Associate Professor Matthew Macauley’s research proposal “Toric partial orders: theory and applications” was recently funded by the Simons Foundation. Professor Macauley describes one aspect of his research in the Algebra and Discrete Mathematics subfaculty:
In 2004, biologist Joel Cohen wrote a paper titled “Mathematics is biology’s next microscope, only better; biology is mathematics’ next physics, only better.” Though the first part of this sentence has been well-documented alongside high-throughput technology, genomic data, and the emergence of fields such as bioinformatics and biostatistics, the second part is just as much ahead of us as it is in the rear-view mirror. Physics transformed mathematics in the 20th century with revolutionary ideas such as quantum mechanics and general relativity, carving out a new field now known as “mathematical physics”. Similarly, the 21st century is seeing new areas of mathematics being created from biological problems. One example of this is an area loosely called “algebraic biology”.
For an illustrative example of algebraic biology, we turn to molecular biology. A cell’s ability to regulate gene expression is crucial for maintaining homeostasis. Classically, this has been modeled using systems of differential equations derived from the laws of mass-action kinetics. Though these models are inherently quantitative, the fact that there can be over a dozen rate constants that are often not known within orders of magnitude, means that in practice, they are essentially qualitative models. In the last few decades, scientists have modeled a number of molecular networks using logical expressions. For example, “gene A is transcribed if enzyme B is present but repressor protein C is not present”. These logical expressions can be easily translated into a system of multivariate Boolean polynomials. This opens the door to using the toolbox of computational algebra to study biological questions. For example, computing fixed points can be done using a Gröbner basis. Network structure can be inferred by constructing a certain monomial ideal and computing its primary decomposition. If the “sign” of such interactions wants to be captured (e.g., activator vs. repressor), then one can use a new algebraic object called a “pseudonomial ideal” over a larger finite field.
Some people working in algebraic biology try to model or simulate biological systems using discrete or algebraic methods, with the goal of making and validating new biological predictions. For example, a gene regulatory network might be modeled with a Boolean network, or an unknown protein-protein interaction network might be inferred using techniques from algebraic geometry applied to time-series data. One of Professor Macauley’s masters students modeled the arabinose operon in E. coli with a Boolean network, and recently published this work in the Bulletin of Mathematical Biology. Other research problems involve developing the new mathematics that arises, such as the theory of pseudonomials over finite fields. This was the thesis topic of Professor Macauley’s most recent masters student. Such work answers a famous question of Bernd Sturmfels from the early 2000s: Can biology lead to new theorems? Yes it can!
Se the list of recent Simons Foundation Collaboration Grant awardees here.
Associate Professor Fei Xue’s research proposal “New Preconditioned Solvers for Large and Complex Eigenvalue Problems” was recently funded by the National Science Foundation. Professor Xue describes this research in the Computation subfaculty:
Eigenvalues and related mathematical tools are fundamentally descriptive in a great variety of scientific and engineering applications, including characterization of matrices, separation of variables in solving differential equations, identification of resonance or general heightened response to selected inputs, analysis of asymptotics and stability, and many more. In many circumstances, crucial information of a large complex system can be found by looking at a small number of eigenvalues of physical relevance of the corresponding matrix.
Reliable and efficient computation of such spectral information has been a central topic in computational mathematics and numerical linear algebra in particular for decades. In recent years, eigenvalue-related problems have become more diverse and challenging (mostly from linear to nonlinear), and the scale of the problems has been ever increasing. Classical algorithms have reached their limit and new numerical methods must be developed and analyzed to tackle these emerging new problems.
Our research concerns systematic development and analysis of innovative numerical methods for several important classes of large-scale and complex eigenvalue-related problems. For large linear symmetric eigenvalue problems, variants of preconditioned eigensolvers have been thoroughly investigated and widely used with great success in many applications. Recently, we find that the preconditioning and the solver framework can both be generalized significantly and integrated with great flexibility to solve a much broader class of challenging eigenvalue-related problems.
We will explore the idea of preconditioning in three main goals. The first is to compute eigenvalues through functions of matrices such as matrix exponentials and trignometrics. These functions map the eigenvalues of largest real or imaginary part to those of largest modulus so that they can be found much more easily and reliably. The second aim is to develop block preconditioned solvers that do not require accurate shift-invert matrix-vector multiplications to compute extreme and interior eigenvalues of large nonlinear eigenvalue problems. These constitute a major extension of the well-known techniques such as the preconditioned conjugate gradient and the Jacobi-Davidson methods to the nonlinear setting. The last goal is to compute certain eigenvalues of structured matrices in Kronecker sum or in companion form. The main interest is to fully explore the redundancy intrinsic in the matrix structure to facilitate computation in the original problem dimension much smaller than these structured matrices themselves.
The new algorithms will help enable large-scale eigenvalue-related modeling and simulations in many areas, including linear stability analysis, mechanics of materials, optimization of acoustic emissions, study of superconductivity, vibrations under conditions of uncertainty, and many more. Certain methods are also of significance to other areas; for example, fast exponential matrix-vector product is essential to the exponential integrator for solving stiff time-dependent differential equations.
Read more about Profexxor Xue’s NSF grant here.
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