathematics.

The Interdisciplinary Mathematics (**I-Math**) program at UTC aims to meet the needs in research, education, and training at the
interface between mathematics and other scientific fields. The program encompasses
a breadth of applied and computational mathematics, statistics, biology, computer
science, and engineering. I-Math faculty work on a variety of research areas including,
but not limited to,

- Computational mechanics and engineering
- Data analytics
- Ecology, epidemiology, and microbiology
- Fluid dynamics
- Matrix theory and application
- Numerical analysis
- Statistics

A major goal of I-Math is to train undergraduate and graduate students with strong interdisciplinary knowledge and skills, and provide them with new education and research experiences that will promote their career development in the STEM fields. Toward this goal, we develop and implement innovative course modules, organize special colloquia and seminars for students with varied backgrounds, host summer camps for students interested in mathematics and its applications, engage math students in dialogues with their peers in other fields, and facilitate the joint supervision of students by a matched pair of faculty members (one from mathematics and the other from a partner discipline).

We encourage motivated students with interest in interdisciplinary mathematics to contact us for training and research opportunities. Meanwhile, we always welcome new collaborations from faculty and researchers throughout the UTC campus as well as from other institutions.

### I-Math Core Faculty

**Jin Wang**: Professor and UNUM Chair of Excellence in Applied Mathematics**Lani Gao**: Assistant Professor of Statistics**David Giles**: Assistant Professor of Biology**Yu Liang**: Associate Professor of Computer Science**Roy Liu**: Assistant Professor of Mathematics

The University of Tennessee at Chattanooga (**UTC**) invites high school students both within and outside Chattanooga area to apply for
the Interdisciplinary Mathematics (**I-Math**) Summer Day Camp, to be held on the UTC campus in July 2016. The one-week day camp
is designed to give high school students early exposure to more advanced mathematics
and its applications in various scientific fields, through an informal and fun learning
environment. The camp, led by UTC faculty from the departments of mathematics, biology,
and computer science, will enhance each student’s college-readiness and stimulate
their scientific and mathematical experience as they learn from university professors
and work on projects with groups of peers.

Please see this webpage for more information.

**Dr. Lani Gao**, University of Tennessee at Chattanooga

Schedule: Friday, April 8, 2016, EMCS 422, 2:00-3:00pm.

Title:*Classification of Cancer Types Using Neural Networks and Cross-Species Genomic Data*

Abstract: Microarray based gene expression profiling has been emerged as an efficient technique for cancer classification, as well as for diagnosis, prognosis, and treatment purposes. The aim of this study is to develop a method of detecting the closest animal model of human cancer types based on the gene expression profiling of human and mouse using artificial neural networks(ANNs). First we trained the neural network using gene expression profiling of mouse for classification of four distinct types of Medulloblastoma (a type of brain cancer). Then mapping procedure between animal and human genes is performed to match the orthologous genes (genes in different species that originated from a single gene of the last common ancestor) between human and mouse. After that we apply the ANN model to human expression data to detect the cancer types of 106 human tumor samples. To test ability of the ANN model, a cross validation using human gene expression data is performed. The classification result by ANN model is also compared to other typical statistical method such as ANOVA, logistic regression and the performance of these methods will be discussed. This study demonstrates the potential application of cancer classifications thus it will lead to improvements in early cancer diagnosis and in giving effective chemotherapy treatment.**Dr. Roy Liu**, University of Tennessee at Chattanooga Chattanooga

Conference: Southern Regional Algebra Conference, Auburn University, March 18-20, 2016.

Title:*Generalization of Some Inequalities for Matrix Exponentials to Lie Groups*

Abstract: Kostant’s pre-order on noncompact connected semisimple Lie groups is a generalization of log-majorization for matrices. We generalize some inequalities for matrix exponential to Lie groups in terms of Kostant’s pre-order.**Prof. Junping Shi**, College of William and Mary

Schedule: Friday, February 19, 2016, EMCS 422, 2:00-3:00pm

Title:*Pattern formation and bifurcations in reaction-diffusion-advection ecological models*

Abstract: Spatial-temporal patterns appear often in historical ecosystem data, and the cause of the patterns can be attributed to various internal or external forces. We demonstrate that in spatial ecological models, spatial-temporal patterns can arise as a result of self-organization of the ecosystem. By using bifurcation theory, we show that the spatial-temporal patterns are generated with the effect of diffusion, advection, chemotaxis or time delay.**Prof. Zhilin Li**, North Carolina State University

Schedule: Monday, February 8, 2016, EMCS 216, 2:00-3:00pm.

Title:*Modeling, Analysis, & Simulations of Free Boundary/Moving Interface Problems*

Abstract: Free boundary/moving interface problems are challenging both theoretically and numerically. In this general talk, I will introduce some application examples and corresponding differential equations models. The applications include Stefan problems of unstable crystal growth, drop spreading, and multi-phase flows. Then I will give a brief review of numerical methods for solving those challenging problems, particularly Cartesian grid methods such as Peskin's Immersed Boundary (IB) method, the Immersed Interface Method (IIM), Augmented IIM, and Immersed finite element method (IFEM) developed by myself. Another major component in solving free boundary/moving interface problems is how to evolve the interface. In our approach, both the front tracking and the level set methods are used. The level set method is simple and robust and can handle topological changes for any dimensions. I am going to discuss some issues about how to combine the level set method with IIM to achieve high order accuracy.**Dr. Roy Liu**, University of Tennessee at Chattanooga Chattanooga

Conference: The 5th International Conference on Matrix Analysis and Applications, Nova Southeastern University, December 17-20, 2015

Title:*Generalization of Golden-Thompson Type Inequalities for Normal Matrices*

Abstract: We obtain some Golden-Thompson type log-majorization relations for normal matrices, which are generalizations of the Golden-Thompson inequality, the Araki-Lieb-Thirring inequality, the Heinz inequality, and the Bernstein inequality, respectively.**Prof. Lili Ju**, University of South Carolina

Schedule: Friday, November 6, 2015, EMCS 422, 2:00-3:00pm.

Title:*Centroidal Voronoi Tessellations: Theory, Algorithms and Applications*

Abstract: Centroidal Voronoi tessellation (CVTs) are special Voronoi tessellations having the property that the generators of the Voronoi tessellations are also the centers of mass, with respect to a given density function, of the corresponding Voronoi cells. The CVT methodologies produce high-quality point distributions in volumes/surfaces or within sets of discrete data. CVTs enjoy an optimization characterization so that they turn out to be very useful in many scientific and engineering applications such as quantization and data analysis, image processing, mesh generation, geometric modeling, resource optimization, network design and control, cell biology and physics, model reduction, numerical partial differential equations and so on. This talk will give a brief review on the theory, algorithms and applications of CVTs.

**Prof. Greg Baker**, Ohio State University

Schedule: Friday, October 23, 2015, EMCS 422, 2:05-3:00pm.

Title:*An Effort to Coordinate Conceptual Development in Math and Physics Education for Engineering Students*

Abstract: Despite great strides in teaching pedagogy in the sciences and engineering, demonstrable long-term success in student performance is difficult to find. It is possible that much of the difficulty in making substantial improvements in science and engineering education lies in the deteriorating skills of students in the use of mathematics. To some it seems that math and physics education, the core to a start in engineering education, has pursued studies to improve how the content is taught at the neglect of what content is taught. There is a pressing need for students to learn and understanding how to use mathematics in physics and engineering. From my experience in teaching ODE to engineering students, it is quite clear that students see their mathematical education as simply a vast collection of specific procedures, a view encouraged by math teachers, probably unintentionally. They have little ability to express ideas that arise in physics and engineering in mathematical terms, and then use math problem-solving skills to understand the consequences. The question raised here is whether better coordination of the content in first-year math and physics courses could improve student ability to use math in subsequent engineering courses. If this is so, then the mathematical content used in the physics course and how it is used must be documented before changes in the content in the math course can be planned. At the same time, the physics course could be changed to better illustrate and emphasize important mathematics concepts, helping students to appreciate what they need to know mathematically. This paper presents a first attempt to document the mathematical content in a typical first-year physics course.**Dr. Roy Liu**, University of Tennessee at Chattanooga

Schedule: Friday, September 18, 2015, EMCS 422, 2:05-3:00pm.

Title:*Some New Matrix Inequalities and Their Extensions to Lie Groups*

Abstract: There is a natural and deep connection between the theory of matrices and the theory of Lie groups. On one hand, closed subgroups of the general linear group are Lie groups, which makes it possible to extend matrix results to abstract Lie groups. On the other hand, a realization of abstract Lie group results in closed linear groups provides deeper understanding of matrix results. The extensions and realizations rely on the abundant structures of (semi-simple) Lie groups. They are the various Lie group decompositions, namely, Cartan decomposition, KAK decomposition, Iwasawa decomposition, Jordan decomposition, etc., which correspond respectively in matrix theory polar decomposition, singular value decomposition, QR decomposition, (complete multiplicative) Jordan decomposition, etc. In many cases, inequalities on trace, eigenvalues and singular values can be reformulated in terms of majorization or log majorization. With the help of Lie group decompositions, Bertram Kostant derived a notion of majorization in Lie groups (which can also be expressed in representation theory), whose realization in closed linear groups is exactly the usual log majorization. This makes is possible to extend matrix inequalities to Lie groups in terms of majorization. In this talk, the speaker will introduce some newly obtained results in this spirit.

This conference was held on March 5-6, 2016 on UTC campus, in honoring the career of Dr. Ronald L. Smith for his many important contributions in linear algebra and graph theory.

More than 70 researchers participated in the conference, and there were 44 presentations in total.

Please see the conference webpage for more information.

AMS – American Mathematical Society (http://www.ams.org)

ASA – American Statistical Association (http://www.amstat.org)

SIAM – Society for Industrial and Applied Mathematics (http://www.siam.org)

SMB – Society for Mathematical Biology (http://www.smb.org)

USACM - United States Association for Computational Mechanics (http://www.usacm.org)

UTC SimCenter – National Center for Computational Engineering (http://www.utc.edu/college-engineering-computer-science/research-centers/simcenter/)

UTC Research Dialogues (http://www.utc.edu/research-sponsored-programs/research-day.php)

UTC Provost Student Research Awards (http://www.utc.edu/research-sponsored-programs/funding-opportunities/internal-competitions.php#PSRA)

### We gratefully acknowledge grant supports from

- National Science Foundation
- UNUM Chair of Excellence Endowment Funds
- Tennessee Higher Education Commission (through CEACSE)