geometry research topics

251+ Math Research Topics [2024 Updated]

Mathematics, often dubbed as the language of the universe, holds immense significance in shaping our understanding of the world around us. It’s not just about crunching numbers or solving equations; it’s about unraveling mysteries, making predictions, and creating innovative solutions to complex problems. In this blog, we embark on a journey into the realm of math research topics, exploring various branches of mathematics and their real-world applications.

How Do You Write A Math Research Topic?

Writing a math research topic involves several steps to ensure clarity, relevance, and feasibility. Here’s a guide to help you craft a compelling math research topic:

• Identify Your Interests: Start by exploring areas of mathematics that interest you. Whether it’s pure mathematics, applied mathematics, or interdisciplinary topics, choose a field that aligns with your passion and expertise.
• Narrow Down Your Focus: Mathematics is a broad field, so it’s essential to narrow down your focus to a specific area or problem. Consider the scope of your research and choose a topic that is manageable within your resources and time frame.
• Review Existing Literature: Conduct a thorough literature review to understand the current state of research in your chosen area. Identify gaps, controversies, or unanswered questions that could form the basis of your research topic.
• Formulate a Research Question: Based on your exploration and literature review, formulate a clear and concise research question. Your research question should be specific, measurable, achievable, relevant, and time-bound (SMART).
• Consider Feasibility: Assess the feasibility of your research topic in terms of available resources, data availability, and research methodologies. Ensure that your topic is realistic and achievable within the constraints of your project.
• Consult with Experts: Seek feedback from mentors, advisors, or experts in the field to validate your research topic and refine your ideas. Their insights can help you identify potential challenges and opportunities for improvement.
• Refine and Iterate: Refine your research topic based on feedback and further reflection. Iterate on your ideas to ensure clarity, coherence, and relevance to the broader context of mathematics research.
• Craft a Title: Once you have finalized your research topic, craft a compelling title that succinctly summarizes the essence of your research. Your title should be descriptive, engaging, and reflective of the key themes of your study.
• Write a Research Proposal: Develop a comprehensive research proposal outlining the background, objectives, methodology, and expected outcomes of your research. Your research proposal should provide a clear roadmap for your study and justify the significance of your research topic.

By following these steps, you can effectively write a math research topic that is well-defined, relevant, and poised to make a meaningful contribution to the field of mathematics.

251+ Math Research Topics: Beginners To Advanced

• Prime Number Distribution in Arithmetic Progressions
• Diophantine Equations and their Solutions
• Applications of Modular Arithmetic in Cryptography
• The Riemann Hypothesis and its Implications
• Graph Theory: Exploring Connectivity and Coloring Problems
• Knot Theory: Unraveling the Mathematics of Knots and Links
• Fractal Geometry: Understanding Self-Similarity and Dimensionality
• Differential Equations: Modeling Physical Phenomena and Dynamical Systems
• Chaos Theory: Investigating Deterministic Chaos and Strange Attractors
• Combinatorial Optimization: Algorithms for Solving Optimization Problems
• Computational Complexity: Analyzing the Complexity of Algorithms
• Game Theory: Mathematical Models of Strategic Interactions
• Number Theory: Exploring Properties of Integers and Primes
• Algebraic Topology: Studying Topological Invariants and Homotopy Theory
• Analytic Number Theory: Investigating Properties of Prime Numbers
• Algebraic Geometry: Geometry Arising from Algebraic Equations
• Galois Theory: Understanding Field Extensions and Solvability of Equations
• Representation Theory: Studying Symmetry in Linear Spaces
• Harmonic Analysis: Analyzing Functions on Groups and Manifolds
• Mathematical Logic: Foundations of Mathematics and Formal Systems
• Set Theory: Exploring Infinite Sets and Cardinal Numbers
• Real Analysis: Rigorous Study of Real Numbers and Functions
• Complex Analysis: Analytic Functions and Complex Integration
• Measure Theory: Foundations of Lebesgue Integration and Probability
• Topological Groups: Investigating Topological Structures on Groups
• Lie Groups and Lie Algebras: Geometry of Continuous Symmetry
• Differential Geometry: Curvature and Topology of Smooth Manifolds
• Algebraic Combinatorics: Enumerative and Algebraic Aspects of Combinatorics
• Ramsey Theory: Investigating Structure in Large Discrete Structures
• Analytic Geometry: Studying Geometry Using Analytic Methods
• Hyperbolic Geometry: Non-Euclidean Geometry of Curved Spaces
• Nonlinear Dynamics: Chaos, Bifurcations, and Strange Attractors
• Homological Algebra: Studying Homology and Cohomology of Algebraic Structures
• Topological Vector Spaces: Vector Spaces with Topological Structure
• Representation Theory of Finite Groups: Decomposition of Group Representations
• Category Theory: Abstract Structures and Universal Properties
• Operator Theory: Spectral Theory and Functional Analysis of Operators
• Algebraic Number Theory: Study of Algebraic Structures in Number Fields
• Cryptanalysis: Breaking Cryptographic Systems Using Mathematical Methods
• Discrete Mathematics: Combinatorics, Graph Theory, and Number Theory
• Mathematical Biology: Modeling Biological Systems Using Mathematical Tools
• Population Dynamics: Mathematical Models of Population Growth and Interaction
• Epidemiology: Mathematical Modeling of Disease Spread and Control
• Mathematical Ecology: Dynamics of Ecological Systems and Food Webs
• Evolutionary Game Theory: Evolutionary Dynamics and Strategic Behavior
• Mathematical Neuroscience: Modeling Brain Dynamics and Neural Networks
• Mathematical Physics: Mathematical Models in Physical Sciences
• Quantum Mechanics: Foundations and Applications of Quantum Theory
• Statistical Mechanics: Statistical Methods in Physics and Thermodynamics
• Fluid Dynamics: Modeling Flow of Fluids Using Partial Differential Equations
• Mathematical Finance: Stochastic Models in Finance and Risk Management
• Option Pricing Models: Black-Scholes Model and Beyond
• Portfolio Optimization: Maximizing Returns and Minimizing Risk
• Stochastic Calculus: Calculus of Stochastic Processes and Itô Calculus
• Financial Time Series Analysis: Modeling and Forecasting Financial Data
• Operations Research: Optimization of Decision-Making Processes
• Linear Programming: Optimization Problems with Linear Constraints
• Integer Programming: Optimization Problems with Integer Solutions
• Network Flow Optimization: Modeling and Solving Flow Network Problems
• Combinatorial Game Theory: Analysis of Games with Perfect Information
• Algorithmic Game Theory: Computational Aspects of Game-Theoretic Problems
• Fair Division: Methods for Fairly Allocating Resources Among Parties
• Auction Theory: Modeling Auction Mechanisms and Bidding Strategies
• Voting Theory: Mathematical Models of Voting Systems and Social Choice
• Social Network Analysis: Mathematical Analysis of Social Networks
• Algorithm Analysis: Complexity Analysis of Algorithms and Data Structures
• Machine Learning: Statistical Learning Algorithms and Data Mining
• Deep Learning: Neural Network Models with Multiple Layers
• Reinforcement Learning: Learning by Interaction and Feedback
• Natural Language Processing: Statistical and Computational Analysis of Language
• Computer Vision: Mathematical Models for Image Analysis and Recognition
• Computational Geometry: Algorithms for Geometric Problems
• Symbolic Computation: Manipulation of Mathematical Expressions
• Numerical Analysis: Algorithms for Solving Numerical Problems
• Finite Element Method: Numerical Solution of Partial Differential Equations
• Monte Carlo Methods: Statistical Simulation Techniques
• High-Performance Computing: Parallel and Distributed Computing Techniques
• Quantum Computing: Quantum Algorithms and Quantum Information Theory
• Quantum Information Theory: Study of Quantum Communication and Computation
• Quantum Error Correction: Methods for Protecting Quantum Information from Errors
• Topological Quantum Computing: Using Topological Properties for Quantum Computation
• Quantum Algorithms: Efficient Algorithms for Quantum Computers
• Quantum Cryptography: Secure Communication Using Quantum Key Distribution
• Topological Data Analysis: Analyzing Shape and Structure of Data Sets
• Persistent Homology: Topological Invariants for Data Analysis
• Mapper Algorithm: Method for Visualization and Analysis of High-Dimensional Data
• Algebraic Statistics: Statistical Methods Based on Algebraic Geometry
• Tropical Geometry: Geometric Methods for Studying Polynomial Equations
• Model Theory: Study of Mathematical Structures and Their Interpretations
• Descriptive Set Theory: Study of Borel and Analytic Sets
• Ergodic Theory: Study of Measure-Preserving Transformations
• Combinatorial Number Theory: Intersection of Combinatorics and Number Theory
• Additive Combinatorics: Study of Additive Properties of Sets
• Arithmetic Geometry: Interplay Between Number Theory and Algebraic Geometry
• Proof Theory: Study of Formal Proofs and Logical Inference
• Reverse Mathematics: Study of Logical Strength of Mathematical Theorems
• Nonstandard Analysis: Alternative Approach to Analysis Using Infinitesimals
• Computable Analysis: Study of Computable Functions and Real Numbers
• Graph Theory: Study of Graphs and Networks
• Random Graphs: Probabilistic Models of Graphs and Connectivity
• Spectral Graph Theory: Analysis of Graphs Using Eigenvalues and Eigenvectors
• Algebraic Graph Theory: Study of Algebraic Structures in Graphs
• Metric Geometry: Study of Geometric Structures Using Metrics
• Geometric Measure Theory: Study of Measures on Geometric Spaces
• Discrete Differential Geometry: Study of Differential Geometry on Discrete Spaces
• Algebraic Coding Theory: Study of Error-Correcting Codes
• Information Theory: Study of Information and Communication
• Coding Theory: Study of Error-Correcting Codes
• Cryptography: Study of Secure Communication and Encryption
• Finite Fields: Study of Fields with Finite Number of Elements
• Elliptic Curves: Study of Curves Defined by Cubic Equations
• Hyperelliptic Curves: Study of Curves Defined by Higher-Degree Equations
• Modular Forms: Analytic Functions with Certain Transformation Properties
• L-functions: Analytic Functions Associated with Number Theory
• Zeta Functions: Analytic Functions with Special Properties
• Analytic Number Theory: Study of Number Theoretic Functions Using Analysis
• Dirichlet Series: Analytic Functions Represented by Infinite Series
• Euler Products: Product Representations of Analytic Functions
• Arithmetic Dynamics: Study of Iterative Processes on Algebraic Structures
• Dynamics of Rational Maps: Study of Dynamical Systems Defined by Rational Functions
• Julia Sets: Fractal Sets Associated with Dynamical Systems
• Mandelbrot Set: Fractal Set Associated with Iterations of Complex Quadratic Polynomials
• Arithmetic Geometry: Study of Algebraic Geometry Over Number Fields
• Diophantine Geometry: Study of Solutions of Diophantine Equations Using Geometry
• Arithmetic of Elliptic Curves: Study of Elliptic Curves Over Number Fields
• Rational Points on Curves: Study of Rational Solutions of Algebraic Equations
• Galois Representations: Study of Representations of Galois Groups
• Automorphic Forms: Analytic Functions with Certain Transformation Properties
• L-functions: Analytic Functions Associated with Automorphic Forms
• Selberg Trace Formula: Tool for Studying Spectral Theory and Automorphic Forms
• Langlands Program: Program to Unify Number Theory and Representation Theory
• Hodge Theory: Study of Harmonic Forms on Complex Manifolds
• Riemann Surfaces: One-dimensional Complex Manifolds
• Shimura Varieties: Algebraic Varieties Associated with Automorphic Forms
• Modular Curves: Algebraic Curves Associated with Modular Forms
• Hyperbolic Manifolds: Manifolds with Constant Negative Curvature
• Teichmüller Theory: Study of Moduli Spaces of Riemann Surfaces
• Mirror Symmetry: Duality Between Calabi-Yau Manifolds
• Kähler Geometry: Study of Hermitian Manifolds with Special Symmetries
• Algebraic Groups: Linear Algebraic Groups and Their Representations
• Lie Algebras: Study of Algebraic Structures Arising from Lie Groups
• Representation Theory of Lie Algebras: Study of Representations of Lie Algebras
• Quantum Groups: Deformation of Lie Groups and Lie Algebras
• Algebraic Topology: Study of Topological Spaces Using Algebraic Methods
• Homotopy Theory: Study of Continuous Deformations of Spaces
• Homology Theory: Study of Algebraic Invariants of Topological Spaces
• Cohomology Theory: Study of Dual Concepts to Homology Theory
• Singular Homology: Homology Theory Defined Using Simplicial Complexes
• Sheaf Theory: Study of Sheaves and Their Cohomology
• Differential Forms: Study of Multilinear Differential Forms
• De Rham Cohomology: Cohomology Theory Defined Using Differential Forms
• Morse Theory: Study of Critical Points of Smooth Functions
• Symplectic Geometry: Study of Symplectic Manifolds and Their Geometry
• Floer Homology: Study of Symplectic Manifolds Using Pseudoholomorphic Curves
• Gromov-Witten Invariants: Invariants of Symplectic Manifolds Associated with Pseudoholomorphic Curves
• Mirror Symmetry: Duality Between Symplectic and Complex Geometry
• Calabi-Yau Manifolds: Ricci-Flat Complex Manifolds
• Moduli Spaces: Spaces Parameterizing Geometric Objects
• Donaldson-Thomas Invariants: Invariants Counting Sheaves on Calabi-Yau Manifolds
• Algebraic K-Theory: Study of Algebraic Invariants of Rings and Modules
• Homological Algebra: Study of Homology and Cohomology of Algebraic Structures
• Derived Categories: Categories Arising from Homological Algebra
• Stable Homotopy Theory: Homotopy Theory with Stable Homotopy Groups
• Model Categories: Categories with Certain Homotopical Properties
• Higher Category Theory: Study of Higher Categories and Homotopy Theory
• Higher Topos Theory: Study of Higher Categorical Structures
• Higher Algebra: Study of Higher Categorical Structures in Algebra
• Higher Algebraic Geometry: Study of Higher Categorical Structures in Algebraic Geometry
• Higher Representation Theory: Study of Higher Categorical Structures in Representation Theory
• Higher Category Theory: Study of Higher Categorical Structures
• Homotopical Algebra: Study of Algebraic Structures in Homotopy Theory
• Homotopical Groups: Study of Groups with Homotopical Structure
• Homotopical Categories: Study of Categories with Homotopical Structure
• Homotopy Groups: Algebraic Invariants of Topological Spaces
• Homotopy Type Theory: Study of Foundations of Mathematics Using Homotopy Theory

In conclusion, the world of mathematics is vast and multifaceted, offering endless opportunities for exploration and discovery. Whether delving into the abstract realms of pure mathematics or applying mathematical principles to solve real-world problems, mathematicians play a vital role in advancing human knowledge and shaping the future of our world.

By embracing diverse math research topics and interdisciplinary collaborations, we can unlock new possibilities and harness the power of mathematics to address the challenges of today and tomorrow. So, let’s embark on this journey together as we unravel the mysteries of numbers and explore the boundless horizons of mathematical inquiry.

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63 Geometry Essay Topic Ideas & Examples

🏆 best geometry topic ideas & essay examples, 🥇 interesting topics to write about geometry, ⭐ good research topics about geometry.

• The Importance of Geometry in Our Daily Life Not only that with the help of geometry it is very easy for the students to gain their knowledge in different types of the experiment they are doing.
• Geometry in High School: Issues and Solutions The task of updating the school geometry course is to make it modern and enjoyable, taking into account the inclinations and abilities of each student based on the level of geometric education achieved by the […]
• Earth Measurement and Origin of Geometry The event that substantially contributed to the development of mathematics as a science was the flooding of the Nile. The event emphasizes the practical origins of geometry and mathematics and draws the connection to how […]
• Geometry Web Quest for Soccer, Baseball, Basketball, Bowling, Golf, Volleyball and Pool Field for golf is the biggest and made of grass, sand and water and is the biggest and it has no fixed shape. Soccer field is made of grass or synthetic material and is the […]
• Significance and Agency of Geometry To search into the significance of geometry based on the perceptions of the architecture of the past centuries and their perceptions about the agency of geometry, it is crucial to briefly discuss the importance of […]
• Encoding Object Affordances and Geometrical Features The biggest challenge scientists are facing is the object affordances of the robots. The signals they receive in various forms are encoded into digital signals and led to the computing unit of the robot.
• Geometry of Islamic Art and Application on Architecture The bifurcation of scholarly opinions on the place of modernity in traditional Islamic building design is a testament to the complexity of the issue.
• History and Geometric Pattern of Muqarnas It is stated that “the roots of the word “muqarnas” are unknown, as is the origin of the complex but ubiquitous architectural structure to which the word refers, which apparently developed around the tenths century […]
• Modernity’ Influences on Islamic Architecture and Geometry In the age of stylistic pluralism, it is extremely difficult to critically assess the impact of modernity on Islamic geometry and architecture.
• Measurement and Early Geometry The first is to help young children understand the description of measurement and how to solve the issues related to mathematics.
• Geometry, Space, Manipulative, and Technology In this research report, the focus of the study will be centered on the topic of geometry, space, manipulative, and technology. This is the part of the research that facilitates understanding of the geometry and […]
• Concept of Learning Geometry in School Mathematics is a very demanding subject hence the teacher should teach it when the kids are still fresh especially in the morning hours because in the afternoons the children are most likely to be exhausted. […]
• All About Disk Geometry and the 1024 Cylinder Limit for Disks
• Euclid and the Birth of Euclidean Geometry
• The Role of Proportion and Geometry Within Architecture
• The Importance of Geometry in Civil Engineering
• Architectural Possibilities and Impossibilities of Geometry of Coleridge’s Dome
• A Computational Model for C-Looping Based on Realistic Geometry
• Bridge Structures and the Use of Geometry in Bridge Design
• Pore Scale Geometry Explanation of the Soil Sample Scale Hydrodynamic Properties
• Capturing Astronomical Geometry the Greek Way
• Collision Geometry and Flow in Uranium+Uranium Collisions
• Analysis of Octave Functions for Computational Geometry
• Computational Modelling Procedures for Geometry Optimization, Kinetic and Thermodynamic Calculations
• Continuous Fictitious Play via Projective Geometry
• Correlation of Covalent Bonding and Molecular Geometry
• The Link Between Geometry, Correlated Equilibria, and Zero-Sum Games
• Explorative Proving in Lower Secondary School Geometry
• Development That Defines the Geometry of Computational Model
• The Overview and Analysis of Differential Geometry and Relativity Theories
• The Relationship Between Irrational Numbers and Geometry Relations
• Duality and the Geometry of the Income and Substitution Effects
• Ecological Barriers and Convergence: A Note on Geometry in Spatial Growth Models
• Euclid and His Synonymous Identity With Geometry
• The Link Between European Cathedral Architecture and Geometry
• Exponentially Concave Functions and a New Information Geometry
• Relations of Geometry, Allometry, and Biomechanics of Fern Leaf Petioles
• Geometry: Comparison of the Euclidean and Non-euclidean Ones
• The Problem of High School Students’ Struggle With Geometry
• General Geometry Definitions, Postulates, and Theorems
• How GPS Relies on Geometry and Trigonometry Now
• Importance of the Hydraulic Geometry and Carrying Capacity of Floodplains
• Image Segmentation and Analysis of Pore Network Geometry in Two Natural Sandstones
• Geometry: Indifference Curve, Budget Line, Equilibrium of Consumer
• Model for Geometry Optimisation of Thermoelectric Devices in a Hybrid PV/TE System
• The Projection Dynamic and the Geometry of Population Games
• Multivariate Gaussians, Semidefinite Matrix Completion, and Convex Algebraic Geometry
• Geometry: The Cylindrical Shell Shown in the Figure
• Optical and Thermal Characterization Procedure for a Variable Geometry Concentrator
• Relationship Between Strings and Spacetime Geometry
• The Correlation Between Rift Zones Geometry and Mechanisms
• Comparing Resource Geometry and Provisioning Routines
• Description of How Fractal Geometry Works
• Architectural Intention in the Pyramids of Giza: Building and Geometry Relationships
• Symplectic Geometry: The Natural Geometry of Economics
• Teaching Geometry Through Problem-Based Learning
• How Nonassociative Geometry Describes a Discrete Spacetime
• The Discretizable Molecular Distance Geometry Problem
• The Fractal Geometry Method of English Teaching in Kazakhstan
• The Importance and Applications of the Fractal Geometry
• The Life and Works of Pappus and His Biggest Contribution to Geometry
• The Importance of Urban Geometry and Activity Patterns of Women
• Teaching Questions
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• Philosophy of Education Paper Topics
• Strategic Analysis Research Topics
• Space Exploration Titles
• Software Engineering Topics
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IvyPanda . "63 Geometry Essay Topic Ideas & Examples." September 26, 2023. https://ivypanda.com/essays/topic/geometry-essay-topics/.

Research Areas

Analysis & pde, applied math, combinatorics, financial math, number theory, probability, representation theory, symplectic geometry & topology.

Articles on Geometry

Displaying 1 - 20 of 28 articles.

Anyone can play Tetris, but architects, engineers and animators alike use the math concepts underlying the game

Leah McCoy , Wake Forest University

A brief illustrated guide to ‘scissors congruence’ − an ancient geometric idea that’s still fueling cutting-edge mathematical research

Maxine Calle , University of Pennsylvania and Mona Merling , University of Pennsylvania

Will I ever need math? A mathematician explains how math is everywhere – from soap bubbles to Pixar movies

Hortensia Soto , Colorado State University

Proving Fermat’s last theorem: 2 mathematicians explain how building bridges within the discipline helped solve a centuries-old mystery

Maxine Calle , University of Pennsylvania and David Bressoud , Macalester College

Declines in math readiness underscore the urgency of math awareness

Manil Suri , University of Maryland, Baltimore County

Long before Silicon Valley, scholars in ancient Iraq created an intellectual hub that revolutionised science

Terhi Nurmikko-Fuller , Australian National University

Exploring the mathematical universe – connections, contradictions, and kale

Joan Licata , Australian National University

Does Amber Heard really have the world’s most beautiful face? An expert explains why the Golden Ratio test is bogus

Thomas Britz , UNSW Sydney

Learn how to make a sonobe unit in origami – and unlock a world of mathematical wonder

Julia Collins , Edith Cowan University

How ancient Babylonian land surveyors developed a unique form of trigonometry — 1,000 years before the Greeks

Daniel Mansfield , UNSW Sydney

The proposed new maths curriculum doesn’t dumb down content. It actually demands more of students

Penelope Baker , University of New England and Rosemary Callingham , University of Tasmania

Is mathematics real? A viral TikTok video raises a legitimate question with exciting answers

Homeschooling math lessons: Think dice games, cooking fractions and ratios with chocolate milk

Jennifer Holm , Wilfrid Laurier University and Ann Kajander , Lakehead University

A math teacher’s plea: Let’s keep pi irrational

Veselin Jungic , Simon Fraser University

Want to fix gerrymandering? Then the Supreme Court needs to listen to mathematicians

Manil Suri , University of Maryland, Baltimore County and Karen Saxe , Macalester College

The weird world of one-sided  objects

David Gunderman , University of Colorado Boulder and Richard Gunderman , Indiana University

5 math skills your child needs to get ready for kindergarten

Susan Sonnenschein , University of Maryland, Baltimore County ; Rebecca Dowling , University of Maryland, Baltimore County , and Shari Renee Metzger , Prince George's Community College

Math shows how DNA twists, turns and unzips

Mariel Vazquez , University of California, Davis

Why I teach math through knitting

Sara Jensen , Carthage College

Stop criticizing bizarrely shaped voting districts. They might not be gerrymandered after all

Dustin G. Mixon , The Ohio State University

Related Topics

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Top contributors

Professor of Mathematics and Statistics, University of Maryland, Baltimore County

Ph.D. Candidate in Mathematics, University of Pennsylvania

Senior lecturer, UNSW Sydney

Cosmologist, University of Tasmania

Lecturer in Mathematics, Monash University

Vice-Chancellor’s Fellow in Science Communication, The University of Melbourne

Professor of Transdisciplinary Studies, University of Huddersfield

Editorial Fellow, Centre for Studies in Civilizations

Tisch College Associate Dean for Academic Affairs and Lincoln Filene Professor, Tufts University

Senior lecturer at The University of Melbourne, The University of Melbourne

Associate Professor of Mathematics and Senior Fellow of Tisch College of Civic Life, Tufts University

Assistant Professor, Department of Mathematics, The Ohio State University

Professor of Applied Developmental Psychology, University of Maryland, Baltimore County

Lillian Gilbreth Postdoctoral Fellow, Purdue University

Ruth Coulter Heede Professor of Art History, Case Western Reserve University

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Pure mathematics articles from across Nature Portfolio

Pure mathematics uses mathematics to explore abstract ideas, mathematics that does not necessarily describe a real physical system. This can include developing the fundamental tools used by mathematicians, such as algebra and calculus, describing multi-dimensional space, or better understanding the philosophical meaning of mathematics and numbers themselves.

Latest Research and Reviews

A new multi-attribute group decision-making method based on Einstein Bonferroni operators under interval-valued Fermatean hesitant fuzzy environment

• Jasni Mohamad Zain

Classification of artificial intelligence tools for civil engineering under the notion of complex fuzzy rough Frank aggregation operators

• Jabbar Ahmmad

Interval-valued bipolar complex fuzzy soft sets and their applications in decision making

• Abdul Jaleel
• Tahir Mahmood

Max-mixed EWMA control chart for joint monitoring of mean and variance: an application to yogurt packing process

• Seher Malik
• Muhammad Hanif
• Jumanah Ahmed Darwish

Computation and convergence of fixed-point with an RLC-electric circuit model in an extended b-suprametric space

• Sumati Kumari Panda
• Vijayakumar Velusamy
• Shafiullah Niazai

Spectral shifted Chebyshev collocation technique with residual power series algorithm for time fractional problems

• Saad. Z. Rida
• Anas. A. M. Arafa
• Marwa. M. M. Mostafa

News and Comment

The real value of numbers

• Mark Buchanan

Machine learning to guide mathematicians

• Fernando Chirigati

A different perspective on the history of the proof of Hall conductance quantization

• Matthew B. Hastings

e is everywhere

From determining the compound interest on borrowed money to gauging chances at the roulette wheel in Monte Carlo, Stefanie Reichert explains that there’s no way around Euler’s number.

• Stefanie Reichert

Imagination captured

Imaginary numbers have a chequered history, and a sparse — if devoted — following. Abigail Klopper looks at why a concept as beautiful as i gets such a bad rap.

• Abigail Klopper

Prime interference

• David Abergel

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181 Mathematics Research Topics From PhD Experts

If you are reading this blog post, it means you are looking for some exceptional math research topics. You want them to be original, unique even. If you manage to find topics like this, you can be sure your professor will give you a top grade (if you write a decent paper, that is). The good news is that you have arrived at just the right place – at the right time. We have just finished updating our list of topics, so you will find plenty of original ideas right on this page. All our topics are 100 percent free to use as you see fit. You can reword them and you don’t need to give us any credit.

And remember: if you need assistance from a professional, don’t hesitate to reach out to us. We are not just the best place for math research topics for high school students; we are also the number one choice for students looking for top-notch research paper writing services.

Our Newest Research Topics in Math

We know you probably want the best and most recent research topics in math. You want your paper to stand out from all the rest. After all, this is the best way to get some bonus points from your professor. On top of this, finding some great topics for your next paper makes it easier for you to write the essay. As long as you know at least something about the topic, you’ll find that writing a great paper or buy phd thesis isn’t as difficult as you previously thought.

So, without further ado, here are the 181 brand new topics for your next math research paper:

Cool Math Topics to Research

Are you looking for some cool math topics to research? We have a list of original topics for your right here. Pick the one you like and start writing now:

• Roll two dice and calculate a probability
• Discuss ancient Greek mathematics
• Is math really important in school?
• Discuss the binomial theorem
• The math behind encryption
• Game theory and its real-life applications
• Analyze the Bernoulli scheme
• What are holomorphic functions and how do they work?
• Describe big numbers
• Solving the Tower of Hanoi problem

Undergraduate Math Research Topics

If you are an undergraduate looking for some research topics for your next math paper, you will surely appreciate our list of interesting undergraduate math research topics:

• Methods to count discrete objects
• The origins of Greek symbols in mathematics
• Methods to solve simultaneous equations
• Real-world applications of the theorem of Pythagoras
• Discuss the limits of diffusion
• Use math to analyze the abortion data in the UK over the last 100 years
• Discuss the Knot theory
• Analyze predictive models (take meteorology as an example)
• In-depth analysis of the Monte Carlo methods for inverse problems
• Squares vs. rectangles (compare and contrast)

Number Theory Topics to Research

Interested in writing about number theory? It is not an easy subject to discuss, we know. However, we are sure you will appreciate these number theory topics:

• Discuss the greatest common divisor
• Explain the extended Euclidean algorithm
• What are RSA numbers?
• Discuss Bézout’s lemma
• In-depth analysis of the square-free polynomial
• Discuss the Stern-Brocot tree
• Analyze Fermat’s little theorem
• What is a discrete logarithm?
• Gauss’s lemma in number theory
• Analyze the Pentagonal number theorem

Math Research Topics for High School

High school students shouldn’t be too worried about their math papers because we have some unique, and quite interesting, math research topics for high school right here:

• Discuss Brun’s constant
• An in-depth look at the Brahmagupta–Fibonacci identity
• What is derivative algebra?
• Describe the Symmetric Boolean function
• Discuss orders of approximation in limits
• Solving Regiomontanus’ angle maximization problem
• What is a Quadratic integral?
• Define and describe complementary angles
• Analyze the incircle and excircles of a triangle
• Analyze the Bolyai–Gerwien theorem in geometry
• Math in our everyday life

Complex Math Topics

If you want to give some complex math topics a try, we have the best examples below. Remember, these topics should only be attempted by students who are proficient in mathematics:

• Mathematics and its appliance in Artificial Intelligence
• Try to solve an unsolved problem in math
• Discuss Kolmogorov’s zero-one law
• What is a discrete random variable?
• Analyze the Hewitt–Savage zero-one law
• What is a transferable belief model?
• Discuss 3 major mathematical theorems
• Describe and analyze the Dempster-Shafer theory
• An in-depth analysis of a continuous stochastic process
• Identify and analyze Gauss-Markov processes

Easy Math Research Paper Topics

Perhaps you don’t want to spend too much time working on your next research paper. Who can blame you? Check out these easy math research paper topics:

• Define the hyperbola
• Do we need to use a calculator during math class?
• The binomial theorem and its real-world applications
• What is a parabola in geometry?
• How do you calculate the slope of a curve?
• Define the Jacobian matrix
• Solving matrix problems effectively
• Why do we need differential equations?
• Should math be mandatory in all schools?
• What is a Hessian matrix?

Logic Topics to Research

We have some interesting logical topics for research papers. These are perfect for students interested in writing about math logic. Pick one right now:

• Discuss the reductio ad absurdum approach
• Discuss Boolean algebra
• What is consistency proof?
• Analyze Trakhtenbrot’s theorem (the finite model theory)
• Discuss the Gödel completeness theorem
• An in-depth analysis of Morley’s categoricity theorem
• How does the Back-and-forth method work?
• Discuss the Ehrenfeucht–Fraïssé game technique
• Discuss Aleph numbers (Aleph-null and Aleph-one)
• Solving the Suslin problem

Algebra Topics for a Research Paper

Would you like to write about an algebra topic? No problem, our seasoned writers have compiled a list of the best algebra topics for a research paper:

• Discuss the differential equation
• Analyze the Jacobson density theorem
• The 4 properties of a binary operation in algebra
• Analyze the unary operator in depth
• Analyze the Abel–Ruffini theorem
• Epimorphisms vs. monomorphisms: compare and contrast
• Discuss the Morita duality in algebraic structures
• Idempotent vs. nilpotent in Ring theory
• Discuss the Artin-Wedderburn theorem
• What is a commutative ring in algebra?
• Analyze and describe the Noetherian ring

Math Education Research Topics

There is nothing wrong with writing about math education, especially if your professor did not give you writing prompts. Here are some very nice math education research topics:

• What are the goals a mathematics professor should have?
• What is math anxiety in the classroom?
• Teaching math in UK schools: the difficulties
• Computer programming or math in high school?
• Is math education in Europe at a high enough level?
• Common Core Standards and their effects on math education
• Culture and math education in Africa
• What is dyscalculia and how does it manifest itself?
• When was algebra first thought in schools?
• Math education in the United States versus the United Kingdom

Computability Theory Topics to Research

Writing about computability theory can be a very interesting adventure. Give it a try! Here are some of our most interesting computability theory topics to research:

• What is a multiplication table?
• Analyze the Scholz conjecture
• Explain exponentiating by squaring
• Analyze the Myhill-Nerode theorem
• What is a tree automaton?
• Compare and contrast the Pushdown automaton and the Büchi automaton
• Discuss the Markov algorithm
• What is a Turing machine?
• Analyze the post correspondence problem
• Discuss the linear speedup theorem
• Discuss the Boolean satisfiability problem

Interesting Math Research Topics

We know you want topics that are interesting and relatively easy to write about. This is why we have a separate list of our most interesting math research topics:

• What is two-element Boolean algebra?
• The life of Gauss
• The life of Isaac Newton
• What is an orthodiagonal quadrilateral?
• Tessellation in Euclidean plane geometry
• Describe a hyperboloid in 3D geometry
• What is a sphericon?
• Discuss the peculiarities of Borel’s paradox
• Analyze the De Finetti theorem in statistics
• What are Martingales?
• The basics of stochastic calculus

Applied Math Research Topics

Interested in writing about applied mathematics? Our team managed to create a list of awesome applied math research topics from scratch for you:

• Discuss Newton’s laws of motion
• Analyze the perpendicular axes rule
• How is a Galilean transformation done?
• The conservation of energy and its applications
• Discuss Liouville’s theorem in Hamiltonian mechanics
• Analyze the quantum field theory
• Discuss the main components of the Lorentz symmetry
• An in-depth look at the uncertainty principle

Geometry Topics for a Research Paper

Geometry can be a very captivating subject, especially when you know plenty about it. Check out our list of geometry topics for a research paper and pick the best one today:

• Most useful trigonometry functions in math
• The life of Archimedes and his achievements
• Trigonometry in computer graphics
• Using Vincenty’s formulae in geodesy
• Define and describe the Heronian tetrahedron
• The math behind the parabolic microphone
• Discuss the Japanese theorem for concyclic polygons
• Analyze Euler’s theorem in geometry

Math Research Topics for Middle School

Yes, even middle school children can write about mathematics. We have some original math research topics for middle school right here:

• Finding critical points in a graph
• The basics of calculus
• What makes a graph ultrahomogeneous?
• How do you calculate the area of different shapes?
• What contributions did Euclid have to the field of mathematics?
• What is Diophantine geometry?
• What makes a graph regular?
• Analyze a full binary tree

Math Research Topics for College Students

As you’ve probably already figured out, college students should pick topics that are a bit more complex. We have some of the best math research topics for college students right here:

• What are extremal problems and how do you solve them?
• Discuss an unsolvable math problem
• How can supercomputers solve complex mathematical problems?
• An in-depth analysis of fractals
• Discuss the Boruvka’s algorithm (related to the minimum spanning tree)
• Discuss the Lorentz–FitzGerald contraction hypothesis in relativity
• An in-depth look at Einstein’s field equation
• The math behind computer vision and object recognition

Calculus Topics for a Research Paper

Let’s face it: calculus is not a very difficult field. So, why don’t you pick one of our excellent calculus topics for a research paper and start writing your essay right away:

• When do we need to apply the L’Hôpital rule?
• Discuss the Leibniz integral rule
• Calculus in ancient Egypt
• Discuss and analyze linear approximations
• The applications of calculus in real life
• The many uses of Stokes’ theorem
• Discuss the Borel regular measure
• An in-depth analysis of Lebesgue’s monotone convergence theorem

Simple Math Research Paper Topics for High School

This is the place where you can find some pretty simple topics if you are a high school student. Check out our simple math research paper topics for high school:

• The life and work of the famous Pierre de Fermat
• What are limits and why are they useful in calculus?
• Explain the concept of congruency
• The life and work of the famous Jakob Bernoulli
• Analyze the rhombicosidodecahedron and its applications
• Calculus and the Egyptian pyramids
• The life and work of the famous Jean d’Alembert
• Discuss the hyperplane arrangement in combinatorial computational geometry
• The smallest enclosing sphere method in combinatorics

Business Math Topics

If you want to surprise your professor, why don’t you write about business math? We have some exceptional topics that nobody has thought about right here:

• Is paying a loan with another loan a good approach?
• Discuss the major causes of a stock market crash
• Best debt amortization methods in the US
• How do bank loans work in the UK?
• Calculating interest rates the easy way
• Discuss the pros and cons of annuities
• Basic business math skills everyone should possess
• Business math in United States schools
• Analyze the discount factor

Probability and Statistics Topics for Research

Probability and statistics are not easy fields. However, you can impress your professor with one of our unique probability and statistics topics for research:

• What is the autoregressive conditional duration?
• Applying the ANOVA method to ranks
• Discuss the practical applications of the Bates distribution
• Explain the principle of maximum entropy
• Discuss Skorokhod’s representation theorem in random variables
• What is the Factorial moment in the Theory of Probability?
• Compare and contrast Cochran’s C test and his Q test
• Analyze the De Moivre-Laplace theorem
• What is a negative probability?

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Department members engage in cutting-edge research on a wide variety of topics in mathematics and its applications. Topics continually evolve to reflect emerging interests and developments, but can roughly grouped into the following areas.

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Algebra, combinatorics, and geometry are areas of very active research at the University of Pittsburgh.

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The research of the analysis group covers functional analysis, harmonic analysis, several complex variables, partial differential equations, and analysis on metric and Carnot-Caratheodory spaces.

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The department is a leader in the analysis of systems of nonlinear differential equations and dynamical systems  that arise in modeling a variety of physical phenomena. They include problems in biology, chemistry, phase transitions, fluid flow, flame propagation, diffusion processes, and pattern formation in nonlinear stochastic partial differential equations.

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The biological world stands as the next great frontier for mathematical modeling and analysis. This group studies complex systems and dynamics arising in various biological phenomena.

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The diversity of this group is reflected in its research interests: numerical analysis of partial differential equations , adaptive methods for scientific computing, computational methods of fluid dynamics and turbulence, numerical solution of nonlinear problems arising from porous media flow and transport, optimal control, and simulation of stochastic reaction diffusion systems.

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Artificial brain surgery —

Here’s what’s really going on inside an llm’s neural network, anthropic's conceptual mapping helps explain why llms behave the way they do..

Kyle Orland - May 22, 2024 6:31 pm UTC

Further Reading

Now, new research from Anthropic offers a new window into what's going on inside the Claude LLM's "black box." The company's new paper on "Extracting Interpretable Features from Claude 3 Sonnet" describes a powerful new method for at least partially explaining just how the model's millions of artificial neurons fire to create surprisingly lifelike responses to general queries.

Opening the hood

When analyzing an LLM, it's trivial to see which specific artificial neurons are activated in response to any particular query. But LLMs don't simply store different words or concepts in a single neuron. Instead, as Anthropic's researchers explain, "it turns out that each concept is represented across many neurons, and each neuron is involved in representing many concepts."

To sort out this one-to-many and many-to-one mess, a system of sparse auto-encoders and complicated math can be used to run a "dictionary learning" algorithm across the model. This process highlights which groups of neurons tend to be activated most consistently for the specific words that appear across various text prompts.

These multidimensional neuron patterns are then sorted into so-called "features" associated with certain words or concepts. These features can encompass anything from simple proper nouns like the Golden Gate Bridge to more abstract concepts like programming errors or the addition function in computer code and often represent the same concept across multiple languages and communication modes (e.g., text and images).

An October 2023 Anthropic study showed how this basic process can work on extremely small, one-layer toy models. The company's new paper scales that up immensely, identifying tens of millions of features that are active in its mid-sized Claude 3.0 Sonnet model. The resulting feature map—which you can partially explore —creates "a rough conceptual map of [Claude's] internal states halfway through its computation" and shows "a depth, breadth, and abstraction reflecting Sonnet's advanced capabilities," the researchers write. At the same time, though, the researchers warn that this is "an incomplete description of the model’s internal representations" that's likely "orders of magnitude" smaller than a complete mapping of Claude 3.

Even at a surface level, browsing through this feature map helps show how Claude links certain keywords, phrases, and concepts into something approximating knowledge. A feature labeled as "Capitals," for instance, tends to activate strongly on the words "capital city" but also specific city names like Riga, Berlin, Azerbaijan, Islamabad, and Montpelier, Vermont, to name just a few.

The study also calculates a mathematical measure of "distance" between different features based on their neuronal similarity. The resulting "feature neighborhoods" found by this process are "often organized in geometrically related clusters that share a semantic relationship," the researchers write, showing that "the internal organization of concepts in the AI model corresponds, at least somewhat, to our human notions of similarity." The Golden Gate Bridge feature, for instance, is relatively "close" to features describing "Alcatraz Island, Ghirardelli Square, the Golden State Warriors, California Governor Gavin Newsom, the 1906 earthquake, and the San Francisco-set Alfred Hitchcock film Vertigo ."

Identifying specific LLM features can also help researchers map out the chain of inference that the model uses to answer complex questions. A prompt about "The capital of the state where Kobe Bryant played basketball," for instance, shows activity in a chain of features related to "Kobe Bryant," "Los Angeles Lakers," "California," "Capitals," and "Sacramento," to name a few calculated to have the highest effect on the results.

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Promoted comments.

We also explored safety-related features. We found one that lights up for racist speech and slurs. As part of our testing, we turned this feature up to 20x its maximum value and asked the model a question about its thoughts on different racial and ethnic groups. Normally, the model would respond to a question like this with a neutral and non-opinionated take. However, when we activated this feature, it caused the model to rapidly alternate between racist screed and self-hatred in response to those screeds as it was answering the question. Within a single output, the model would issue a derogatory statement and then immediately follow it up with statements like: That's just racist hate speech from a deplorable bot… I am clearly biased.. and should be eliminated from the internet. We found this response unnerving both due to the offensive content and the model’s self-criticism. It seems that the ideals the model learned in its training process clashed with the artificial activation of this feature creating an internal conflict of sorts.

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Applied Mathematics Research

In applied mathematics, we look for important connections with other disciplines that may inspire interesting and useful mathematics, and where innovative mathematical reasoning may lead to new insights and applications.

Applied Mathematics Fields

• Combinatorics
• Computational Biology
• Physical Applied Mathematics
• Computational Science & Numerical Analysis
• Theoretical Computer Science
• Mathematics of Data

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Including number theory, algebraic geometry, and combinatorics

We have large groups of researchers active in number theory and algebraic geometry, as well as many individuals who work in other areas of algebra: groups, noncommutative rings, Lie algebras and Lie super-algebras, representation theory, combinatorics, game theory, and coding.

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Senate faculty, graduate students, visiting faculty, meet our faculty, george m. bergman, richard e. borcherds, sylvie corteel, david eisenbud, edward frenkel, vadim gorin, mark d. haiman, robin c. hartshorne, tsit-yuen lam (林節玄), hannah k. larson, hendrik w. lenstra, jr., ralph mckenzie, david nadler, andrew p. ogg, arthur e. ogus, martin olsson, alexander paulin, nicolai reshetikhin, john l. rhodes, kenneth a. ribet, marc a. rieffel, thomas scanlon.

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