# Applied and Interdisciplinary Mathematics

Applied and interdisciplinary mathematics plays an important role in our graduate program. We currently run two professional masters degree programs:

The students in these MS programs can participate in the optional co-op program or the optional internship program.

There are also opportunities for graduate students to join research projects in applied and interdisciplinary mathematics, including at the PhD level.

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The Math Department has a small but increasing number of faculty who are engaged in various aspects of applied and interdisciplinary mathematics. These include:

Chris Beasley — Mathematical physics, quantum field theory, string theory

Adam Ding — Statistical applications: survival analysis, dynamic systems, side-channel analysis, machine learning.

Christopher King — Quantum information theory, switching systems, computer networks (More info)

Mike Malioutov — Statistics, probability, experimental design, information theory (More info)

Robert McOwen — Partial differential equations, inverse problems

Jayant Shah — Computer vision

Alex Suciu — Topology, symbolic computation, applications to chemistry

Gilead Tadmor (joint with ECE)–Systems theory, optimal control, dynamic estimation, medical imaging

Petar Topalov — Dynamical systems, fluid dynamics

There are also faculty in other departments at Northeastern whose research involves applied mathematics and who have connections with our Department:

Dana Brooks (Electrical and Computer Engineering)– Biomedical signal and image processing, applied inverse problems

Anthony Devaney (Electrical and Computer Engineering)– Electromagnetic wave propagation, Inverse scattering tomography

Alain Karma (Physics)–Nonlinear dynamics, pattern formation, cardiac dynamics

Dagmar Sternad (Biology, Electrical and Computer Engineering and Physics)–Motor control and neuroscience

Applied and interdisciplinary mathematics is playing an increasing role in our undergraduate education as well. At the forefront is our PRISM program which is run by the Math Department in conjunction with the Biology and Physics Departments: this offers biweekly seminars in the Fall and intensive research-based courses in the Spring and Summer. In addition to PRISM, several of our faculty members are interested in supervising undergraduate research projects. Interested students should contact any of the faculty listed above.

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**Chris King’s Applied and Interdisciplinary Mathematics**

Chris King’s main area of research is quantum information theory. Research in this field centers around the question of how the unique properties of quantum systems (such as state superpositions and entanglement) can be exploited to enhance the efficiency of tasks involving information storage and transfer. Chris also conducts research in pure mathematics (matrix analysis) and classical dynamical systems (stability of switching systems). There are several ongoing research projects which interested graduate students could join:

- The classical information capacity of a generic high-dimensional quantum channel is non-additive over tensor products (part of the fun will be learning the meaning of each of these terms). However there is no known explicit example of a non-additive channel. The quest for such an example has led to many interesting and novel applications of mathematics in quantum information theory.
- Resonant energy transfer is the main mechanism whereby energy moves through molecular complexes, and plays a vital role in processes like photosynthesis. The equations describing this process (Schrodinger’s equation for multi-electron systems) are too complicated to solve exactly so approximation schemes are used to derive the efficiency and duration of the energy transfers. The analysis of these approximations provides many fascinating and challenging mathematical problems.
- Research throws up numerous small but interesting problems which can offer a way into research — recent examples have included a consensus model using nearest neighbor voting, a new class of solutions for a hybrid switching system, and oscillatory solutions of a differential equation with delay. Problems like these can provide the opportunity for a motivated graduate student to get involved with mathematical research directed toward applications.

References:

- C. King, B. Barbiellini, D. Moser, V. Renugopalakrishnan, “Exactly soluble model of resonant energy transfer between molecules”, Phys. Rev. B 85, 125106 (2012).
- C. King and D. Moser, “Average output entropy for quantum channels”, J. Math. Phys. vol. 52, 112202 (2011); selected for the November 2011 issue of Virtual Journal of Quantum Information.
- C. King, “Conditions for quadratic stability of a multi-parameter switching system”, Journal of Nonlinear Systems and Applications, vol. 2, 26 — 34 (2011).
- M. Fukuda and C. King, “Entanglement of random subspaces via the Hastings bound”, J. Math. Phys. vol. 51, 042201 (2010).
- M. Fukuda, C. King and D. Moser, “Comments on Hastings’ Additivity Counterexamples”, Commun. Math. Phys., vol. 296, no. 1, 111 (2010).

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**Mike Malyutov’s Applied and Interdisciplinary Mathematics**

For a long time, Mikhail Malyutov’s main research interest has been in developing information theory inspired by statistical methods. Recent research efforts have been concentrated on two interrelated problems:

- Statistical analysis of multi-dimensional systems under the so-called sparsity assumption, and
- Robust nonparametric statistical analysis of stationary time series using universal compression and related Variable Order Markov Chains (VOMC) training.

Three typical exciting challenging mathematical problems are:

- Finding the capacity of recovering sparse active inputs of additive regression (so-called Forged Coins) model under Linear Programming analysis, see [1],
- Proving the Asymptotic Normality of the universally compressed length increment for a long training stationary sequence, see [2].
- Developing the fastest Change-Point detection multi-scale algorithm for unknown stationary processes.

References:

- “Recovery of sparse active inputs in general systems: A review”, Plenary paper, in Proceedings, International Conference on Computational Technologies in Electrical and Electronics Engineering, IEEE Region 8, SIBIRCON 2010, 1, 15-22, available via IEEXplore..
- “Compression based homogeneity testing”, Proceedings (Doklady) of Russian Academy of Sciences, 443, No. 4, 2012.