Graduate Degree in
Computing + Mathematical Sciences

The Computing and Mathematical Sciences (CMS) PhD program is a unique, new, multidisciplinary program at Caltech involving faculty and students from computer science, electrical engineering, applied math, economics, operations research, and even the physical sciences. The program sets high standards for admission and graduation, and boasts a broad collection of world-class faculty (any faculty at Caltech from any of the areas above can advise students).

Overview

Graduate Option Rep

Prof. Joel A. Tropp
jtroppcms.caltech.edu

Options Manager

Maria Lopez
mlopezcms.caltech.edu

Disciplines across the information sciences are experiencing an unprecedented convergence. As different areas interact, new fields are emerging. For example, combining Computer Science with...

...Optimization and Statistics has led to machine learning, "big data," and the field of data science.
...Control and Electrical Engineering has led to the smart grid, smart buildings, and the internet of things.
...Physics has led to quantum computing and quantum information theory.
...Economics has led to algorithmic game theory, privacy, and the field of network science.
...Biology and Electrical Engineering has led to bioinformatics, molecular programming, and biomolecular circuits.

Because of this convergence, a new intellectual core is emerging in the information sciences. The core contains material from a spectrum of disciplines: algorithms, networks, machine learning, statistics, optimization, signal processing, and the underlying mathematics. But each area is enriched by the broader context. For instance, the study of algorithms now encompasses the traditional discrete problems of computer science, the continuous problems of applied mathematics, as well as worst-case and average-case perspectives.

The CMS PhD program is designed around the new information science core. This core provides the ideal foundation for future applications across the sciences, engineering, and beyond. Our approach requires the mastery of the following ways of thinking about information science:

  • Interpret "information" and "computation" broadly. We study mechanisms that communicate, store, and process information. These structures might be etched in silicon and called hardware or written in code and called software. But the same mechanisms may be expressed in nucleotides and called DNA. They also arise in our society, where they are called social networks or markets. Our view encompasses all of these manifestations.
  • Algorithmic thinking is the foundation. The modern world demands the ability to think algorithmically. Algorithms are not just the basis for advanced computer systems, but they help us understand biological organisms and auction design and more.
  • Data is central. Data is being collected at an unprecedented speed and scale. Every area of science and society will be transformed as researchers learn to use this data to develop and test new hypotheses. To unlock this potential, we need to develop reliable algorithms for extracting information and making decisions based on data.
  • Seek rigor and relevance. The CMS Program focuses on the theoretical core of information science. We believe that principled and rigorous methods provide the only solid basis for progress. But we also insist on research that is relevant to applications, and we train students to work at the interface of information science and other disciplines.

Students may select a research adviser from any of the 30+ faculty affiliated with the CMS Department, including specialists in Applied & Computational Mathematics, Biological Engineering, Computation & Neural Systems, Computer Science, Control & Dynamical Systems, Economics, Electrical Engineering, Mechanical Engineering, Philosophy, and Physics.

Who should apply?

The CMS program is meant for students who have interdisciplinary interests and do not want to be pigeonholed. Instead we are looking for students that want to be at the boundaries of disciplines, where ideas overlap and intersect and where new fields emerge. Many CMS students have studied more than one field during their undergraduate programs. Typically CMS students have undergraduate training in mathematics, computer science, electrical engineering, or economics, but some have completed programs in other parts of engineering or the mathematical sciences. The common factor in the students is that they are curious about how areas interact with and influence each other.

The CMS core requirement

The CMS program builds on a set of core classes that students take together during the first year. These classes teach foundational ideas in computer science, economics, electrical engineering, mathematics, and statistics that support research in a wide variety of other disciplines.

optimization

Fall: Mathematical Fundamentals

  • CMS/ACM 107. Introduction to Linear Analysis with Applications.
  • CMS/ACM 113. Mathematical Optimization.
  • CMS/ACM 117. Introduction to Probability and Random Processes.
signal

Winter: Computing Fundamentals

  • CMS/CS 139. Analysis and Design of Algorithms
  • CMS/CS/EE 144. Networks: Structure & Economics
  • CMS/CS/CNS 155. Machine Learning & Data Mining.

Depth requirement: Research focus areas

In addition to the core classes, each CMS student is required to take three classes in a focus area, which is chosen in consultation with a faculty adviser. Example focus areas include:

  • Algorithms & complexity: Approximation algorithms, online algorithms, complexity theory, and computability.
  • Algorithmic economics: Auctions and mechanism design, algorithmic game theory, and privacy.
  • Biological circuits: Organic substrates for computation, including neuronal computing and DNA computing.
  • Feedback & control: Robust control, feedback, dynamical systems theory.
  • Inference & statistics: Statistical decision theory, information theory, and adaptive signal processing.
  • Information systems: Information theory, coding theory, communication, and signal processing.
  • Machine learning & vision: Algorithmic, mathematical, and biological perspectives on computational models for learning and vision.
  • Networked systems: The study of complex networks, in fields ranging from biology, social science, communications, and power.
  • Optimization: Convex optimization, conic and discrete optimization, and numerical methods for largescale optimization.
  • Quantum information theory: Quantum algorithms and complexity, convex optimization, and operator theory.
  • Scientific computing: Computational methods for problems arising in the physical sciences, partial differential equations.
  • Uncertainty quantification: Markov chains and martingales, stochastic system analysis, and convex optimization.

Other degree requirements

The CMS Graduate Program is a research oriented course of study leading to the PhD degree in Computing and Mathematical Sciences. The requirements are designed to train students to undertake transformational research in a range of application areas.

Advising. Incoming students will be paired with an academic adviser who will provide guidance on coursework and help students make the transition into research. During the first year, each student will choose a research adviser from CMS, Economics, Electrical Engineering, Mathematics, or an allied field.

Breadth requirement. In addition to the core courses and depth requirements above, students must complete three other graduate courses from mathematics, science, engineering, or economics.

CMS Colloquium. During their first year students are required to attend the CMS Colloquium (CMS 290abc), a biweekly technical seminar series featuring distinguished researchers from around the world who specialize in computing and mathematical sciences.

CMS preliminary examination. In the winter term of the first year, each student will complete an oral examination focused on his or her coursework in mathematics, computer science, and engineering.

CMS candidacy examination. By the end of the third year, each student will complete an oral candidacy examination designed to assess the breadth and depth of preparation for undertaking research in the chosen area.

The PhD dissertation. Each student will complete a PhD dissertation documenting independent research. To complete the degree, each student will given an oral presentation on the dissertation work to the thesis committee.

PhD thesis defenses consist of a public presentation with an opportunity for the audience to ask questions. This is followed by a private examination with only the thesis committee and the candidate present. Thesis defenses will be announced and the CMS community as a whole is encouraged to attend.

Complete details about the PhD requirements can be found in the Caltech Catalog. An overview of the typical progression through the program and advice for current students can be found in the Option Guidelines.