Focus Areas

The Computing + Mathematical Sciences department pursues numerous research interests covering a wide array of application areas. We take full advantage of Caltech's unique interdisciplinary character by drawing on research expertise not only from our own department, but from throughout the Institute. Research efforts within the department evolve at a fast pace, and cover currently six discernible focus areas:

Discrete Differential Modeling

(Mathieu Desbrun; Peter Schröder; Alan H. Barr)

The focus area of Discrete Differential Modeling is based around the insight that numerical simulation and modeling on a computer require discrete versions of the continuous mathematical models describing these systems. For these computations to be truly predictive, reliable, and efficient the underlying continuous structures must be transferred to the discrete systems. The goal of this focus area is to identify the relevant mathematical and computer science tools to lay the foundation for the rational construction of such discrete differential modeling tools which preserve the relevant structures. The study of geometry in a broad sense forms the core of this area but it also draws considerably on fields ranging from algebraic topology to computational geometry, graph theory, combinatorics, applied mathematics, and computer science. Application areas include computer graphics, variational mechanics, and biological systems. Labs within this research focus include the Graphics Group, the Multires Modeling Group, and the Applied Geometry lab.

DNA Computing and Molecular Programming

Underneath the computer revolution that has changed our lives are the fundamental principles of computer science---they have allowed us to master electronic systems with billions of components and software with millions of lines of code to do amazingly complex tasks. This focus area develops new computer science principles for programming information-bearing molecules like DNA and RNA to create artificial biomolecular programs of similar complexity.

The biomolecular programs of life give inspiration that this task is possible, from the low-level operating system controlling cell metabolism, to the high-level code for development, the process by which a single cell becomes an entire organism. The team aims to create analogous molecular programs using non-living chemistry, in which computing and decision-making will carried out by chemical processes themselves. Through the creation of molecular programming languages, theory for analyzing them, and experiments for validating them, our long-term vision is to establish "molecular programming" as a subdiscipline of computer science---one that will enable a yet-to-be imagined array of applications from chemical circuitry for interacting with biological molecules to molecular robotics and nanoscale computing. Check out the Pierce Lab, DNA and Natural Algorithms Group, as well as the Molecular Programming Project for more details.

Perceptual and Machine Learning for Autonomous Systems

(Yaser Abu-Mostafa; Andreas Krause; Richard Murray. See also: Joel Burdick; Christof Koch; Pietro Perona)

Perception and action are the salient characteristics of autonomous systems. How does one build sensory systems, such as vision, hearing and olfaction, that are able to extract `information’ from massive, multidimensional and noisy data? What is the best set of actions to be taken to achieve a given goal? How can this be learnt from example? How can we design machines that interact successfully and productively both with humans and other machines?

Answering these questions will allow us to build autonomous intelligent systems and machines that can improve our lives. Autonomous and safe road vehicles, networks of sensors for environmental monitoring, engineered bacteria for drug synthesis and medical treatment, robust financial systems, neural prosthetics are examples of what could be achieved.

We combine knowledge and techniques from machine learning, statistics, applied mathematics, computer science, controls, signal processing, robotics, the social sciences and neuroscience. Interested students will apply to the PhD programs in CNS, CDS, CS, ME, EE depending on their specific interests and background. Labs within this research focus include the Machine Learning and Adaptive Systems, Learning Systems, Robotics Lab, Computational Vision, and Koch Lab.

Rigorous Systems Research

(Yaser Abu-Mostafa; Mani Chandy; John Doyle; Tracey Ho; Gerard Holzmann; Andreas Krause; Steven Low; Alain Martin; Richard Murray; Adam Wierman. See also: Jehoshua (Shuki) Bruck; Michelle Effros; Babak Hassibi; John Ledyard)

The Rigorous Systems Research Group (RSRG) studies the design of computer systems; but it's not your ordinary systems group. RSRG is distinguished by three characteristics:

1. Theory is the foundation. Everybody in the group develops new theoretical results that inform system design and performance analysis.
2. Get your hands dirty. Everybody in the group builds, or uses measurements from, systems and prototypes.
3. Be truly interdisciplinary. Everybody in the group uses ideas from disciplines outside computer science (such as operations research, economics, and control theory) or develops complex systems that are used in varied disciplines (such as space exploration or control of power grids).

Consequently, the group develops theory, puts theory into formal tools and systematic methodologies, puts tools and methodologies into practice, and develops theory from practice, thus closing the loop. For convenience, RSRG also has labs (subgroups) that work on more focused problems such as networking, machine learning, system design, or computer architecture, including Learning Systems, Infospheres, NASA/JPL Laboratory for Reliable Software, PerfLab, Machine Learning and Adaptive Systems Group, Asynchronous VLSI Group, and Netlab.

Scientific Computing and Applied Analysis

(Oscar Bruno; Emmanuel Candes; Thomas Hou; Houman Owhadi; Peter Schröder; Joel Tropp. See also: James L. Beck, Dan Meiron; Niles Pierce.)

The interwoven fields of applied and computational mathematics are among the most interdisciplinary research areas in science and engineering, encompassing modeling, analysis, algorithm development, and simulation for problems arising throughout the pure sciences and engineering. CMS provides a uniquely small and interactive research environment to explore the mathematical properties of systems in physics, chemistry, biology, geology, astronomy, materials science, fluid mechanics, and a number of other disciplines. Focused labs studying sparse approximation, spectral methods, uncertainty quantification, theoretical and computational fluid mechanics, multiscale and stochastic processes, stochastic modeling of dynamic systems, probability theory, statistics, randomized numerical methods, computational molecular biology, homogenization theory, computational electromagnetism, and image processing constitute the core of this research focus.

Theory of Computation

(Alexei Kitaev; Katrina Ligett; Leonard Schulman; Chris Umans. See also: John Preskill)

What problems are computationally tractable? How is the answer to this question affected by the use of randomness as a resource? Or even more importantly -- by the fact that we live in a quantum mechanical world? What mathematics do we need to understand and develop in order to answer such questions? What happens when several computational agents interact -- how do they convey information to each other, hide information from each other, or combine their data or computational resources? In pursuing these questions, research in Theory of Computation in CMS spans Algorithms (randomized, quantum, algebraic, and on-line); Computational Complexity; Algorithmic Game Theory; Coding and Information Theory; Combinatorics and Explicit Construction; Discrete Probability; Combinatorial Geometry; Quantum Computation and Quantum Information Theory; and Data Privacy.

To learn more on the nature of each person's research projects and expertise, click on a professor's name or browse through the links of our CMS website.