# Vanderbilt Math Research Paper

### SHOULD YOU MAJOR IN MATH?

Some students who are good at math and enjoy solving math problems don’t seriously consider majoring in the subject because they are unsure of how a math major could be of use to them.

One common misconception is that a major in math is of no use unless you plan to teach math. But the truth is that there are a multitude of interesting and rewarding jobs for people with degrees in math. We’ll list some of those jobs in a moment, but first let’s consider what math can do for you regardless of your career choice.

By studying math you develop analytical skills and an analytical attitude. You learn to pay attention to all the assumptions involved in a given problem or situation, and you learn to break down a complicated problem into a series of tractable steps. You develop the habit of critical thinking: testing your conclusions-and the conclusions of others-to make sure they’re based on adequate data and accurate reasoning.

Such skills and attitudes are highly valued by employers as well as graduate and professional schools (architecture, engineering, business, law, medicine, pharmacy, etc.). A degree in mathematics, especially if accompanied by a transcript showing good grades in math courses, signals to a prospective employer or admissions officer that you are capable of learning the kinds of analytical procedures that are required in a given job or profession, even if those procedures are not specifically mathematical.

Employers recognize that if you can do math, you can do any job that calls for precise analysis and careful deduction.

The website http://weusemath.org/ provides very useful information about career perspectives for math majors.

WHO HIRES MATH GRADUATES?

People with math degrees are in demand in the following places:

- Accounting firms
- Actuarial firms
- Airlines and other transportation companies
- Banks and other financial institutions
- Chemical and pharmaceutical manufacturers
- Computer manufacturers
- Consulting firms
- Educational institutions
- Engineering firms
- Government agencies
- Health services
- Insurance agencies
- Insurance companies
- Investment firms
- Real estate firms
- Research and development firms
- Securities firms
- Software development firms
- Technical publishing firms
- Telecommunications companies
- Textbook publishing companies
- Utility companies

In government, education, manufacturing, and service industries of many kinds, math majors get good jobs.

### JOB TITLES COMMONLY HELD BY MATH GRADUATES

- Accountant
- Actuary
- Auditor
- Biostatistician
- Budget Analyst
- Cartographer
- Compensation/Benefits Specialist
- Computer Applications Specialist
- Computer Programmer
- Computer Systems Analyst
- Credit Analyst
- Cryptographer
- Data Analyst
- Data Base Manager
- Data Processing Manager
- Economic Analyst
- Financial Analyst
- Financial Consultant
- Financial Manager
- Financial Services Sales Representative
- Information Systems Analyst
- Insurance Agent
- Insurance Underwriter
- Inventory Control Specialist
- Investment Manager
- Loan Counselor/Officer
- Market Research Analyst
- Meteorologist
- Operations Research Analyst
- Production Planner
- Professor
- Psychometrician
- Purchasing Manager
- Real Estate Appraiser
- Risk Manager
- Securities Analyst
- Statistician
- Surveyor
- Teacher
- Traffic Technician
- Urban Planner

### FURTHER INFORMATION

For more information, or to set up an appointment, contact the Math Department’s Director of Undergraduate Studies.

*Updated GS, September 2011*

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Nonnegativity constraints for spanning systems, (joint with A.H.Spaeth), Transactions of the AMS, 368 (2016), 5783-5806.

Fusion frames and randomized subspace actions, (joint with X. Chen), Constructive Approximation 43 (2016), 103-134.

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Perturbations of measurement matrices and dictionaries in compressed sensing, (joint with A. Aldroubi and X. Chen), Applied and Computational Harmonic Analysis, 33 (2012), no. 2, 282-291.

A note on finite dual frame pairs, (joint with O. Christensen, X.C. Xiao), Proceedings of the AMS, 140 (2012), no. 11, 3921-3930.

Quantization and finite frames, (joint with R. Saab and O. Yilmaz), Chapter 8 (pages 267-302) in "Finite frames" (edited by P. Casazza and G. Kutyniok), Birkhauser, Boston, 2012.

Coarse quantization for random interleaved sampling of bandlimited signals, (joint with J. Tanner, Y. Wang, O. Yilmaz), ESAIM Mathematical Modelling and Numerical Analysis, 46 (2012), 605-618.

Time-frequency concentration of generating systems, (joint with P. Jaming), Proceedings of the AMS, 139, (2011), 3279-3290.

Mean squared error bounds for the Rangan-Goyal soft thresholding algorithm, Applied and Computational Harmonic Analysis, 29, (2010), 251-271.

Sobolev duals in frame theory and sigma-delta quantization, (joint with J. Blum, M. Lammers and O. Yilmaz), Journal of Fourier Analysis and Applications, 16, (2010), 365-381.

Alternative dual frames for digital-to-analog conversion in Sigma-Delta quantization, (joint with M. Lammers and O. Yilmaz), Advances in Computational Mathematics, 32, (2010), 73-102.

Regularity for complete and minimal Gabor systems on a lattice, (joint with C. Heil), Illinois Journal of Mathematics, 53, (2009), 1077-1094.

Uncertainty principles for orthonormal sequences, (joint with P. Jaming), Journal of Functional Analysis, 243, (2007), 611-630.

Gabor Schauder bases and the Balian-Low Theorem, (joint with C. Heil), Journal of Mathematical Physics, 47, 113506 (2007), 21 pages.

An optimal example for the Balian-Low uncertainty principle, (joint with J. Benedetto and W. Czaja), SIAM Journal on Mathematical Analysis, 38, (2006), 333-345.

Recent developments in the Balian-Low Theorem, (joint with W. Czaja), in Harmonic Analysis and Applications (edited by C. Heil), Birkhauser, Boston, 2006.

Sigma-Delta quantization and finite frames, (joint with J. Benedetto and O. Yilmaz), IEEE Transactions on Information Theory, 52 (2006), 1990--2005.

An endpoint (1,infinity) Balian-Low Theorem, (joint with J. Benedetto, W. Czaja, J. Sterbenz), Mathematical Research Letters, 13 (2006), 467-474.

A (p,q) version of Bourgain's theorem, (joint with J. Benedetto), Transactions of the AMS, 358 (2006), 2489--2505.

Second order Sigma-Delta quantization of finite frame expansions, (joint with J. Benedetto and O. Yilmaz), Applied and Computational Harmonic Analysis, 20 (2006), 126--148.

Time-frequency mean and variance sequences of orthonormal bases, Journal of Fourier Analysis and Applications, 11 (2005), 375--387.

Orthonormalized coherent states, in Wavelets, Frames, and Operator Theory (C. Heil, P. Jorgenson, D. Larson, editors), AMS Contemporary Mathematics, 345 (2004), 283--297.

The Balian-Low Theorem and regularity of Gabor systems, (joint with J. Benedetto, W. Czaja, and P. Gadzinski), Journal of Geometric Analysis, 13 (2003), 239--254.

Sigma Delta quantization for compressed sensing, 44th Annual Conference on Information Sciences and Systems (Princeton, NJ, March 2010), joint with C.S. Gunturk, M. Lammers, R. Saab, O. Yilmaz.

Sobolev duals of random frames, 44th Annual Conference on Information Sciences and Systems (Princeton, NJ, March 2010), joint with C.S. Gunturk, M. Lammers, R. Saab, O. Yilmaz.

Estimation algorithms with noisy frame coefficients, invited, Wavelets XII (San Diego, CA, 2007), SPIE Proc., SPIE Bellingham, WA, 2007.

On quantization of finite frame expansions: sigma-delta schemes of arbitrary order, invited, (joint with M. Lammers and O. Yilmaz), Wavelets XII (San Diego, CA, 2007), SPIE Proc., SPIE Bellingham, WA, 2007.

Analog to digital conversion for finite frames, invited, (joint with J. Benedetto and O. Yilmaz), Wavelets XI (San Diego, CA, 2005), SPIE Proc., SPIE Bellingham, WA, 2005.

Sigma-Delta quantization and finite frames, invited, (joint with J. Benedetto and O. Yilmaz), Proceedings of ICASSP-2004, IEEE International Conference on Acoustics, Speech, and Signal Processing, Montreal, May 2004.

MRI signal reconstruction by Fourier frames on interleaving spirals, (joint with J. Benedetto and H.-C. Wu), Proceedings of the 2002 IEEE International Symposium on Biomedical Imaging, Washington, DC, July 2002.

Academia Sinica Institute of Mathematics (Taipei, Taiwan). Summer Course Co-Organizer: ``Mathematical signal processing and data analysis." July 6, 2015 - August 14, 2015.

AMS Short Course on Finite Frame Theory (San Antonio). Lecturer: ``Quantization for finite frames." January 9, 2015.

Academia Sinica Institute of Mathematics (Taipei, Taiwan). Summer Course Co-Organizer: ``Mathematical signal processing and data analysis." July 7, 2014 - August 15, 2014.

Technical University of Denmark (Lyngby, Denmark). Short Course: "Going digital with redundant signal representations." May 31 - June 4, 2010.

Hong Kong University of Science and Technology, Seminar. May 27, 2016.

Fudan University, International Workshop on Mathematical Aspects of Data Science. May 21, 2016.

Georgia Institute of Technology, Analysis Seminar. March 16, 2016.

National Central University (Jhongli, Taiwan). Seminar. June 24, 2015.

International Conference on Harmonic Analysis and Applications (New York, NY). June 2, 2015.

Kent State University, Colloquium. April 30, 2015.

Georgetown University, Colloquium. April 24, 2015.

AMS Central Spring Sectional Meeting (East Lansing, MI), Special Session on Approximation Theory in Signal Processing and Computer Science. March 14, 2015.

Colloquium, University of Michigan at Dearborn. March 13, 2015. Consistent Reconstruction and Randomized Recovery Algorithms.

Turfe Lecture, University of Michigan at Dearborn. March 12, 2015. Algorithms for Analog-to-Digital Conversion.

February Fourier Talks. February 19, 2015. Error Bounds for Consistent Reconstruction.

AMS Joint Mathematics Meetings, Special Session on Frames and Their Applications. January 10, 2015. Error bounds for consistent reconstruction.

International Conference on Learning and Approximation (Shanghai). December 9, 2014. Randomized subspace actions for fusion frames.

AMS Fall Eastern Sectional Meeting (Halifax, Canada), Special Session on Sampling Theory. October 18, 2014. Fusion frames and randomized subspace actions.

Taida Institute for Mathematical Sciences (Taipei, Taiwan), TIMS Seminar in Statistical Methodology. June 13, 2014. Algorithms for analog-to-digital conversion.

National University of Singapore, ACM Seminar. March 12, 2014. Algorithms for analog-to-digital conversion.

AMS Fall Central Sectional Meeting, Washington University at St. Louis, Special Session on Wavelets, Frames, and Related Expansions, October 18, 2013. Consistent reconstruction and random polytopes.

Academia Sinica, Institute of Mathematics (Taipei, Taiwan), Colloquium, July 25, 2013. Means and variances of bases and the uncertainty principle.

Universite Bordeaux, Analysis Seminar. April 29, 2013. Consistent reconstruction and some geometry of random polytopes.

Oberlin College. April 18, 2013. Recovering information from noisy measurements.

DARPA workshop on Big Data and Large Scale Analytics. March 20, 2013. Randomized algorithms for signal reconstruction.

University of Michigan, Analysis/Probability Learning Seminar. February 7, 2013. Consistent reconstruction and some geometry of random polytopes.

University of Michigan, Analysis/Probability Seminar. February 6, 2013. Consistent reconstruction and some geometry of random polytopes.

Probability Seminar. Academia Sinica Insitute of Mathematics (Taipei, Taiwan), June 25, 2012. Title: Compressed sensing and Sobolev duals of Gaussian random matrices.

Hong Kong Polytechnic University, Colloquium, May 16, 2012. Title: Consistent reconstruction and random polytopes.

Second Midwest Conference on Mathematical Methods for Images and Surfaces, Michigan State University, August 27, 2011. TItle: Consistent reconstruction and the geometry of random polytopes.

Research Seminar in Probability, Academia Sinica Insitute of Mathematics (Taipei, Taiwan), July 1, 2011. Title: Randomized solvers for linear systems with and without noise, Part II.

Research Seminar in Probability, Academia Sinica Insitute of Mathematics (Taipei, Taiwan), June 17, 2011. Title: Randomized solvers for linear systems with and without noise.

Fourth International Conference on Computational Harmonic Analysis, City University of Hong Kong, May 26, 2011. Title: Sigma-Delta quantization for compressed sensing measurements.

University of Wisconsin at Milkwaukee, Analysis Seminar, April 22, 2011. Title: Quantization for compressed sensing measurements.

INSPIRE 2010, Conference on Information Representation and Estimation, University College London, September 6, 2010. Title: Coarse quantization for random interleaved oversampling of bandlimited signals.

AMS-MAA Joint Mathematics Meetings, San Francisco, Special Session on Optimal Frames and Operator Algebras, January 16, 2010. Title: Regularity for exact Gabor systems on a lattice.

City University of Hong Kong (Hong Kong, SAR, China), Mathematical Analysis and its Applications Colloquium, October 28, 2009. Title: Rounding algorithms and digital representation.

Academia Sinica, Institute of Mathematics (Taipei, Taiwan), Probability Seminar, July 27, 2009. Title: Error bounds in the Rangan-Goyal estimation algorithm.

Strobl09 Conference on Time-Frequency, (Strobl, Austria), June 15, 2009. Title: Regularity for complete and minimal Gabor systems on a lattice.

Sampling Theory and Applications (SampTA), special session on "Sampling and quantization," (Marseille, France), May 22, 2009. Title: Noncanonical reconstruction for quantized frame coefficients.

Vanderbilt University, Workshop on "Constructive Approximation: City University of Hong Kong and Vanderbilt University Workshop," April 29, 2009. Title: Reconstruction algorithms for quantized data.

University of Minnesota, Applied Mathematics Seminar, April 23, 2009. Title: Reconstruction algorithms for quantized redundant representations.

Michigan State University, Applied Mathematics Seminar, April 10, 2009. Title: Reconstruction algorithms for quantized overcomplete signal representations.

Banff International Research Station (Banff, Canada), Workshop on ``Frames from first principles: error correction, symmetry goals, and numerical efficiency," March 17, 2009. Title: Error bounds for consistent reconstruction by soft thresholding.

University of Edinburgh (Edinburgh, Scotland), Colloquium, November 26, 2008. Title: Mathematical aspects of data representation: harmonic analysis and uncertainty principles.

University of Tennessee (Knoxville), Colloquium, October 3, 2008. Title: Rounding algorithms for redundant signal expansions.

Foundations of Computational Mathematics (FoCM) conference, workshop on Signal and Image Processing, City University of Hong Kong, June 25, 2008. Title: Consistent reconstruction from quantized frame coefficients.

Academia Sinica, Institute of Mathematics (Taipei, Taiwan), Colloquium, June 12, 2008. Title: Rounding algorithms for redundant signal expansions.

National Chung Hsing University (Taichung, Taiwan), Seminar, May 30, 2008. Title: Error bounds for Sigma-Delta algorithms.

Providence University (Taichung, Taiwan), Seminar, May 29, 2008. Title: Redundancy and digital signal representation.

Illinois/Missouri Applied Harmonic Analysis Seminar (Bloomington, IL), April 19, 2008. Title: Signal estimation from noisy frame coefficients.

Washington University at St. Louis, Colloquium, April 18, 2008. Title: Schauder bases and the Balian-Low uncertainty principle.

AMS Southeastern Meeting, Louisiana State University, March 29, 2008. Title: Schauder bases of translations and modulations.

SIAM Southeastern-Atlantic Section Conference, University of Central Florida, March 15, 2008. Title: MSE bounds for the RG-algorithm.

University of Michigan, Applied and Industrial Mathematics Seminar, October 19, 2007. Title: Rounding algorithms for redundant signal expansions.

University of Michigan, Applied and Industrial Mathematics Student Seminar, October 19, 2007. Title: Redundancy in analysis and signal processing.

AMS Western Section Meeting, University of New Mexico, October 13, 2007. Title: Error estimates for recursive consistent estimation.

SPIE, International Society on Optical Engineering, Wavelets XII, San Diego, CA. Title: Estimation algorithms with noisy frame coefficients, August 27, 2007.

International Conference on "Trends in Harmonic Analysis," Strobl, Austria, June 20, 2007. Title: "Signal estimation from noisy dense measurements."

National Taiwan University (Taipei, Taiwan), Department of Mathematics, June 1, 2007. Title: ``Quantization for overcomplete signal expansions.''

Chinese Academy of Sciences (Beijing, China), Institute of Computational Mathematics and Scientific/Engineering Computing, May 24, 2007. Title: ``The Balian-Low uncertainty principle for Gabor bases.''

Chinese Academy of Sciences (Beijing, China), Institute of Computational Mathematics and Scientific/Engineering Computing, Institute Seminar, May 22, 2007. Title: ``Quantization for overcomplete signal expansions.''

SIAM Southeastern-Atlantic Section Conference, University of Memphis, May 4, 2007. Title: ``Signal estimation from noisy dense measurements."

Vanderbilt University, Computational Analysis Seminar, January 24, 2007. Title: "Finding good dual frames for reconstructing quantized frame expansions."

Norbert Wiener Center Seminar, University of Maryland, September 28, 2006. Title: "Alternative duals for linearly reconstructing sigma-delta quantized frame coefficients."

Texas A&M University, Concentration Week on "Frames, Banach Spaces and Signal Processing," August 8, 2006. Title: "Alternate duals for sigma-delta quantization of finite frame expansions."

University of Colorado (Boulder), Conference on "Current trends in harmonic analysis and its applications: wavelets and frames," May 20, 2006. Title: "Alternate dual frames for Sigma-Delta quantization."

Cornell University, Analysis Seminar, April 10, 2006. Title: The Balian-Low uncertainty principle: sharpness, non-symmetry, and other extensions.

University of Alberta, Department of Mathematics, March 17, 2006. Title: Localization of Gabor bases.

Tennessee State University, Math and Physics Research Seminar. Title: Digital representations for oversampled data, February 23, 2006.

Vanderbilt University, Undergraduate Seminar in Mathematics. Title: Secret periodicity in the Fibonacci numbers, February 8, 2006.

AMS Western Section Meeting, University of Oregon, Special Session on Wavelets, Frames, and Related Expansions. Title: Time-frequency mean and variance sequences of orthonormal bases, November 13, 2005.

Vanderbilt University, Computational Analysis Seminar. Title: Analog to digital conversion for finite frame expansions, November 1, 2005.

University of Illinois at Urbana-Champaign, Harmonic Analysis and Mathematical Physics Seminar. Title: Uncertainty principles for time-frequency expansions, October 25, 2005.

University of Illinois at Urbana-Champaign, Applied/Interdisciplinary Mathematics Seminar. Title: Sigma-Delta modulation and finite frames, October 24, 2005.

Vanderbilt University, Analysis and Biomathematics Seminar. Title: Uncertainty principles for time-frequency expansions, September 21, 2005.

SPIE, International Society on Optical Engineering, Wavelets XI, San Diego, CA. Title: Error estimates and stability for Sigma-Delta quantization of finite frame expansions, August 2, 2005.

Erwin Schrodinger Institute, Vienna, Austria, Workshop on ``Non-Orthogonal Expansions and Greedy Algorithms.'' Title: Sigma-Delta quantizers for finite frames: error estimates and stability, June 6, 2005.

University of Vienna, Vienna, Austria, NuHAG Seminar. Title: Recent work on the Balian-Low theorem, June 2, 2005.

International conference on ``Modern Methods of Time-Frequency Analysis,'' Strobl, Austria, May 23, 2005. Title: Time-Frequency Mean and Variance Sequences of Orthonormal Bases.

Center for Scientific Computation and Mathematical Modeling (CSCAMM), University of Maryland, Workshop on ``Oversampling and coarse quantization of signals,'' invited speaker, April 14, 2005. Title: Finite frames and Sigma-Delta quantization - Part 2.

Princeton University, Time-Frequency Seminar. Title: Recent work on the Balian-Low theorem, March 29, 2005.

United States Naval Academy, Colloquium. Title: Finite frames and Sigma-Delta quantization, March 2, 2005.

Vanderbilt University, Colloquium. Title: Finite frames and Sigma-Delta quantization, February 24, 2005.

Syracuse University, Colloquium. Title: Finite frames and Sigma-Delta quantization, February 17, 2005.

Texas A&M University, Colloquium. Title: Finite frames and Sigma-Delta quantization, February 10, 2005.

San Francisco State University, Colloquium. Title: Finite frames and Sigma-Delta quantization, February 7, 2005.

Lehigh University, Colloquium. Title: The uncertainty principle in time-frequency analysis, February 2, 2005.

University of Central Florida, Colloquium. Title: Sigma-Delta quantization for finite frames: error estimates and stability, January 31, 2005.

Princeton University, Time-Frequency Seminar. Title: Finite frames and Sigma-Delta quantization, October 26, 2004.

MathFest 2004, Providence, Special Session on ``SMALL'' Mathematics. Title: Finite frames and Sigma-Delta quantization, August 14, 2004.

Second International Conference on Computational Harmonic Analysis, Vanderbilt University. Title: Sigma Delta quantization of finite frame expansions: refinements and 2nd order schemes, May 24, 2004.

Joint AMS-SMM International Meeting, Houston, Special Session on Designing Frames and Wavelets: From Theory to Digitization. Title: Finite frame Sigma-Delta quantization, May 14, 2004.

Joint AMS-MAA Mathematics Meeting, Baltimore, AMS Special Session on Wavelets, Frames and Operator Theory. Title: A (p,q) weighted version of a theorem of J. Bourgain, January 17, 2003.

IEEE Signal Processing Letters, Associate Editor (2012 - current).

- NSF DMS Grant 1521749. co-PI. "Collaborative Research: Computational methods for ultra-high sensitivity magnetometry of geological samples," July 2015 -- July 2018.
- NSF DMS Grant 1211687. PI. "High Dimensional Data Representations: Compressed Sensing, Randomized Row-Action Methods, and Quantization," July 2012 -- July 2016.
- NSF DMS Grant 0811086. PI. "Accurate digital representation and recovery for redundant frames," July 2008 -- July 2012.
- NSF DMS Grant 0504924. co-PI. "Accurate digital representations of overcomplete data expansions," July 2005 -- July 2008.

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