This site is devoted to mathematics and its applications. Created and run by Peter Saveliev.

# Peter Saveliev's Academic Portfolio

### From Mathematics Is A Science

As of 2011...

## Contents

## 1 Bio

Dr. Saveliev is a professor of mathematics at Marshall University since 2002.

Prior to graduate school, he worked as an applied mathematician and software engineer at a large R&D institute in Moscow, Russia. At the time he developed numerical methods of image matching for missile navigation/guidance systems. He also worked at a start-up company on an automated fingerprint identification system.

Since his mathematics Ph.D. from the University of Illinois at Urbana-Champaign, he has conducted research in algebraic topology and several other fields. During this time a transition to applied mathematics ha taken place. The new areas of interest are

## 2 Teaching and Advising

### 2.1 List of courses

Courses taught and new courses developed during the review period

Fall 2011

- MTH 450 Modern Algebra I -- Fall 2011
- MTH 335 Differential Equations -- Fall 2011
- MTH 130 College Algebra -- Fall 2011

Spring 2011

- MTH 331. Linear algebra, TeX lecture notes
- MTH 225. Introductory statistics (first time)

Fall 2010

- MTH 691. Differential geometry (new course), TeX lecture notes
- MTH 229. Calculus with Analytic Geometry I, TeX lecture notes

Fall 2009 - Spring 2010

- MTH 430-431, 630-631 Topology I and II (new course), TeX lecture notes
- MTH 427-428, 527-528 Advanced Calculus I and II, TeX lecture notes

Spring 2009

- MTH 203 Calculus for Business
- MTH 229 Calculus I

Spring 2008

- MTH 229 Calculus I
- MTH 130 College Algebra.
- New course developed The Mathematics of Computer Vision (canceled)

Fall 2007

- MTH 229 Calculus I
- HON 396 Problem Solving in Sciences and Engineering, 8 students

Spring 2007

- MTH 203 Calculus for Business
- MTH 231 Calculus with Analytic Geometry III

Fall 2006

- MTH 335 Differential Equations
- MTH 127 College Algebra.

Spring 2006

- MTH 335 Advanced Calculus II
- HON 396 Problem Solving in Sciences and Engineering (new course) – seminar, 9 students

### 2.2 Teaching with technology

During my lectures, I write on the screen of a tablet PC with Windows Journal and a digital pen and the writing instantly appears on the screen for the students to see. I started doing this in 2009 and I really love the results: bright, colorful slides like in PowerPoint but with the spontaneity and flexibility of the chalkboard.

Unlike with the chalkboard,

- I write in bright blue on white, with other colors used for emphasis;
- I draw a lot of colorful illustrations;
- I insert images, run programs (Excel etc), go to the internet;
- I don’t have to deal with chalk everywhere.

Unlike with PowerPoint,

- I don't need to type the text (including the formulas) for the slides;
- I don't risk going too fast;
- I can change the lecture plan at any time.

Once the class is over, the whole lecture is immediately published in the pdf format on the class web page.

### 2.3 Online textbook

The content comes directly from my lectures. I transcribe the lectures into TeX, put them in articles on my site, and insert the illustrations as images.

The wiki format was chosen for its simplicity. There is no intent to make this site, which is an online textbook, match Wikipedia's goals of completeness, definitiveness, etc. On the other hand, "trivial" details, "elementary" examples, "easy" exercises, etc are never omitted. Indeed, the source of articles is always a *complete* set of lectures. In fact, not only the set is complete, each lecture is complete too -- everything that appears on the board is included in the articles.

This requires a multistage procedure:

- prepare for the lecture by creating the initial notes;
- deliver the lecture (with a tablet) which always ends up very different from either the textbook or the notes;
- save the notes as a Journal file (and also in pdf, for the class page);
- edit the Journal notes: rewrite in full sentences to make the text more readable, clean up illustrations, proofread;
- transcribe the notes into TeX, put the result online with illustrations inserted;
- edit this text: make it more narrative, improve formatting, add links, proofread again.

Some statistics:

- over 850 articles,
- over 2100 illustrations,
- 7 one-semester courses,
- about 1000 daily visitors.

I think this approach has significant advantages over the common practice of posting video lectures online: read-ability, search-ability, edit-ability, link-ability, the speed of download, etc. As a bonus, I can rearrange the content into any course I like.

### 2.4 Teaching the discrete

The data nowadays comes in a digital format. This data is simply large tables of numbers. For example, it might come from either:

- experimental data: thousands of experiments each producing a list of hundreds of measurements, or
- digital images: 2d and 3d arrays of pixels each with a different vector value.

Sometimes there is a continuous function or process behind the numbers but often there isn't. The issues one has to deal with are the same though: max/min, increasing/decreasing behavior, rate of change, lengths, areas, volumes, etc.

What is commonly done is to go back to continuous functions via approximation, interpolation, curve fitting, etc. This approach introduces errors in your analysis and to handle those errors one needs to learn even more advanced math. Meanwhile, a typical graduate has never seen discrete functions and now he doesn't recognize that what he does with Excel every day has something to do with what he learned in his calculus class.

The bottom line is: the numerical aspect should be built in the math that we teach.

The answer to how this can be done is known: discrete differential forms and discrete exterior calculus. I have been trying to introduce discrete forms in a gentle manner in some of my courses, such as Differential Geometry (2010), Differential Equations (2011), as well as students' research.

### 2.5 Students' research supervision

Students research projects during the review period:

- Fall 2011:
*Modeling vector fields with discrete exterior calculus*by George Chappell, senior capstone project. - Summers 2010-2012: NSF grant
*"REU Site: Computational Science Training at Marshall University for Undergraduates in the Mathematical and Physical Sciences"*. The Departments of Mathematics, Physics, and Chemistry at Marshall University jointly host twelve students for ten weeks of instruction and research in computational science.- Julie Lang,
*Modeling Heat Transfer on Various Grids with Discrete Exterior Calculus*; - Misha Dowd,
*Evaluating the quality of image-to-image search*; - James Molchanoff,
*3D image analysis*; - Joseph Snyder,
*The topology of data*.

- Julie Lang,
*Using MATLAB to process images for the analysis of plant organ growth and curvature*, Harrison, Marcia; Silver, Donald; Saveliev, Peter; Sarra, Scott, 19th International Conference on Arabidopsis Research, Montreal, July 2008.- Fall 2007: HON 396 Problem Solving in Sciences and Engineering.
- Maggie Chenoweth,
*JAVA Based Robotic Vision*; - Kevin Leatherman,
*Laminar Flow Over a Flat Plate With MATLAB*; - Tyson Lipscomb,
*Digital Image Analysis with Medical Images*; - Clyde Meador,
*Guitar Chord Calculator Prototype Using C++*; - Shawn Larkin,
*Motion analysis with Matlab*; - Zachary T. Tackett,
*Development of a Phylogenetic Analysis Software for the Evaluation of Genetic Data Derived Using the Triangulation Identification for Genetic Evaluation of Risk (TIGER)*.

- Maggie Chenoweth,
- Summer 2007: Research for the US Navy grant
*Autonomous maritime navigation*. Three Marshall University computer science students and faculty worked on the project. - Fall 2006: HON 396 Problem Solving in Sciences and Engineering.
- Lindsay Dale, Kevin Forget, Amber Lilly,
*Virtual Stock Broker*; - John Stonestreet, Charlie Lowe,
*Geometric Study of the Pattern of Microscopic Hair Development on the Wings of the Mutated Fruitfly*; - Jared Marsh, Camden Clutter,
*Effects of Ethylene on Stem Gravitropic Curvature*; - Shane Gillies; Richard Kuykendall,
*Blended Learning Model And Implementation of An Individual Diagnosis Tool*.

- Lindsay Dale, Kevin Forget, Amber Lilly,

## 3 Research and Scholarly Activity

### 3.1 Description

During the review period I have transitioned from pure to applied mathematics.

Since my PhD and until about 2006 my main interest was in fixed point theory. This theory is a part of algebraic topology concerned with what amounts to the solvability problem for topological maps. Even though it has applications is differential equations, optimization, and game theory, this theory is firmly in the pure mathematics realm. After publishing a paper on applications of fixed point theory in control I became more and more interested in the areas of pure mathematics with more immediate consequences in sciences and engineering.

Algebraic topology has a long history in pure mathematics. However, its applied and computational aspects have been dormant until the 1990s. I attended the *Computational topology* short course in 2004. It was run by Herbert Edelsbrunner and John Harer in the Institute for Mathematics and Its Applications located on campus of the University of Minnesota. After the course I became convinced that algebraic topology would have serious industrial applications. It took me another year to find my personal interest - digital
image analysis, and later also data analysis. The main tool used in computational topology is homology theory and its extension specifically designed for applications that face uncertainty and noise, persistent homology.

Applications of these ideas in digital image analysis has lead to the creation of my own image analysis algorithm and, eventually, to the development of Pixcavator, image analysis software. The purpose of this software is to help researchers and engineers without a significant computing background solve mathematical problems they encounter when dealing with the deluge of digital images from their microscopes and cameras. These problems are mainly concerned with counting: blood cells, seeds, fungi, fiberglass particles, even stars; and measuring: cancer tumor, blood vessel, vegetation coverage, or just fish.

There have been about 25,000 downloads every year. Pixcavator is not only "as simple as a calculator", it's also as universally applicable as a calculator. It is not designed for any particular field of science or a particular type of images. It's based on math and math only. That's what makes Pixcavator as broadly used as it is. Pixcavator has been used in both industrial and academic environment.

Pixcavator has been used by such corporations as Alcoa, BASF, Ferris Manufacturing Corp., Hunter Fan Company, Landis International Inc., Owens Corning, Siemens Medical, Weber Metals Inc., and many others.

There are 30 research publications that credit Pixcavator that I am aware of. Some of the title of the journals are: Cancer Research, Journal of Chromatography, Research in Vision and Ophthalmology, Catalysis Letters, Colloids and Surfaces, The Plant Journal, Journal of Alloys and Compounds, The Minerals, Metals & Materials Society, Breast Cancer Research and Treatment, Pharmacy&Technology, Pflügers Archiv European Journal of Physiology, Geotechnical Engineering, Clinical Biochemistry and Nutrition, Diabetes, Applied Physics, Polymer Science, Advanced Biotechnology, Skin Research and Technology, Physics Procedia, American Society of Agricultural and Biological Engineers.

Another interest of mine is using discrete exterior calculus and discrete differential forms for modeling physical and other processes. The approach follows the intriguing idea that calculus is topology.

The research is presented in full in website Computer Vision and Math (inperc.com).

### 3.2 Publications

Peer reviewed:

*Robustness of topology of digital images and point clouds*, 23rd Canadian Conference on Computational Geometry, 2011, pp. 411-416.*A graph, non-tree representation of the topology of a gray scale image*, in Image Processing: Algorithms and Systems IX, edited by Jaakko T. Astola, Karen O. Egiazarian, Proceedings of SPIE Vol. 7870 (SPIE — The International Society for Optical Engineering, Bellingham, WA 2011) 78700O, O1-O19.*Image analysis for high content screening with cellAnalyst*(with J. Meyer and A. Pahwa), IEEE International Conference on Bioinformatics and Bioengineering, Institute of Electrical and Electronics Engineers, June 2009.*Topology based method of segmentation of gray scale images*, Proceedings of International Conference on Image Processing, Computer Vision, and Pattern Recognition, pp. 620-626, 2009.*A topological approach to cell counting*(with A. Pahwa), Proceedings of Workshop on Bio-Image Informatics: Biological Imaging, Computer Vision and Data Mining, 2008, Center for Bio-Image Informatics, University of California - Santa Barbara, USA, January 17-18, 2008.*Applications of Lefschetz numbers in control theory*, SIAM Journal of Control and Optimization, Society for Industrial and Applied Mathematics, 44 (2005) 5, 1677-1690.*Higher order Nielsen numbers*, Fixed Point Theory and Its Applications, 2005:1 (2005) 47-66.*Removing coincidences of maps between manifolds of different dimensions*, Topological Methods in Nonlinear Analysis, 22 (2003) 1, 105-114.*Lomonosov's invariant subspace theorem for multivalued linear operators*, Proceedings of the American Mathematical Society, 131 (2003) 3, 825-834.*Lefschetz coincidence theory for maps between spaces of different dimensions*, Topology and Its Applications, 116 (2001) 1, 137-152.*Fixed points and selections of set valued maps on spaces with convexity*, International Journal of Mathematics and Mathematical Sciences, 24 (2000) 9, 595-612.*A Lefschetz-type coincidence theorem*. Fundamenta Mathematicae, 162 (1999) 1-2, 65-89.*Dissipativity in the plane*, Russian Academy of Sciences. Sbornik. Mathematics, 75 (1993) 1, 125-135.*The dissipativity of the generalized Lienard equation*, Differential Equations, 28 (1992) 6, 794-800.*On Poincare-Bendixson theorem and dissipativity on a plane*, Moscow University Mathematics Bulletin, 46 (1991) 3, 54-55.*The dissipativity of systems of ordinary differential equations in the plane*, Moscow University Mathematics Bulletin, 43 (1988) 4, 93-96.

Pending:

*A graph representation of the topology of color images*, accepted to Proceedings of SPIE.*Topology based method of partition, analysis, and simplification of dynamical images*, US patent application, August 2006, App. No. 20070036434.

In preparation:

Other:

*Fixed Points and Coincidences*, Ph.D. Thesis. University of Illinois at Urbana-Champaign, 1999.*Accuracy estimation of the multilateral scheme*(with N. Chmutin), Raketno-Kosmicheskaya Tekhnika (Rocket and Space Technology), 6 (1989) 3, 62-65.

### 3.3 Conferences and workshops

Conferences and workshops attended with talks indicated

- 23rd
*Canadian Conference on Computational Geometry*, Toronto, Canada, August 2011. (Talk) *Computational topology*short course conducted by the American Mathematical Society, New Orleans, January 2011.*Ohio Valley Affiliates for Life Sciences*(OVALS) conference, Louisville, KY, April 2010.*Joint AMS-MAA Mathematics Meeting*, Washington, DC, January 2009.*International Conference on Image Processing, Computer Vision, and Pattern Recognition*, Las Vegas, Nevada, July 2009. (Talk)- IMA New Directions Short Course
*Applied Algebraic Topology*at the Institute for Mathematics and Its Applications, University of Minnesota, June 2009. *Bio-Image Informatics Workshop*, Santa Barbara, CA, January 2008. (Poster)- National meeting of the
*American Society for Cell Biology*, Washington, DC, November 2007. - Workshop
*Applications of topology in science and engineering*, at Mathematical Sciences Research Institute, Berkeley, CA, September 2006. (Talk*Homology of color images*).

### 3.4 Consulting

Consultations on how to solve specific image analysis problems with image analysis software:

- 2010:
- Matt Giovanni, School of Natural Resources, University of Nebraska-Lincoln
- Robert Lee, President, Accushape, Inc.
- Mike Irish, Delphi Corp
- Doug Johnson, Cambria Corporation

- 2009:
- Michael Peterson, GoDigital Media Group
- Gregory Trude, Koehler-Bright Star, Inc.
- Craig A. Downs, Haereticus Environmental Laboratory
- Alexander V. Ruban, School of Biological and Chemical Sciences, Queen Mary, University of London
- Eduardo Nicolas Schulz, Instituto de Ingeniería Electroquímica & Corrosión, Argentina
- Richard Molnar, MIT Lincoln Laboratory
- Christopher L. Schwab, Industrial Hose Solutions
- Matt Giovanni, School of Natural Resources, University of Nebraska-Lincoln

- 2008:
- Don Wise, Owens Corning Science & Technology Center
- Dr. Nikolay Makarenko, Pulkovo Observatory, Russian Academy of Sciences
- Carl Schwarz, Osteotech Inc.
- David Oliver, Landis International Inc.
- Dr. Nalin Mehta, Poudre Valley Hospital, Fort Collins, CO
- Dr. Joy Mammen, Henry Ford Health System
- John Sherwood, Zerosum Inc.

## 4 Service to University and Community

### 4.1 Service to Marshall University

2011:

- Member of the Executive Committee of the Mathematics Department.
- Member of the Lecture and Colloquium Committee of the Mathematics Department.
- Organizing and assisting with SCORES, Marshall Mathematics Field Day, etc.

2010:

- Member of the Undergraduate Committee of the Mathematics Department.
- Member of the Executive Committee of the Mathematics Department.

2009:

- Member of the Undergraduate Committee of Mathematics Department.
- Organizing and assisting with SCORES, Marshall Mathematics Field Day, etc.

2008:

- Member of the Faculty Senate.
- Member (and the Senate liaison) of the University Research Committee.
- Member of the Undergraduate Committee of Mathematics Department.

2007:

- Member of the Faculty Senate
- Member (and the Senate liaison) of the University Research Committee.
- Member of Faculty Search Committee of Mathematics Department.
- Member of the Undergraduate Committee of Mathematics Department.
- Organizing and assisting with SCORES, Marshall Mathematics Field Day, etc.
- March 2007. Talk,
*Elementary Computer Vision*, PiMuEpsilon, Marshall University.

2006:

- Member of the Undergraduate Committee of Mathematics Department.
- Organizing and assisting with SCORES, Marshall Mathematics Field Day, etc.
- Talk, Low level vision through topological glasses, Mathematics Colloquium.

### 4.2 Service to the science community

- 7 reviews for
*Mathematical Reviews*published by the American Mathematical Society. - 2006: referee for
*Topology and Its Applications*. - 2007: review of a grant proposal for the Department for Natural and Technical Sciences of the Austrian Science Fund.
- Web site
*Computer Vision and Math*: 850 articles with 2100 illustrations, about 1000 daily visitors. *Pixcavator*image analysis software: about 25,000 yearly downloads.*cellAnalyst*image analysis software: about 500 registered users.- Membership in American Mathematical Society (AMS).

### 4.3 Grants

Grants and grant proposals during the review period

- 2009-2012: NSF grant
*REU: Computational Science Training at Marshall University for Undergraduates in the Mathematical Sciences and Physics*.

Over the summers of 2010–2012, the Departments of Mathematics, Physics, and Chemistry at Marshall University will jointly host twelve students for ten weeks of instruction and research in computational science. Each student will extend some carefully-selected and delimited aspect of his or her mentor’s research. Weekly informal meetings will be held to discuss progress and problems. Students will present the results of their research in a symposium to conclude the summer program; afterwards, they will present their research at an appropriate professional conference and will receive co-author credit for any eventual publication. In addition to performing research in a specific area, students will be instructed in practices and issues that are common to all areas of computational science.

- 2008: NSF grant proposal
*REU: Computational Science Training at Marshall University for Undergraduates in the Mathematical Sciences and Physics*.

- 2007. Research for grant for the US Navy -
*Autonomous maritime navigation*.

Covered by Huntington *Herald-Dispatch*. Marshall U Team Works on Computer Vision Navigation.

Three Marshall University computer science students and faculty are working on a project to build a sensor suite for the United States Navy to be used on autonomous marine vehicles.

The work is being done under a $2.7 million contract from the Navy with Spatial Integrated Systems of Rockville, MD. Marshall University is a subcontractor on the project, according to a report in the Huntington, WV Herald Dispatch.

The goal of the Autonomous Maritime Navigation (AMN) project is to develop integrated hardware and software to enable ships to autonomously navigate in waterways.

The solution requires software-based data fusion from an array of sensors, including sonar, radar, GPS, and digital cameras, according to Venkat Gudivada, a professor of computer science at Marshall University's Huntington, WV campus.

The team is focusing on ways to generate 3D reference points using stereo vision to estimate the distance of obstacles, such as ocean vehicles and coastlines. The resulting system would constitute a form of computer vision that would enable a marine vehicle to steer itself clear, according to the researchers.

The three Marshall professors involved in the project are Gudivada; Joe Fuller, a professor of computer science; and Peter Saveliev, an associate professor of mathematics. The three Marshall computer science students on the project are Camden Clutter of Clarksburg, WV, Shawn Cotton of Huntington, and Brad Fitzwater of Eleanor, WV.

- 2006: grant proposal NSF
*BioMath*.