Applied + Computational Mathematics

Applied and Computational Mathematics at the California Institute of Technology offers a rigorous interdisciplinary curriculum designed to equip students with a solid foundation in mathematical methods, computational techniques, and their applications across various scientific and engineering disciplines. This program emphasizes the development of analytical skills, problem-solving abilities, and computational proficiency necessary for tackling complex real-world challenges. Students engage in coursework that covers differential equations, numerical analysis, mathematical modeling, and scientific computing, all tailored to prepare them for research, industry, or further academic pursuit. The flexible structure of the program allows students to customize their education through electives spanning physics, computer science, engineering, and applied mathematics, fostering an integrated understanding of how mathematical principles underpin modern technological innovations. Hands-on laboratory components and computational projects are integral parts of the curriculum, providing practical experience in applying theoretical concepts to practical problems. The program also encourages collaboration and research, often involving projects that address current scientific questions, such as modeling biological systems, optimizing engineering processes, or analyzing data in emerging fields like artificial intelligence. Students benefit from the institute’s world-renowned faculty, cutting-edge resources, and vibrant research environment, enabling them to participate in groundbreaking work. Upon graduation, students are well-prepared for careers in academia, industry, or entrepreneurship, contributing to advancements in technology, science, and engineering. The Applied and Computational Mathematics program at Caltech is committed to cultivating intellectually curious individuals who will become leaders in their fields, pushing the boundaries of knowledge and innovation through mathematical and computational excellence.

All ACM students are required to take a total of 18 nine-unit courses at the graduate level (or the equivalent of 162 units) during their graduate study at Caltech. Among these 18 courses, the following core courses, typically taken during the student’s first year of study, are required: ACM 101 ab, ACM 105, CMS/ACM 107, Ma 108 abc ACM 106 ab, CMS/ACM 117, and an application elective course. The application elective course in the first year is selected, with the recommendation of the student’s adviser, from among a wide range of courses offered by an outside option within the Institute. In addition, CMS 290 is required for all first year CMS/ACM/CDS/CS graduate students during each term (fall/winter/spring). In the second and third years, students are expected to take graduate-level courses appropriate to their chosen research area. The remaining courses towards the 162 unit requirement would normally include graduate-level ACM or CMS courses such as CMS/ACM 113, 201 ab, 210 ab, 216, 217, CDS 140, etc., as deemed appropriate to the student’s research program, and which must be selected in consultation with the student’s research adviser.

Students who have already taken some of the required courses may use them to satisfy the course requirements, even though the units may not be used to satisfy the total unit requirement for the Ph.D. degree.

Master's Degree

Students are not admitted to work toward the master’s degree. The master’s degree may be awarded to an ACM student only in exceptional circumstances. Of the 135 units of graduate work required by Institute regulations, at least 81 units of advanced graduate work should be in applied mathematics.

Degree of Doctor of Philosophy

• The Preliminary Examination. The student must pass an oral or written candidacy examination administered by a faculty committee. This examination is offered during the third term of the first year of graduate residence at the Institute. The material covered in this examination is based on the three core sequences described above. For students who have already taken the required courses before coming to Caltech, the examination can also be based on the substituted courses taken in the first year.
• The Candidacy Examination. To be recommended for candidacy for the Ph.D. degree in Applied and Computational Mathematics, all students must, in addition to meeting the general Institute requirements and passing the preliminary examination, pass a candidacy examination administered by a committee that consists of at least four faculty, is approved by the option representative, and is chaired by the student’s research adviser. The examination will ascertain the student’s breadth and depth of preparation for research in the chosen area. The examination must be taken within the first three years of residence.
• Advising and Thesis Supervision. Upon passing the preliminary examination, the student is required to choose a thesis supervisor who assumes the major responsibility in supervising the Ph.D. thesis. At the same time, an advising committee consisting of three faculty members is formed to help oversee the advising process. This committee should be formed no later than the third year of graduate study. The student’s supervisor is part of this committee, but does not chair the committee. The student is encouraged to meet with the committee members informally for advice or suggestions. Joint supervision between two faculty members is also possible, as is seeking a thesis adviser outside the core applied and computational mathematics option, although in this case it is mandatory that an applied mathematics faculty member be nominated as a co-adviser.
  
  Should a disagreement of any kind occur between the student and his or her supervisor as regards the timely completion of the thesis, the student is encouraged to direct his or her concerns to the committee chair. If this is not workable, the student should feel free to consult with the option representative, the executive officer, or an applied and computational mathematics faculty member of the student’s choice. If the student’s concerns cannot be resolved through consultation with these individuals, the student is encouraged to pursue resolution of his or her concerns through other channels as outlined in Student Problem Resolution Process on page 50.
• Submission of Thesis. On or before the first Monday in April of the year in which the degree is to be conferred, a candidate for the degree of Ph.D. in applied and computational mathematics must deliver a typewritten or printed copy of the completed thesis to his or her research supervisor.
• Final Examination. The final oral examination is held within four weeks after the submission of the thesis. The examination covers the thesis and related areas.

Subject Minor in Applied and Computational Mathematics

The group of courses must differ markedly from the major subject of study, and must include 54 units of advanced courses in applied mathematics and must not be simultaneously used for fulfilling a requirement of the second option. The qualifying courses exclude ACM95/100, although some flexibility is allowed depending upon the option of origin. The student must pass an oral examination whose subject is directly related to the material covered in the qualifying courses. This oral examination will be waived if the student has received a grade of A in every course.

Requirements

• Each new graduate student admitted to work for the Ph.D. in applied and computational mathematics is given an informal interview on Thursday or Friday of the week preceding the beginning of instruction for the fall term. The purpose of this interview is to ascertain the preparation of the student and assist him or her in mapping out a course of study. The work of the student during the first year will usually include some independent reading and/or research.
• Statement of Purpose
• Include payment of the application fee or submit a fee waiver request. 
• Three letters of recommendation from individuals familiar with your academic and/or work performance are required for all applicants. 
• Transcripts from all colleges or universities attended are required. Transcripts may be sent directly from the Registrar of the institution you attended or transcripts issued directly to the applicant, and so marked, are acceptable, and can be submitted online following the electronic submission process. If you attended Caltech as an undergraduate, you must request an official transcript from the Office of the Registrar. We cannot obtain one on your behalf. 
• Attach a CV or resume
• TOEFL

The California Institute of Technology (Caltech) offers a range of financial aid options to support students enrolled in its programs, including the Applied + Computational Mathematics major. Undergraduate students typically receive financial assistance through a combination of grants, scholarships, work-study opportunities, and loans. Caltech is committed to meeting 100% of demonstrated financial need for admitted students, ensuring that financial barriers do not prevent talented individuals from pursuing their studies.

Financial aid packages are determined based on individual circumstances, including family income, assets, and overall financial situation. The institute's financial aid program emphasizes need-based assistance, allowing students from diverse economic backgrounds to attend. Students are encouraged to submit the Free Application for Federal Student Aid (FAFSA) and the CSS Profile to determine their eligibility for available aid programs.

Caltech also offers merit-based scholarships for exceptional academic achievement and potential, which may be awarded in addition to need-based aid. These scholarships are highly competitive and are granted based on academic records, research accomplishments, and recommendations.

In addition to institutional aid, students may explore external sources of funding such as government grants, private scholarships, and fellowships relevant to mathematics and computational sciences. The university provides resources and counseling to assist students in identifying and applying for external funding opportunities.

Students are responsible for understanding the terms of their financial aid packages, including any borrowing obligations and repayment terms for loans. Caltech’s Financial Aid Office provides ongoing support, guidance, and transparency regarding financial planning, ensuring students can make informed decisions about funding their education.

Overall, the financial studies at Caltech aim to make education accessible and affordable, enabling students to focus on their academic and research pursuits in applied and computational mathematics without undue financial stress.

The California Institute of Technology offers a unique and interdisciplinary academic program focused on Applied and Computational Mathematics. This program is designed to equip students with a strong foundation in mathematical principles, computational techniques, and their applications to solving real-world problems across various scientific and engineering disciplines. Students enrolled in this program gain a comprehensive understanding of both theoretical and practical aspects of mathematics, emphasizing computational methods, numerical analysis, and the development of algorithms to address complex challenges in science, technology, and industry. The curriculum typically includes courses in differential equations, linear algebra, probability, statistics, numerical methods, scientific computing, and applied modeling. Students are often encouraged to participate in research projects, internships, or collaborations with faculty to deepen their understanding and gain hands-on experience.

The program aims to foster analytical thinking, problem-solving skills, and the ability to develop innovative solutions using mathematical and computational approaches. Graduates of the Applied and Computational Mathematics program are well-prepared for careers in academia, research institutions, technology companies, finance, and consulting firms, or for pursuing advanced graduate studies. The program also emphasizes collaboration and communication skills, preparing students to work effectively in multidisciplinary teams. With access to Caltech's state-of-the-art laboratories, computer facilities, and a vibrant academic community, students have many opportunities to engage in cutting-edge research and internships. Overall, the Applied and Computational Mathematics program at Caltech strives to produce versatile professionals capable of addressing some of the most pressing scientific and technological challenges of our time through a rigorous, innovative educational approach.