M.Sc. | Mathematics

Duration : 2 yrs. (4 Semesters)

M.Sc. Mathematics is another industry-oriented course meant to enhance knowledge, creativity and computational skills in core mathematical subjects and their applications in real life. The course is the beginning to pursue a wide range of mathematics-based careers. The modules are designed as per the guidelines of UGC and are accessible, relevant, interesting, and challenging. The program includes a wide range of lecture modules in Operation research, Analysis, Topology, Algebra, Number Theory, Solid mechanics, Fluid mechanics, Integral Transforms, and Differential Geometry but is not limited to these topics, equipping the students for a variety of roles in employment and research.

PROGRAMME Objective

To cultivate a Mathematical aptitude and nurture the interests of the students toward problem solving aptitude
To motivate young minds to do research in mathematical sciences
To train computational scientists who can work on real-life challenging problems

PROGRAMME Outcome

Students are able to pursue research or careers in industry in mathematical sciences and allied fields
Gaining effective scientific and/or technical skills in both oral and writing communication
Acquiring relevant knowledge and skills appropriate to professional activities
Demonstrating the highest standards of ethical issues in mathematical sciences with critical thinking to carry out scientific investigation objectives without being biased

rankings

PLacement Highlights

1st Semester

Advanced Abstract Algebra

Objectives
This course will provide information about groups, sub groups, characteristics of a field, prime subfield and ideal theory in the polynomial ring. It also imparts knowledge about modules.

Outcomes
After completion of the course, the student will be able to:

Explain the fundamental concepts of advanced algebra such as groups and rings and their role in modern mathematics and applied contexts.
Demonstrate accurate and efficient use of advanced algebraic techniques.

Real Analysis

Objectives
This course will provide information about the properties of real numbers, series of real numbers. It also imparts knowledge about convergence and divergence of series, differentiability of real functions and related problems.

Outcomes
After completion of the course, the student will be able to:

Understand the theoretical structures of basic concepts in analysis.
Learn the foundational results in the fields of Real Analysis.
Understand the theoretical foundation and properties of the Riemann Stieltjes integral, Pointwise convergence, Uniform convergence.
Know the definition of Riemann Stieltjes integral, Pointwise convergence, Uniform convergence and how to determine the components of the convergence of a sequence as well as series.
Apply the different type of tests to find the convergence.
Understand Functions of several variables, Partial derivatives, directional derivatives, Jacobian and Lagrange's multiplier method.
Students should know the importance of Real Analysis and its applications in various fields.

Topology

Objectives
This course will provide information about intense foundation in fundamental concepts of point-set topology

Outcomes
After completion of the course, the student will be able to:

Work basic problems (proofs, construction of examples, counter-examples, or argue that a claim is false) in the Topology , Topology of Metric Spaces, Moore Spaces, Tychonoff spaces, and Hausdorff spaces.
Familiar with separability, completeness, connectedness, compactness, densityand basis.

Advanced excel lab

Objectives
This course will provide information about the excel software which help to manage the data and presenting the data in both mathematical and picture form. The representation of data in different forms of graphs and maintaining the data using advance option will be discussed.

Outcomes
After completion of the course, the student will be able to:

Familiar with the software that helps to organize a vast amount of raw data into well organized meaningful information in minimal time frame.

Seminar

Outcomes
Seminar enhance the skills of students in presentation, discussion, listening, critical thinking, studying major work.

Elective I

Complex Analysis

Objectives This course will provide information about the concepts of analyticity,Cauchy integral theorem in different domains. Students will be equippedwith the understanding of the fundamental concepts of complex variable theory. In particular,students will acquire the skill of contour integration to evaluate complicated real integrals viaresidue calculus. Outcomes After completion of the course, the student will be able to:

Understand about the Cauchy-Riemann equations, analytic functions, entire functionsincluding the fundamental theorem of algebra
Evaluate complex contour integrals and apply the Cauchy integral theorem in itsvarious versions, and the Cauchy integral formula.
Analyze sequences and series of analytic functions and types of convergence
Represent functions as Taylor and Laurent series, classify singularities and poles, find residues and evaluate complex integrals using the residue theorem.

Commutative Algebra

Objectives
This course will provide information about algebraic structures. It also imparts knowledge about Noetherian and Aritinian rings and modules over such rings.

Outcomes
After completion of the course, the student will be able to:

Knows basic definitions concerning elements in rings, classes of rings, and ideals in commutative rings.
Know constructions like tensor product and localization, and the basic theory for this.
Know basic theory for noetherian rings and Hilbert basis theorem.
Know basic theory for integral dependence, and the Noether normalization lemma.
Have insight in the correspondence between ideals in polynomial rings, and the corresponding geometric objects: affine varieties.
Know basic theory for support and associated prime ideals of modules, and know primary decomposition of ideals in noetherian rings.
Know the theory of Gröbner bases and Buchbergers algorithm.
Know the theory of Hilbert series and Hilbert polynomials.
Know dimension theory of local rings.

Elective II

Differential Equations

Objectives
This course will provide information about the various results and methods for solving Ordinary Differential equation. It includes many theorems and significant results of first, second and higher order differential equations. Also, provide a brief introduction to Initial value and boundary value problems, Sturm-Liouville problems and autonomous system.

Outcomes
After completion of the course, the student will be able to:

Familiar with various methods of solving Ordinary differential equations.
Solve many Initial and Boundary value Problem exists in different fields of science.
Study of autonomous system will helpful for qualitatively analysis of different types of Ordinary differential equations.

Spherical Trigonometry and Astronomy

Objectives
This course will provide information about the Spherical easel technology tool to form different Geometric objects on the surface of a sphere and give knowledge about some objects in Spherical Geometry.
Outcomes
After completion of the course, the student will be able to:

Know planetary phenomena, geocentric motion of a planet,
determination of longitude and latitude, sextant, dip of the horizon , effect of refraction on sun-rise and disc of the sun , aberration in longitude and latitude .

2nd Semester

Measure theory and Integration

Objectives
This course will provide information about Measurable sets, Measurable functions, Lebesgue Integral, Differentiation and Integration ,the Lebesgue Lp Spaces , their properties and also some of their fruitful applications.

Outcomes
After completion of the course, the student will be able to:

understand how the Lebesgue measure on R is defined.
understand basic properties are measurable functions.
understand how the measures may be used to construct integrals.
know the basic convergence theorems for the Lebesgue integral.
understand the relation between differentiation and Lebesgue integration.

Differential Geometry

Objectives
This course will provide information about the concepts of curvature of a space curve, the fundamental theorem for plane curves, the curvature and torsion of space curves, the fundamental theorem for space curves, the concept of a parameterized surface with the help of examples, the idea of first fundamental form/metric of a surface..To get introduced to geodesics on a surface and their characterization.

Outcomes
After completion of the course, the student will be able to:

Define the equivalance of two curves.
Find the derivative map of an isometry and analyse the equivalence of two curves by applying some theorems.
Define surfaces and their properties express definition and parametrization of surfaces.express tangent spaces of surfaces and
Explain differential maps between surfaces and find derivatives of such maps.integrate differential forms on surfaces

Partial differential Equation

Objectives
This course will provide information about the various types of method for solving partial differential equations and difference equations. Many problems based on the Formation of Difference equations and their solution using Z transforms will be discussed.

Outcomes
After completion of the course, the student will be able to:

Familiar with various methods of solving Partial differential equations.
Solve many difference equation and find out the results using Z-transform

Seminar

Seminar enhance the skills of presentation, discussion, listening, critical thinking etc.

Elective I

Discrete Mathematics

Objectives
This course will provide information about basic concepts of set theory, logic, proof techniques, binary relations, graph and trees.
Outcomes
After completion of the course, the student will be able to:

Construct mathematical arguments using logical connectives and quantifiers.
Validate the correctness of an argument using statement and predicate calculus.
Understand how graphs and trees are used as tools and mathematical models in the study of networks.
Learn how to work with some of the discrete structures which include sets, relations, functions, graphs and trees.

Cryptography

Objectives
This course will provide information about the techniques and some of the foundations of the cryptographic methods.
Outcomes
After completion of the course, the student will be able to:

Understand Cryptography-Encryption schemes, Cryptanalysis, Block ciphers, Stream ciphers and the basics of RSA security
Break the simplest instances.

Elective II

Wavelets

Objectives
This course will provide information how to pursue the more theoretical aspects such as Fourier Analysis.
Outcomes
After completion of the course, the student will be able to:

Understand the properties of various scaling functions and their wavelets.
Understand the properties of multiresolution analysis.
Construct the scaling functions using infinite product formula and iterative procedure.
Implement wavelets in various problems like image compression, denoising etc.

Analytical Mechanics & Variational Methods

Objectives This course will provide information about the variation technique and mechanics that exists in different fields of science. Outcomes After completion of the course, the student will be able to:

Solve the various Scientific problems based on the variations
Familiarize with various theories of classical mechanics.
Solve the various physics problems based on the classical mechanics

Elective III

MAT LAB

Objectives This course will provide information about MATLAB software, to enable the student to carry out simple numerical computations and analysis using MATLAB Outcomes After completion of the course, the student will be able to:

Understand the syntax, semantics, data-types and library functions.
Solve the mathematical equations using different commands and functions.
Interpret and visualize simple mathematical functions and operations thereon using plots/display.

SPSS/R

Objectives This course will provide information about the SPSS/R software that helps to analyze the data, representation of data by using various statistical tools. Outcomes After completion of the course, the student will be able to:

Learn the technique how analyze the data
Representation of data by various graphs.

3rd Semester

Functional Analysis

Objectives
This course will provide information about pure and applied Mathematics, with countless applications to the theory of differential equations, engineering, and physics. The students will be exposed to the theory of Banach spaces, the concept of dual spaces, the Hahn-Banach theorem, the axiom of choice and Zorn's lemma, Open mapping theorem, closed graph theorem. Inner product spaces, Hilbert spaces and their examples
Outcomes
After completion of the course, the student will be able to:

Understand the fundamentals of functional analysis and the concepts associated with the dual of a linear space.
Understand mathematical applications of Functional analysis in pure mathematics such as representation theory.

Solid Mechanics

Objectives
This course will provide information about stress and strain of materials, properties of area, principal axes and moments of inertia, tension and compression, strain energy, torsion.
Outcomes
After completion of the course, the student will be able to:

Understand the concepts of stress at a point, strain at a point, and the stress-strain relationships for linear elastic, homogeneous, isotropic materials.
To determine principal stresses and angles, maximum shearing stresses and angles, and the stresses acting on any arbitrary plane within a structural element.
To utilize basic properties of materials such as elastic moduli and Poisson's ratio to appropriately to solve problems related to isotropic elasticity.
Understand affine transformations and geometrical interpretation of the components of strain and terms related to strain tensor.
Understand the generalized Hooke’s law, reduction of elastic constants to different elastic models from the most general case.
Develop equilibrium and dynamical equations of an isotropic elastic solid.
Solve a problem of strain analysis

Optimization Techniques

Objectives
This course will provide information about solution of real life problems related to business using various optimization techniques.
Outcomes
After completion of the course, the student will be able to:

Model engineering minima/ maxima problems as optimization problems.
Use MATLAB to implement optimization algorithms.

Essentials of Information Technology-I

Objectives
This course will provide information about the Python software. It includes basics of python and formation of python program using python lists, python tuples and how to build the python files using special functions.
Outcomes
After completion of the course, the student will be able to:

Familiar with the basics of python software.
Prepare many programs with the use of python commands.

Python Software

Objectives
This course will provide information about the python environment. It includes the installation of python, basic and advanced programs based on mathematical form that helpful to understand the python formats.
Outcomes
After completion of the course, the student will be able to:

Prepare many mathematical programs with the use of python software.
learn plotting of many types of graphs in python environment.

Seminar

Seminars enhance the skills of students in resentation, discussion, listening, critical thinking etc.

Elective I

Mathematical Statistics

Objectives
This course will provide information about various statistical tools for data analysis.

Outcomes
After completion of the course, the student will be able to use various statistical methods for data analysis

Fuzzy Sets

Objectives
This course will provide information about fuzzy sets, arithmetic operations on fuzzy sets, fuzzy relations, possibility theory, fuzzy logic, and its applications

Outcomes
After completion of the course, the student will be able to:

Construct the appropriate fuzzy numbers corresponding to uncertain and imprecise collected data.
Find the optimal solution of mathematical programming problems having uncertain and imprecise data.
Deal with the fuzzy logic problems in real world problems.

Elective II

Integral Transforms

Objectives
This course will provide information about Integral transforms so that the knowledge can be used in different fields of Science and Engineering.
Outcomes
After completion of the course, the student will be able to:

Apply Laplace Transformation to solve initial and boundary value problems.
Learn Fourier transformation and their applications to relevant problems.
Understand Hankel's Transformation to solve boundary value problem.

Mathematical Finance and Insurance

Objectives
This course will provide information about theoretical and practical aspects of Mathematical models and Analytical methods.
Outcomes
After completion of the course, the student will be able to:

Display basic knowledge and key technical skills in advanced calculus and linear algebra and processes of probability, statistics and stochastic processes as appropriate to financial and insurance mathematics;
Interpret the mathematics that arises across a range of problems in financial and insurance mathematics, including financial and risk models;
Demonstrate skills in the written presentation of a mathematical argument that enable mathematical, financial and insurance concepts, processes and results to be communicated effectively to diverse audiences.

4th Semester

Integral Equation

This course will provide information about the integral equation that exists in different fields of science.

Operation Research

This course will provide information how to solve real life problems related to business using various optimization techniques. Outcomes: After completion of the course, the student will be able to:

Identify and develop operational research models from the verbal description of the real system,
Understand the mathematical tools that are needed to solve optimization problems,
Use mathematical software to solve the proposed models.
Develop a report that describes the model and the solving technique, analyze the results and purpose recommendations in language understandable to the decision-making processes in Management Engineering.

Theoretical Seismology

This course will provide information about basic concepts in Theoretical Seismology, including plane waves, harmonic wave, P-waves, SV-waves, progressive waves and stationary waves. This course will present and emphasize those topics in order to aid the student in his future mathematical studies.

Outcomes:

After completion of the course, the student will be able to:

Discuss the concept and limitations of basic theory for seismic wave propagation.
Understand mathematical representation of waves, elastic waves, and seismic sources.
Understanding of the theory on the basis of examples of application.
Use abstract methods to solve problems. Ability to use a wide range of references and critical thinking.
Obtain the solutions of the wave equations and Lamb’s problems

Latex

This course will provide information about the latex software that helps to prepare the high quality document typesetting which is preferably used for mathematical and scientific papers for various journals. It Includes the basics of Latex software and various options used to prepare the manuscripts or chapters as the requirement of the journals and thesis formats.

Outcomes:

After completion of the course, the student will be able to:

Familiar with the Latex software that helps to prepare their documents.
Prepare the typesetting of journal article, technical report, thesis, books and slide presentation etc.

Seminar

Seminars enhance the skills of students in resentation, discussion, listening, critical thinking etc.

Elective I

Fluid Mechanics

Objectives This course will provide information about nature of a fluid, general analysis of fluid motion, the analysis of the behavior of fluids is based on the fundamental laws of mechanics which relate continuity of mass and energy with force and momentum together with the familiar solid mechanics properties. Outcomes After completion of the course, the student will be able to:

Understand the basic principles of fluid mechanics, such as Lagrangian and Eulerian approach, conservation of mass etc.
Use Euler and Bernoulli's equations and the conservation of mass to determine velocity and acceleration for incompressible and inviscid fluid.
Understand the concept of rotational and irrotational flow, stream functions, velocity potential, sink, source, vortex etc.
Analyse simple fluid flow problems( motion of Cylinders ) with Navier - Stoke's equation of motion.

Theory of vibration with applications

Objectives This course will provide information about advanced multi-degree of freedom system analysis techniques for vibroacoustic systems, including modal analysis, statistical energy analysis and finite element analysis. Introduction to mechanical signature analysis; vibration measurement and instrumentation; signal processing and analysis; filtering; frequency domain analysis; vibration monitoring; introduction to condition monitoring and fault diagnosis.. Outcomes After completion of the course, the student will be able to:

Understand vibration phenomenon.
Model of vibration problems encountered in application
Examine vibration response.
Establish relation between real system and physical model,
Form mathematical model from physical model, methods used examining of vibrations and its usage fields.
Experience in solution of mathematical model and to be interpreted of its results.

Programming and its Applications

After completion of the course, the student will be able to:
Use their knowledge of programming to find the solution for various concepts.

Elective II

Graph Theory

Objectives
This course will provide information about graphs as a powerful tool that can be used to solve practical problems in various fields.
Outcomes
After completion of the course, the student will be able to:

Demonstrate knowledge of the syllabus material
Write precise and accurate mathematical definition of objects in graph theory.
Use of combination of theoretical knowledge and independent mathematical thinking in creative investigation of questions in graph theory.

Number Theory

Objectives
This course will provide information about basic concepts of number theory. Students are able to apply theoretical knowledge to problems related to computer security.
Outcomes
After completion of the course, the student will be able to:

Prove results involving divisibility and greatest common divisors.
Solve systems of linear congruences
Find integral solutions to specified linear Diophantine Equations;
Apply Euler-Fermat’s Theorem to prove relations involving prime numbers.Apply the Wilson’s theorem.

Elective III

Dissertation/Project

Objectives This course will provide information about to develop an understanding of the processes and skills required to undertake a supervised research project at masters level of study. To develop research skills commensurate with the accomplishment of a masters degree ,to develop skills in independent inquiry ,produce a coherent and logically argued piece of writing that demonstrates competence in research. Outcomes After completion of the course, the student will be able to:

Design and manage a piece of original project work.
Synthesize knowledge and skills.
Previously gained and applied to an in depth study.
Establish links between theory and methods within their field of study.
Select from different methodologies, methods and forms of analysis to produce a suitable research.
Present the findings of their project in a written report.

Eligibility Criteria

Duration: 2 yrs.

(1) B.Sc (Hons.) in concerned subject with at least 50% marks in aggregate from any recognized University

(2) B.Sc in full subjects with (Hons.) in concerned subject, obtaining at least 50% marks in aggregate of Hons. Examination from any recognized University

(3) B.Sc. with concerned subject securing at least 50% marks in aggregate from any recognized University

(4) BA with Mathematics as one the main subject (for M.Sc. Mathematics only) securing at least 50% marks in aggregate from any recognized University.

Key Recruiters of the departmental students

Fees

Course

Course Fee

Indian (INR)

International (USD)

Tuition Fee

Development Fee

M.Sc. Mathematics

40,000

5,000

2500

Note:

The Fee will have to be deposited at the time of Counselling.
University Charges will be extra as per University norms.
Fee once paid shall not be refunded under any circumstances.
There will be an annual increase of 7% in tuition fees.
Students Fund for all the students is Rs.2,000/- p.a.
Lifetime Alumni Membership fee is Rs. 4,000/-.
Hostel & Transport Charges will be extra.
Students who wish to leave the Institute mid-stream, will have to deposit the fee for the remaining period of the course as per judgement of the Hon'ble Supreme Court in the case of Islamic Academy of Education and others Vs State of Karnataka & PA lnamdar case.

Merit Scholarships

(Scholarships on the basis of % of Marks in Qualifying Exam)

% Marks	Scholarship Criteria (% Tuition Fee Waiver)
90% & above	100%
80 to 89.99%	40%
70 to 79.99%	30%
60 to 69.99%	20%

Alumni Speaks

Good Teachers are the reason why ordinary students dream to do extraordinary things.Thank you to all my MM(DU) teachers and mentors for guiding me,inspiring me and making me what I am today.

MandeepAssistant Administrative officer(LIC of India)

My two years at MM(DU) have been a wonderful experience of learning with prolific industry exposure. I appreciate the faculty’s dedication towards the students. Thank you for continually inspiring me to do my best!

Veenu Saini Lecturer (Sardar Ajeet inter college Ambetha,Saharanpur)

MM(DU) did wonders to my brain. The education I received here has helped me a lot to inculcate a practical approach. I am thankful to our hard working faculty for helping me in my career.

KritikaAssistant teacher (Sanjay Gandhi School, Ladwa)

Being a student of MM(DU) is one of the best phases of my life. I appreciate the learning environment and excellent infrastructure which motivated me to grow as a better person and achieve my goals. I am extremely thankful to the wonderful and supportive faculty.

Priyanka SoodSoftware Asset Management Associate (software one)

I feel happy studying at an institution of such great reputation like MM(DU) because the institutions follows the philosophy of helping and guiding its students with grooming sessions. Faculty always motivated us to achieve more than what we deserve. Their constant efforts are highly appreciated

Amandeep RawatSubject Expert (Innovalance Learning systems, Delhi)

For me, MM(DU) was the stepping stone to my future. I am so thankful to my professors and this campus for shaping my life.

Ujjawal VermaAssistant Teacher (Ursuline Convent Girls’s Senior Secondary School Purnea, Bihar)