Computer Science and Mathematics

Ramdas B. SonawaneNess Wadia College of Commerce

Pune

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Outline• Introduction• Influence of Math on Computer Science• Influence of Computer Science on Math• Computational Mathematics• Career Opportunities

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Introduction• Computer science and mathematics

are currently extensively used in many fields

• Both have influenced the other and led to many advancements in each

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What is Decision Mathematics?

Decision Mathematics - finding efficient solutions to real life problems

e.g. what would be the most efficient route for delivering postaround a network of streets?This topic uses algorithms whichare vital in computer science.

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Influence of Math on C.S.• Many computer science programs have

spawned from mathematics departments• During their time under mathematics,

they slowly built up reputation and students, with most eventually becoming their own department, then later, their own school

• Many noted computer scientists have degrees in mathematics, including Knuth, Cook, Lamport, Backus (FORTRAN, BNF), Kay (GUI), and McCarthy (LISP)

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Influence of Math on C.S.Why do C.S. students need Math?• Many people believe that mathematics

is fundamental to computer science • Real world problems are often

expressed mathematically and require problem solving skills

• Studies have shown that mathematical ability has a strong correlation with success in introductory computer science courses

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Influence of Math on C.S.• When the human brain is subjected to

extended educational experience permanent, physical changes occur, creating neural pathways which aid in learning new things

• The more repetitive the learning process, the strong and longer lasting the changes

• The mind finds it extremely difficult to accept formal abstractions, yet at some point, they seem very real and the developer does not see them as abstract

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Influence of Math on C.S.• Actual information learned in a class may go

away, but the learning abilities are still there

• True goal of education is to improve minds• Should enable the student to acquire

abilities and skills to accomplish new things• Main benefit of learning and doing

mathematics is to develop the ability to reason precisely and analytically about abstract structures, which is what computing deals with

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Influence of Math on C.S.• An example of math playing a vital role in

computer science occurs when using public key cryptography

• A published, publicly available key (the public key) is provided in an open directory

• The person keeps a private key• To send an encrypted file or email

message, the sender encrypts using the recipient’s public key

• The private key decrypts the message

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Influence of Math on C.S.• The keys are formed and secured based on

the properties of prime numbers• Basically, the public key is a product of two

random, large primes• The private key is the two primes

themselves• This algorithm is secure because of the

almost impossible task of factorizing the large number into the two primes

• To break this encryption, all prime numbers less than the product must be checked

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Cryptology• Cryptography

– Inventing cipher systems; protecting communications and storage

• Cryptanalysis– Breaking cipher systems

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Cryptology

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Cryptanalysis

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What is used in Cryptology?• Cryptography:

– Linear algebra, abstract algebra, number theory

• Cryptanalysis:– Probability, statistics, combinatorics,

computing

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Need more than secrecy….• Need reliability!

• Enter coding theory…..

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What is Coding Theory?• Coding theory is the study of error-

control codes• Error control codes are used to detect

and correct errors that occur when data are transferred or stored

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What IS Coding Theory?• A mix of mathematics, computer

science, electrical engineering, telecommunications– Linear algebra– Abstract algebra (groups, rings, fields)– Probability & Statistics– Signals & Systems– Implementation issues– Optimization issues– Performance issues

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Influence of Math on C.S.Samples of Math in C.S. areas• Relational Databases – Rely on the

ideas of set theory to understand usage• NP-Completeness – Heavy math basis• Network – Topology of networks,

routing, and load analysis problems deal with graph theory

• Cryptography – Encryption/decryption schemes rely on mathematical ideas and methods to provide security

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Influence of C.S. on MathComputer algebra system (CAS)• Software program that allows symbolic

mathematics, the manipulation of equations and expressions in symbolic format

• Typical CASs store expressions as directed acyclic graphs, DAGs are also used in the parse trees for compilers

• Popular examples include Maple, MATLAB, and Mathematica

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Influence of C.S. on MathMaple• General purpose CAS• Developed in 1981 by a group at the

University of Waterloo• Sold commercially since 1988• Interpreted, dynamically typed

language• Allows static scoping, where a variable

always refers to its nearest binding

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Influence of C.S. on Math

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Influence of C.S. on MathMATLAB• Numerical computing environment and

programming language• Has a toolbox to interface with Maple

engine which turns it into a CAS• Invented in late 1970s by Cleve Moler,

chairman of computer science at the University of New Mexico

• Used mainly for linear algebra and numerical analysis

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Influence of C.S. on Math

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Influence of C.S. on MathMathematica• CAS and powerful programming language• Written by Stephen Wolfram• First version released in 1988• Uses a kernel which does all calculations

and feeds results to multiple front ends• Based on term rewriting • Supports functional and procedural

programming

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Influence of C.S. on Math

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The Four Color Theorem

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The Four Color Theorem• Long standing conjecture proven with aid

of the computer• Originally posed in the 1850s• Proven by Kenneth Appel and Wolfgang

Haken in 1976• States that the chromatic number (least

number of colors required to color a graph) of a planar graph is no greater than four

• Relies on case-by-case analysis carried out by computer

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The Four Color Theorem• Showed that if the theorem was false, a

counterexample would have to exist in one of 1,936 reducible configurations (later reduced to 1,476)

• A later enhancement by Robertson, Sanders, Seymour, and Thomas reduced that number to 633

• Many disproofs use one region touching all others, not realizing that the surrounding regions can be colored with only three

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The Four Color Theorem

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Computational Mathematics• Considered to be at the intersection of

computer science and mathematics• Application of computer methods to

simulate computer models and analyzing the results

• Historically, the software and models were developed by people already working in the application area (engineers, scientists, etc.)

• Developing and analyzing these models requires more than classic mathematics and elementary computer science

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Computational Mathematics• Deals with issues such as

– the implications of finite precision arithmetic

– the efficiency, accuracy, and stability of numerical computations

– the development and maintenance of mathematical software

– the effects of modern developments in computer architectures and networks

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Computational Mathematics• Students in these programs study

– Asymptotics, analysis, numerical analysis, and signal processing

– Discrete mathematics, combinatorics, algorithms, computation geometry, and graphics

– Mechanics and field theories– Optimization– Partial and ordinary differential equations– Stochastic modeling, probability, statistics,

and information theory

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Computational Mathematics• Sample applications include

– Tornado tracking for meteorologists– Numerical simulation of flow in

aeronautical engineering– Derivative pricing in computational finance– Modeling breaking waves– Medical imaging with functional magnetic

resonance imaging (fMRI)

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What are the career opportunities?

There is a huge shortage of people with STEM(Science, Technology, Engineering and Math education) skills needed to enter the workforce.

There are many new applications of mathematics in technology:

•Games Design•Internet Security•Programming•Communications

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What are the career opportunities?On-going applications in engineering, such as Aircraft Modelling Fluid Flows Acoustic Engineering Electronics Civil Engineering.

…and new scientific processes such as modelling populations and Diseases Quantum Physics,

Astronomy,Forensics andDNA sequencing.

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What are the career opportunities?

Financial systems and online purchasing systems are also underpinned by mathematics, relying heavily on online security and encryption.

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What are the career opportunities?

Other areas include

FinanceActuarial Work, Accountancy,

Banking, Financial Modelling

StatisticsMedical Statistics, Market

Research, Government Policy Advice

BusinessLogistics, Traffic Planning, Project

Management, Consultancy

Law

Teaching/Lecturing

MedicineDentistry, Pharmacy,

Doctors, Nursing

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Computer Science PG Programs• IIT Bombay – M.Sc. Applied Statistics

and Informatics• IIT Guwahati – M.Sc. Mathematics and

Computing• IIT Kharagpur – M.Sc. Mathematics and

Computing• All IITs - M. Tech. Computer Science

(after M.Sc.)• IIT Madras – M. Tech. Industrial

Mathematics and Scientific computing

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Computer Science PG Programs• SPPU – M.Sc. Scientific Computing, M.

Tech. Modelling and Simulation• ISI – M. Tech. Computer Science (after

M.Sc.)• IIT Kharagpur - Computer Science &

Data Processing

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Conclusions• Computer science and mathematics

have both influenced each other for many years

• A number of advancements in each field can be directly linked to the other

• Computational mathematics has successfully combined the two fields into one which is being used to produce quality software for many important fields

41Dr. Raymond Greenlaw – The Synergy of Computer Science and Mathematics