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• Arts & Sciences
• Graduate Studies in A&S

Undergraduate Honors

Undergraduate honors in mathematics and statistics.

The Department of Mathematics and Statistics has several different levels of honors and prizes in the department. 

• You can find the requirements of honors at the Washington University Bulletin (scroll to bottom of the page).
• You can follow the newest requirements, or find your entry year and follow those requirements: Washington University Bulletin Prior editions

Distinctions for Majors

For most of the majors in the department, there are three levels of distinction awarded to graduating students:

• Distinction: Usually more advanced coursework at a sufficiently high level.
• High Distinction: Completing an honors project.
• Highest Distinction: Passing graduate qualifer examinations or additional course work.

Latin Honors

Students who graduate with High Distinction are recommended to the College of Arts and Science for Latin Honors (cum laude, magna cum laude, or summa cum laude).  Awards of Latin Honor are controlled by the College and there may be additional requirements that are needed outside of Mathematics and Statistics.

The Honors Thesis

Arts & Sciences mathematics majors who want to be candidates for Latin Honors, High Distinction, or Highest Distinction must complete an honors thesis. Writing an honors thesis involves a considerable amount of independent work, reading, creating mathematics, writing a paper that meets acceptable professional standards, and making an oral presentation of results.

Types of Projects

An honors thesis can take three forms: 

• A thesis that presents significant work by the student on one or more nontrivial mathematics problems.
• A project in mathematical or applied statistics that involves an in-depth analysis of a large data set. To do an honors thesis involving data analysis, it is usually necessary to have completed 3200-493-494 by the end of the junior year, and to have an ability to work with statistical software such as SAS, R or Python.
• A substantial expository paper that follows independent study on an advanced topic under the guidance of a department faculty member. Such a report would involve careful presentation of ideas and synthesis of materials from several sources.

Process and Suggested Timeline

• Junior Year Fall Semester: Talk with your faculty advisor about possible projects.
• Junior Year Spring Semester: Complete the  Honors Proposal Form  and submit it to  Blake Thornton .
• Senior Year, end of January:   Give your advisor a draft abstract and outline of the paper.
• Senior Year, end of February: Give your advisor a rough draft, including your abstract.
• Senior Year, end of March: Complete your final draft and present your work. (Deadline is March 31.)

Finding a Project and an Advisor

Start with the Mathematics and Statistics undergraduate research page .  Then, talk to your major advisor, your instructors and other faculty in the department about your interests, your background.  Be sure to let them know you are interested in doing a honors project.

Departmental Prizes

Each year the department considers graduating majors for three departmental prizes. Recipients are recognized at an annual awards ceremony in April, where they each receive a certificate and a set of honors cords to be worn as part of the academic dress at Commencement. Awards are noted on the student's permanent university record. 

Ross Middlemiss Prize

The Ross Middlemiss Prize is awarded to a graduating math major with an outstanding record. The award was established by former Professor Ross Middlemiss, who taught at Washington University for forty years. From 1936 through the 1960s, Middlemiss authored several books, including a widely popular calculus text that was used in University College courses until the late 1970s.

Putnam Exam Prize

The Putnam Exam Prize is awarded to a graduating senior who has participated regularly in the Putnam Exam Competitionand done exceptionally well throughout his/her time at Washington University.

Martin Silverstein Award

The Martin Silverstein Award was established in memory of Professor Martin Silverstein who, until his death in 2004, was a pioneer in work at the interface of probability theory and harmonic analysis. Each year the department considers for this award students in any major track, but especially those with strengths in probability or statistics.

Brian Blank Award

The Brian Blank Award was established in memory of Professor Brian Blank who passed away in 2018. Each year the Mathematics Department will select distinguished junior(s), majoring in mathematics and statistics.

Honors Program

Please note that the requirements for graduating with honors in Math and Applied Math are as follows:

In addition to completing the requirements for the major in mathematics or applied mathematics, students in the Honors Program must:

(a) earn a GPA of at least 3.5 in upper division and graduate courses in the major and at least 3.3 in all courses taken at the University (As a COVID-term policy modification please note that up to 3 of the 8 upper-division major requirements can be satisfied with P grades as long as those P grades were from a COVID-affected term: Spring 2020, Fall 2020, Spring 2021, Summer 2021.)

(b) complete one of the following:

• Math 196 (senior honors thesis)
• At least 2 graduate classes with grades of A- or higher in each
• At least 1 graduate class with an A- or higher plus 1 graduate class with a Pass grade if that "P" grade is earned in one of the COVID-affected terms of Spring 2020, Fall 2020, Spring 2021 or Summer 2021.  This modification applies to students graduating in the Spring 2020 and in subsequent terms (including 2020-21 and the 2021-22 academic years)

(c) receive the recommendation of the Head Major Advisor [ Note: This is not required in most cases and is only needed if there is some ambiguity in the completion of requirements (a) through (b). ] 

If you have additional questions about the honors requirements and designation, please consult with a  Math Staff Advisor .

Related Resources:

• Petition to Enroll in Math 196 (Honors Thesis)
• Make an appointment with a Math Staff Advisor

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Undergraduate Honors & Senior Thesis

Students excelling in their major coursework and interested in pure math should consider Departmental Honors. Departmental Honors means you will graduate “With Distinction” as opposed to College Honors which is “With Honors”.  The most important component of graduating with Departmental Honors is researching and writing a Senior Thesis.

Requirements for Departmental Honors:

• Must complete a B.S. Mathematics Degree.
• Must satisfactorily complete at least one three-quarter sequence 402-3-4, 424-5-6, or 441-2-3; or two two-quarter sequences from this list. Exceptions must be approved by the chair of the Departmental Honors Committee.
• Must earn a GPA of 3.5 or better in Math coursework completed at the UW.
• Must write a senior thesis (earn a numerical grade for MATH 496).
• Must have a 3.3 minimum cumulative GPA at UW.

Please note: If you are not interested in the College Honors or Departmental Honors in Mathematics, you may still write a Senior Thesis. The process is the same as above, but it does not need to be approved by the Honors Committee.

Research credit (Math 498) may be available with faculty permission.

Beginning of your final year at the UW : think about a thesis topic and seek out a faculty supervisor.  Read below for more details about selecting a topic.

First week of classes the quarter before you expect to graduate: submit a thesis proposal form to the Dept. Honors Committee.  The form is online here: Math Dept. Honors Thesis Proposal Form

Last day of your final quarter: Once your advisor approves the thesis, email it to [email protected] and cc your faculty advisor.  You may also wish to upload it to the University Libraries archive .

Nature of the thesis

The senior thesis shall be an expository account of a topic in pure or applied mathematics related to the student’s area of interest. (Original results or proofs are welcome but are definitely not expected.) The thesis must contain some nontrivial mathematical arguments. (E.g., a non-technical essay on “fractals in nature” would not be acceptable.) The thesis should normally be about 20 to 30 pages in length (double spaced, Times New Roman 12pt font, 1” margins). These figures are guidelines, not rigid requirements. The topic should be something that cannot simply be read out of a standard textbook. Writing the thesis should involve:

• obtaining material from the periodical literature, or
• consulting several books and synthesizing material from them, or
• reading an account of a topic in a book that is substantially more advanced than the student’s regular coursework, digesting it, and putting it into readable form.

Choosing a topic

Finding a topic is the students’ responsibility, although consultation with faculty members is encouraged. The topic must be approved by a faculty member of the Mathematics Department who will supervise the work (the “supervisor”) and by the chair of the Departmental Honors Committee. A Senior Thesis Topic Proposal form can be found at the link above, and should be filled out by the student with the supervisor's support (the Dept. will check in with your supervisor). The topic proposal must be submitted to the chair of the Departmental Honors Committee no later than the end of the first week of classes the quarter preceding the quarter in which the student expects to graduate. Exceptions to this deadline may be granted only by the chair of the Departmental Honors Committee. Students contemplating writing a thesis are strongly encouraged to start thinking about a topic in the autumn quarter of their senior year.

Writing the thesis

The student must register for Math 496 (Honors Senior Thesis) during the last quarter of thesis work. The student may receive three credit hours of W-course credit for writing the thesis. Normally, the students will register for a reading course (Math 498) with the supervisor during the preceding quarter (s). The student will receive three hours of credit for each of these courses, but in exceptional cases, with the approval of the supervisor, the number or credit hours may be increased. The supervisor may allow the student to replace Math 498 with a suitable topics course; however, it is still expected that the student will meet periodically with the supervisor.

There is no specific required thesis template for an undergraduate thesis.  Some students may choose to use a modified version of the graduate thesis templates, but this is not required.

Approval of thesis

The student shall submit a draft of the thesis to the supervisor for comments and criticisms, and then shall submit a final version with appropriate revisions. The supervisor shall read the thesis and certify its acceptability with respect to both content and exposition. In order to ensure sufficient time for these things, the student must submit the first draft no later than three weeks before the last day classes of the quarter in which the student expects to graduate, and the final draft no later than the last day of classes. Exceptions to these deadlines may be granted only by the chair of the Departmental Honors Committee.

Once the thesis has been approved by your faculty supervisor, you will need to email the document to [email protected] (required) as well as submit it to the ResearchWorks archive , part of the University Libraries (optional but strongly recommended).  Submission to the archive will allow your thesis to be included in the dissemination and preservation of scholarly work.  Your thesis will be made publicly available.

Interdisciplinary theses

Theses which are concerned with the application of some part of mathematics to some others field are acceptable, as long as they contain some substantial mathematics. In exceptional cases the student may wish to work most closely with a faculty member in another department in preparing the thesis. However, in such cases the thesis topic and the thesis itself must still be approved by a member of the Mathematics Department.

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Home > Mathematics and Statistics > Math Honors

Honors Theses in Mathematics and Statistics

Honors theses from 2023 2023.

Elliptic Curves Over Finite Fields , Christopher S. Calger

Honors Theses from 2022 2022

Representation Theory and its Applications in Physics , Jakub Bystrický

Decoding Cyclic Codes via Gröbner Bases , Eduardo Sosa

Decomposing Manifolds in Low-dimensions: from Heegaard Splittings to Trisections , Suixin "Cindy" Zhang

Honors Theses from 2021 2021

A Generalized Polar-coordinate Integration Formula, Oscillatory Integral Techniques, and Applications to Convolution Powers of Complex-valued Functions on $\mathbb{Z}^d$ , Huan Q. Bui

Counting Conjugacy Classes of Elements of Finite Order in Compact Exceptional Groups , Qidong He

Honors Theses from 2019 2019

Basis Reduction in Lattice Cryptography , Raj Kane

Primes in Arithmetical Progression , Edward C. Wessel

Honors Theses from 2018 2018

Parametric Polynomials for Small Galois Groups , Claire Huang

Algebraic Number Theory and Simplest Cubic Fields , Jianing Yang

On Spectral Theorem , Muyuan Zhang

Honors Theses from 2017 2017

Chow's Theorem , Yohannes D. Asega

Tying the Knot: Applications of Topology to Chemistry , Tarini S. Hardikar

Normal Surfaces and 3-Manifold Algorithms , Josh D. Hews

Some Examples of the Interplay Between Algebra and Topology , Joseph D. Malionek

Honors Theses from 2015 2015

The Central Hankel Transform , Matthew J. Levine

Quantization of Analysis , Kelvin K. Lui

Honors Theses from 2013 2013

The Eichler-Selberg Trace Formula for Level-One Hecke Operators , Alex Barron

A Monte Carlo Simulation Study of the Performance of Hypothesis Tests Under Assumption Violations , Gareth Cleveland

Unknotting Number and Combinatorial Sutured Manifold Theory , Alexander Rasmussen

Honors Theses from 2012 2012

The Radon Transform and the Mathematics of Medical Imaging , Jen Beatty

Recurrence Relations, Fractals, and Chaos: Implications for Analyzing Gene Structure , Sarah. M. Harmon

Odd or Even: Uncovering Parity of Rank in a Family of Rational Elliptic Curves , Anika Lindemann

Honors Theses from 2010 2010

Describing Gray Wolf Movement Using Brownian Motion and PDEs , Ashley M. Blum

No, I’m really, really bad at math: Competition for self-verification , Alexandra E. Wesnousky

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Mathematics Honors Program presentations are held each May. These presentations were based on theses submitted for examination to the Mathematics Department Honors Committee.

The Mathematics Department Honors Committee determines the level of departmental honors awarded (Honors with Distinction, Honors with High Distinction, or Honors with Highest Distinction), based on the student's GPA in the major and the quality of the honors work.

If you are interested in attending the Honors Program Presentations, please send an email to [email protected].

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Home > Arts and Sciences > Mathematics > MATHEMATICSHONORS

Mathematics Undergraduate Honors Theses

Honors theses from 2023 2023.

Automorphisms of a Generalized Quadrangle of Order 6 , Ryan Pesak

Demographic Noise and Fragmentation in Stochastic Extinction , Cameron Curtis

Design and Application of Surrogate Models for Hypersonic Inlet Physics , Owen Guch

Examining Factors Using Standard Subspaces and Antiunitary Representations , Paul Anderson

Merging Cross-Platform Gene Expression Data , Jiayi Xu

Minimal Network Structure for Turing Instability , Brendan Millis

Modeling the Effects of Seasonal Births and Predation on Disease Spread , Ally Introne

The Probability Distribution of the Kaplan-Meier Product-Limit Estimator and its Application to Bias and Interval Estimation , Yuxin Qin

Honors Theses from 2022 2022

Approximating Star-Discrepancy with a Genetic Algorithm , Isabel Agostino

Counting Holes in Physical Systems - Applications of Computational Homology to Systems in Physics - , Sage Stanish

Enumerating Switching Isomorphism Classes of Signed Graphs , Nathaniel Healy

Investigating Text Mining Techniques Within the Context of Politicized Social Media Data , Grace Smith

Investigating the Effectiveness of GARCH(1,2) and COGARCH(1,2) Models in Estimating Volatility in the S&P500 Index , Ethan Hackett

Modeling and Analyses of Mechanisms Underlying Network Synaptic Dynamics in Two Neural Circuits , Linda Ma

Modern Theory of Copositive Matrices , Yuqiao Li

Period Doubling Cascades from Data , Alexander Berliner

Schur Class Power Series over the Quaternions , Rongbiao Wang

The Enumeration of Minimum Path Covers of Trees , Merielyn Sher

The Minimum Number of Multiplicity 1 Eigenvalues Among Real Symmetric Matrices Whose Graph is a Tree , Jacob Zimmerman

Voting Rules and Properties , Zhuorong Mao

Honors Theses from 2021 2021

A Survey of Methods to Determine Quantum Symmetry of Graphs , Samantha Phillips

Blockchain in Healthcare: a New Perspective from Social Media Data , Andrew Caietti

Determining Quantum Symmetry in Graphs Using Planar Algebras , Akshata Pisharody

Reality and Strong Reality in Finite Symplectic Groups , Spencer Schrandt

The Minimum Number of Multiplicity 1 Eigenvalues among Real Symmetric Matrices whose Graph is a Tree , Wenxuan Ding

Toward a Holographic Transform for the Quantum Clebsch-Gordan Formula , Ethan Shelburne

Using Machine Learning to Track the Location of the Shock Train in Hypersonic Engines , Alison Reynolds

Honors Theses from 2020 2020

A Mathematical Model for the Trend and Prediction of Movie Revenue , Yuxin Shang

Analysis of Modelling Deficiencies that Contributed to the High Unanticipated Loan Losses Incurred During the Housing Price Collapse of the Great Recession , Jennifer Shulman

Dynamics of Sensory Integration of Olfactory and Mechanical Stimuli Within the Response Patterns of Moth Antennal Lobe Neurons , Harrison Tuckman

Learning & Planning for Self-Driving Ride-Hailing Fleets , Jack Morris

Non-linear Modifications of Black-Scholes Pricing Model with Diminishing Marginal Transaction Cost , Kaidi Wang

Nonnegative Matrix Factorization Problem , Junda An

RMSE-Minimizing Confidence Intervals for the Binomial Parameter , Kexin Feng

Stage-structured Blue Crab Population Model with Fishing, Predation and Cannibalism , Fangming Xu

Honors Theses from 2019 2019

A General Weil-Brezin Map and Some Applications , Benjamin Bechtold

Continuous Opinion Dynamics on an Adaptive Network , Xinyu Zhang

Disentanglement of Whisker Deflection Velocity and Direction , Srijan Bhasin

Eventually Positive Matrices and Tree Sign Patterns , Madellyne Waugh

Minimal Principal Series Representations of SL(3,R) , Jacopo Gliozzi

Partial Difference Sets in Nonabelian Groups and Strongly Regular Cayley Graphs , Gabrielle Tauscheck

Rankin-Cohen Brackets and Fusion Rules for Discrete Series Representations of SL(2,R) , Emilee Cardin

Spectrogram Analysis of Blood Pressure on Neonates with Hypoxic Ischemic Encephalopathy (HIE) , Tianrui Zhu

Totally Positive Completable Matrix Patterns and Expansion , David Allen

Honors Theses from 2018 2018

A Mathematical Study of Competition and Adoption of Two Consumer Products , Chengli Huang

A New Upper Bound for the Diameter of the Cayley Graph of a Symmetric Group , Hangwei Zhuang

Counting Real Conjugacy Classes in Some Finite Classical Groups , Elena Amparo

Implementation and Analysis of the Nonlinear Decomposition Attack on Polycyclic Groups , Yoongbok Lee

Modeling Social Interactions of Yeast Biofilms with a Stochastic Spatial Simulation , Aparajita Sur

Potential Stability of Matrix Sign Patterns , Christopher Hambric

Strongly Real Conjugacy Classes in Unitary Groups over Fields of Even Characteristic , Tanner N. Carawan

The Doubly Stochastic Single Eigenvalue Problem: An Empirical Approach , John Wilkes and Charles Royal Johnson

TP Matrices and TP Completability , Duo Wang

Ultra-High Dimensional Statistical Learning , Yanxin Xu

Honors Theses from 2017 2017

A Mathematical Model of Economic Growth of Two Geographical Regions , Xin Zou

Center Manifold Theory and Computation Using a Forward Backward Approach , Emily E. Schaal

Involutions and Total Orthogonality in Some Finite Classical Groups , Gregory K. Taylor

Saving Babies Using Big Data , Evan Dienstman

Spatial Analysis with Applications on Real Estate Market Price Prediction , Yujing Zheng

TP and TN Completability of Border Patterns , Haoge Chang

Honors Theses from 2016 2016

Computing All Isolated Invariant Sets at a Finite Resolution , Martin Salgado-Flores

Graph packing with constraints on edges , Fangyi Xu

Growing Networks with Positive and Negative Links , Corynne Smith Dech

Normal Matrices Subordinate to a Graph , Morrison Turnansky

Relaxed Coloring of Sparse Graphs , Michael C. Kopreski

Row and Column Distributions of Letter Matrices , Xiaonan Hu

(Un)Stable Manifold Computation via Iterative Forward-Backward Runge-Kutta Type Methods , Dmitriy Zhigunov

Honors Theses from 2015 2015

On the Non-Symmetric Spectra of Certain Graphs , Owen Hill

Relaxation of Planar Graphs With d∆≥2 and No 4-Cycles , Heather A. Hoskins

Honors Theses from 2014 2014

Basins of Attraction for Pulse-Coupled Oscillators , Ryan Gryder

Combinatorially Derived Properties of Young Tableaux , James R. Janopaul-Naylor

Linear and Nonlinear Trees: Multiplicity Lists of Symmetric Matrices , Eric Wityk

Local Zeta Functions over p-Adic Fields , Stephen P. Cameron

Nonlinear Models of Zooplankton Communities , Catherine King

Honors Theses from 2013 2013

A Population Density Model of Domain Calcium-Mediated Inactivation of L-Type Ca Channels , Kiah Hardcastle

Finding Open Locating Dominating Sets on Infinite and Finite Graphs , Allison Oldham

On Almost Normal Matrices , Tyler J. Moran

Statistical Inference Based on Upper Record Values , Daniel J. Luckett

Honors Theses from 2012 2012

A Model for Blue Crab Population in the Chesapeake Bay , Timothy J. Becker

Analysis and Simulation of an Optimal Control Model of an Oyster Population Displaying an Allee Effect , Timothy Raymond McDade

Circadian Oscillations of the Intestinal Stem Cell Lineage , Brian Waldman

Finding the Minimum Randic Index , Sarah Joyce Kunkler

Fixed Points of Pick and Stieltjes functions: A Linear Algebraic Approach , Nicholas Andrew Woods

Global Dynamics of Pulse-Coupled Oscillators , Allison Leslie Corish

Perfect Partitions of Some (0,1)-Matrices , Jeffrey Soosiah

Permutations with Extremal Routings on Cycles , Luis Alejandro Valentin

Strongly Real Conjugacy Classes of the Finite Unitary Group , Zach Gates

Synchronous Oscillatory Solutions in a Two Patch Predator-Prey Model , Matthew H. Becker

The Laplacian on Isotropic Quantum Graphs: Solutions and Applications , Patrick King

Honors Theses from 2011 2011

A New Lower Bound on the Minimum Density of Vertex Identifying Codes for the Infinite Hexagonal Grid , Ariel J. Cukierman

Basins of Attraction in Stage Structured Populations , Georgia Waite Pfeiffer

Critical Exponents: Old and New , Olivia J. Walch

Factoring Banded Permutations and Bounds on the Density of Vertex Identifying Codes on the Infinite Snub Hexagonal Grid , Chase A. Albert

Persistent Activity in Assortative Networks of Integrate and Fire Neurons , Matthew D. Peppe

Poles and Zeros of Generalized Carathéodory Class Functions , Yael Gilboa

Solution Theory for Systems of Bilinear Equations , Dian Yang

Topological Characterization of Extinction in a Coupled Ricker Patch Model , Benjamin Robert Holman

Honors Theses from 2010 2010

Bistability in Differential Equation Model of Oyster Population and Sediment Volume , William Crowell Jordan-Cooley

Equitable and Defective Coloring of Sparse Graphs , Harold Lee Williams II

Finding a Sparse Solution of a Linear System with Applications to Coding Theory and Statistics , Andrew Gordon Wilcox

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Honours Mathematics

Mathematics is everywhere. It's in your digital alarm clock, the weather, the Internet, your retail purchases and so much more. Mathematics is at the core of everything we do and is the foundation of many fields of study, such as commerce, computing, engineering, and science — and essential for asking many fundamental questions about how our world works.

If you enjoy mathematics, whether it's through numbers, patterns, puzzles or symbols, then studies in mathematics may be right for you. 

Honours Mathematics allows you to explore and consider a number of different mathematical fields. If you're not sure which mathematics you like best, you have time to decide. All of our current students in Honours Mathematics take a general first year and study calculus, linear algebra and computer science — with the option to take more mathematics courses to discover their interests. Starting as early as second year, they'll choose a major in Honours Mathematics to focus their math studies.

There are 14 majors for you to choose from within Honours Mathematics. You can study one of these programs by applying to  Mathematics. We do not offer direct entry for these majors. 

It is possible to customize your studies by combining majors and adding a minor or two. A conversation with an academic advisor will be needed to declare your major and to add a second major or minor. Some majors will require specific courses to be taken to prove eligibility to declare.  

Explore each of our majors to learn more about their courses, co-op and career opportunities and student experiences. 

• Actuarial Science
• Applied Mathematics
• Biostatistics
• Combinatorics and Optimization
• Computational Mathematics
• Data Science
• Mathematical Economics
• Mathematical Finance
• Mathematical Optimization
• Mathematical Physics
• Mathematical Studies
• Mathematics/Teaching
• Pure Mathematics

Looking for more specialized programs?

Explore studies in our Math and Business and Computer Science programs. 

Mathematics/Bridge to Academic Success in English (BASE)

If you meet all of our  admission requirements , except the  English language requirement , you may receive an alternative offer to the Bridge to Academic Success in English program, known as  Math/BASE . You won't be able to apply to this program. You'll be automatically considered, if you're eligible. 

Applied Mathematics

• Undergraduate Program

Each of the Applied Mathematics concentrations allows exceptional students to pursue honors, which involves in-depth project work with faculty.

Outline of Honors Requirements

• Excellence in grades
• Completion of an in-depth, original research project carried out under the guidance of a Brown-affiliated faculty advisor
• Completion of an honors thesis describing this research 
• Completion of two semesters of independent study courses while working on the honors thesis

Deadlines and requirements for honors differ between joint concentrations due to the different needs and scales of each program. Check the details for each concentration below:

• Be in good academic standing by the end of the seventh (or penultimate) semester.
• Earn grades of A or S-with-distinction in at least 70% of the Brown University courses used for concentration credit, excluding calculus and linear algebra, or be in the upper 20% of the student's cohort (as measured by the fraction of grades of A or S-with-distinction among courses used for concentration credit, excluding calculus and linear algebra) by the end of the seventh (or penultimate) semester. (Since S-with-distinctions do not appear on the internal academic record or the official transcript, the department will consult directly with the Registrar’s Office to confirm a student’s grades in concentration courses.)
• Secure a faculty advisor and at least one second reader for the proposed honors thesis project. One of the advisors/readers must be an Applied Mathematics faculty member. 
• Meet regularly, as agreed upon, with their honors thesis advisor and provide regular written drafts on the thesis project.
• Complete an honors thesis that is approved by the faculty advisor and second reader(s) prior to the deadline in the students eighth (or final) semester. Deadlines and guidance about the honors thesis are described below. 
• Email a copy of their approved thesis to Student Affairs Manager Candida Hall ( [email protected] ) in Applied Math for archival purposes prior to the deadline.
• Give an oral presentation of the honors thesis at an approved venue, usually the senior thesis day in Applied Mathematics.
• Complete two semesters of independent study courses while working on the honors thesis, such as APMA 1970/1971 or BIOL 1950/1960 or ECON 1960/1970 or CS 1970. One of these courses can be used to fulfill the senior seminar requirement of the APMA ScB, but they cannot be used to fulfill other concentration requirements. 
• Obtain permission to pursue honors from the department by submitting a completed Honors Declaration Form to our Student Affairs Manager Candida Hall by the deadline (usually at the beginning of the seventh semester). The honors declaration form requires signatures from the thesis advisor, second reader, and concentration advisor, as well as, a brief description of the proposed thesis research. It also requires a preliminary check of eligibility requirements, including the fraction of quality grades.
• Secure a faculty advisor and at least one second reader for the proposed honors thesis project. One of the advisors/readers must be an Applied Mathematics faculty member and one must be a Biomed-affiliated faculty member. 
• Email a copy of their approved thesis to our Student Affairs Manager, Candida Hall ( [email protected] ) in Applied Math for archival purposes prior to the deadline.
• Give an oral presentation of the honors thesis at an approved venue, usually at the senior thesis day in Applied Mathematics or in Biology.
• Complete two semesters of independent study courses while working on the honors thesis, such as APMA 1970/1971 or BIOL 1950/1960. One of these courses can be used to fulfill the research course requirement of the concentration, but they cannot be used to fulfill other concentration requirements. These extra independent study courses are included in the calculation of quality grades described above.
• Obtain permission to pursue honors from the department by submitting a completed Honors Declaration Form to our Student Affairs Manager, Candida Hall ( [email protected] ) by the deadline (usually at the beginning of the seventh semester). The honors declaration form requires signatures from the thesis advisor, second reader, and concentration advisor, as well as, a brief description of the proposed thesis research. It also requires a preliminary check of eligibility requirements, including the fraction of quality grades.
• APMA-CS concentrators can choose to pursue honors within either APMA or CS, but their primary thesis advisor must be in the department that they choose. Students wishing to do honors research with a non-APMA or CS advisor should contact the Directors of Undergraduate Studies in APMA and CS to discuss options.
• Students pursuing honors within APMA should follow the APMA requirements described above.
• Students pursuing honors within CS should follow the CS requirements and deadlines described here: http://cs.brown.edu/degrees/undergrad/concentrating-in-cs/honors/
• Students pursuing honors in CS must also email a copy of their approved thesis to our Student Affairs Manager, Candida Hall ( [email protected] ) in Applied Math for archival purposes.
• APMA-Econ concentrators can choose to pursue honors within either APMA or Econ, but their primary thesis advisor must be in the department that they choose.
• Students pursuing honors within APMA should follow the APMA requirements described above. 
• Students pursuing honors within Econ should follow the Econ requirements and deadlines described here: https://economics.brown.edu/academics/undergraduate/honors-and-capstones/thesis
• Students pursuing honors in Econ must also email a copy of their approved thesis to our Student Affairs Manager, Candida Hall ( [email protected] ) in Applied Math for archival purposes.

APMA Deadlines for Honors Program

Applied Mathematics Honors Declaration Form

Honors Thesis Guidelines:

Mathematical Content :

• Research problem: The thesis should be written on a mathematical problem or on an application that is approached using mathematical techniques. The thesis should demonstrate that the research question is significant and important.
• Thoroughness: The thesis should put the research problem into a broader context, address it in a convincing and thorough manner, and use mathematical approaches that are sound, feasible, and appropriate to the research problem.
• Depth: The thesis should involve mathematics at the level of 1000-level APMA courses and should demonstrate a solid understanding of the mathematics used in the thesis.

Writing Quality :

• Organization: The thesis should have a clear and coherent organization that effectively develops the central idea. There is an introduction that includes a clear statement of the research problem and an outline of the research method. Throughout the paper, arguments are presented clearly and in logical order, and the conclusions are precise and concise. The thesis does not contain awkward or unexpected transitions.
• Clarity: The thesis must be clearly written; in particular, the mathematical content must be clear to the intended audience. It should be clear from the writing that the student has a correct and complete understanding of the mathematical content of the thesis. Assertions are clearly stated and well supported.
• Citations: All sources used in the thesis should be referenced and cited completely and correctly: it should become clear what information from other sources has been integrated into the thesis and where that information came from. The bibliography should also contain an accurate and reasonably complete list of related works and papers.
• Grammar and Orthography: The thesis should be properly formatted and free of errors of grammar, spelling, and punctuation. The tone should be professional.

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School of Mathematics and Statistics - Theses

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• Item No Preview Available The mathematics of individual-based modelling: developing a realistic model of simple epithelial tissue Germano, Domenic Paul Joe ( 2023-05 ) Simple epithelia are the functioning components of many tissues found throughout the body. However, they are susceptible to disruption, which can lead to diseases such as cancer, asthma, cardiac disease, and viral infections. Before we can understand how these diseases occur, we must first understand how these tissues are normally maintained. Individual-based modelling is one such way to study simple epithelia. This thesis aims to gain a better understanding of the mathematics and mechanisms that underpin realistic individual-based models. We use these findings to develop a realistic model of simple epithelia. This research consists of two key parts. The first part focuses on understanding the fundamental mathematical constructions of individual-based models. In this research, we investigate three individual-based models of tissue dynamics: Overlapping Spheres, Voronoi Tessellations and Vertex Models. We investigate how particular modelling assumptions made at the tissue and cell boundaries affects both tissue growth and tissue collision. We find that all models are sensitive to their boundary description, with Overlapping Spheres models being highly sensitive to evolutionary time-scale, tissue structures of Voronoi Tessellation models being highly sensitive to their tissue shape, and Vertex Models being the lest sensitive description. This research emphasises the importance of thorough mathematical understanding to undertake model selection for specific problems, as to ensure macroscopic tissue behaviours are not artefacts of model selection. Upon understanding the importance of model selection, we then consider the sensitivity of the Centre-based models of Overlapping Spheres and Voronoi Tessellation models. By investigating the models’ parameters, we demonstrate how they contain two independent time scales of tissue evolution. We also provide a guide for numerically solving the equations of motion and demonstrate how naive parameter choices can result in unstable behaviour. Finally, to ensure biologically realistic dynamics in the model, a degree of Brownian motion should be incorporated, unless a tissue maintains high cell renewal. After understanding the fundamental mathematics of individual-based models, the second part of this research introduces a novel three-dimensional model of simple epithelia. Our description of the simple epithelia is deformable and consists of multiple layers. The movement of cells within the tissue is regulated by minimising a bending potential, cell-cell adhesion, and cell viscosity. We demonstrate that this model is robust to tissue relaxation and dynamic homoeostasis while undergoing renewal. Lastly, we also show how the description is capable of maintaining the structure at dynamic homoeostasis under regeneration via cell migration and removal, and we show the model is comparable to that of a fixed geometry, without the need for the unrealistic limitations. Finally, we show how our novel model can describe tissues with curved surfaces, applying the model to describe spherical organoids under regimes of relaxation and renewal, showing that dynamic homoeostasis is maintained. We propose a novel extension that is capable of maintaining actively deforming structures, in specified regions within the tissue, to describe highly generalised tissue structures. We demonstrate that this extended model exhibits robustness under tissue relaxation and renewal, while undergoing active tissue deformations. Finally, we show our description of general simple epithelial can describe tissue regeneration via cell migration and removal, while undergoing active tissue deformations. The results and findings of this research will prove valuable to better understanding the mechanisms that contribute to simple epithelial tissue maintenance and homoeostasis within the human body.

• Item No Preview Available Modelling high-dimensional spatial-temporal data: a focus on nonstationary and nonlinear phenomena for a future-focused landslide early-warning system Zheng, Hangfei ( 2023-11 ) Rainfall-induced landslides are seeing an increase as the by-products of climate change and present significant damage to the environment, society, and human lives. These landslides are nonstationary and nonlinear phenomena often recorded as high-dimensional vector time series manifesting spatiotemporal dependence. Modelling and forecasting landslides are difficult, with the challenges coming from the complexity of the underlying time series and the remote-sensing techniques used in obtaining these monitoring data. Also, these time series may be of irregular frequency and contain missing values. We tackle these challenges by developing statistical forecasting tools for a future-focused landslide early-warning system (LEWS). These forecasting tools include our developed statistical models characterising complex time series, dimension reduction for efficient dynamic data representation, and three complementary risk assessment prongs to turn the derived forecasts into early-warning predictions of slope failure. Our proposed models are based on a novel spatial dimension reduction technique called empirical dynamic quantiles (EDQ). The idea behind this technique is to use a small number of representative EDQ series from the observed time series to surmise the whole dataset. We then perform various statistical analyses based on these representative EDQ series which will be computationally feasible. The general form of our time series model combined two advanced econometric methods error-correction cointegration (ECC), vector autogregression (VAR) and a nonlinear function $\boldsymbol{c}(t)$ with the EDQ method named ECC-VAR-$\boldsymbol{c}(t)$-EDQ model. We use this model to deal with these high-dimensional, spatial-temporal dependent vector time series with nonstationary and nonlinear phenomena. For different purposes in practice, we provide two methods to estimate the nonlinear function $\boldsymbol{c}(t)$ and further improve the forecasting accuracy. One is the \emph{empirical function-based method} and the other is \emph{physical-based method}. Once the form of $\boldsymbol{c}(t)$ has been determined, we can use the generalised least square (GLS) to estimate the unknown parameters involved in this ECC-VAR-$\boldsymbol{c}(t)$-EDQ model after performing our developed nonlinear cointegration test. The above-mentioned model is in a situation where there are no missing values and for high-frequency data. To apply the general time series analysis for low-frequency data with some missing values, we develop a model that combines the stochastic differential equations (SDE) and Markov chain Monte Carlo (MCMC) approach with the EDQ technique named SDE-MCMC-EDQ model. The basic idea behind this model is that we can convert these low-frequency time series to high-frequency by introducing some additional data between every two consecutive observations implies the estimation of these unknown data in addition to the SDE model parameters, where both these imputed data and the parameters in SDE are treated as random variables. The reproducibility and robustness of all these developed models are assessed by the application of different real-world ground motion data or simulation studies. Results found well fitted these different slope data with the goodness of fit statistic ($0.94\leq R^2\leq0.99$) close to 1. In addition to the forecast values derived from our developed models, we use three risk assessments in parallel to predict where, when, and risk of failure for supporting a complete future-focused LEWS which more accessible to the public.

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Honours in the School of Mathematics and Statistics

General information.

This page contains brief information on Honours Programs in the School of Mathematics and Statistics at the University of Sydney.

Applied Mathematics

• Financial Mathematics and Statistics

Pure Mathematics

While the school's research groups offer courses separately, the honours programs are flexible and students are encouraged to consult their program coordinator if they are interested in taking a course from another discipline.

Lists of the lecture units offered in 2023 by Applied Mathematics , Pure Mathematics and Statistics appear below.

Students should be aware of the University's plagiarism policy .

Students have the right to appeal any academic decision made by the School or Faculty. For further information , see the Science Faculty web site.

Announcements

The school's annual the honours information session.

You can learn more about our honours program in general, and about our specific programs by attending our school's annual Honours Information Session on Tuesday 12/9/2023.

Worried about not having two majors? Don't be!

If you are not on track to qualify for two majors before starting your Honours studies then you should apply for BSc (Honours).

Enrolment advice

Enrolment advice, FAQs and Student Policies can be found on the School of Mathematics and Statistics Student Portal , on the Canvas website, or on the Enrolment Advice page on this server.

Master of Mathematical Sciences

Information about this postgraduate degree can be found on the page https://sydney.edu.au/courses/courses/pc/master-of-mathematical-sciences.html

2021 AMSI Summer School

The School will run from 11/1/2021 to 5/2/2021 as a virtual event hosted by The University of Adelaide. More information is available .

Applying for Honours

When applying for Honours through the faculty of science web interface (BAS / BSc ) you will be asked to upload a "proof of contact" / "expression of interest" document. For this purpose, please download and fill the School's Expression of Interest form .

Please keep in mind the Faculty of Science honours application deadlines (15/1 for semester 1 and 25/6 for semester 2) .

In particular potential students from other states should be aware of the Honours Relocation Scholarship. This scholarship valued at $6000 is to facilitate students from interstate enrolling in an honours degree in the Faculty of Science at the University of Sydney by assisting to defray costs incurred in moving interstate in order to enrol in the honours year. The student must have a minimum SCIWAM of 80 and with the current terms must be moving from outside of NSW. See the Faculty of Science honours scholarships information page.

Program Coordinators

The following people are available to discuss the honours program.

The handbooks contain detailed information including assessment details, important dates, course outlines, an outline of the project requirement including a list of possible projects and supervisors. Students should ensure they have a copy of the relevant handbook.

You should start with a general overview of the Honours Program . Then, for more details, consult the handbooks for Honours in Applied Mathematics ( 2022 , 2023 ), Financial Mathematics and Statistics ( 2022 , 2023 ), Pure Mathematics ( 2024 , 2023 ), Statistics ( 2022 , 2023 ), Data Science 2024 .

Although each handbook contains advice on how you should prepare and give your talk, you may find the following sources particularly useful Seminar advice , Seminar advice summary .

University of Sydney Honours Scholarships

These $6,000 Honours Scholarships are awarded annually on the basis of academic merit and personal attributes such as leadership and creativity.

Writing proficiency

The honours essay is also assessed based on the quality of the writing. This does not mean we look for the next Shakespeare, but you should make sure you express your ideas in an organized manner using a clear and grammatically correct English. The university offers several resources that can help you achieve this goal. The Learning Centre offers workshops for students that need help with extended written work, and a trove of online resources for improving your writing skills is also available . Make sure you make use of these resources as early as possible as writing skills develop slowly over time and with much practice.

Honours Lecture Courses

Following is a list of honours courses offered by the divisions of Applied Mathematics, Pure Mathematics and Statistics. Students in Mathematics may also take advanced third year units – see the Senior Mathematics page for a list of these. Consult your program coordinator if you are interested in taking a course from another division.

   First Semester

• MATH4411: Applied Computational Mathematics ( Georg Gottwald )
• MATH4413: Applied Mathematical Modelling (Not offered in 2024) ( Peter Kim and Robert Marangell )

   Second Semester

• MATH4412: Advanced Methods in Applied Mathematics ( Geoff Vasil )
• MATH4414: Advanced Dynamical Systems (Not offered in 2024) ( Geoff Vasil )

Financial Mathematics

• MATH4511: Arbitrage Pricing in Continuous Time ( Zhou Zhou )
• STAT4528: Probability and Martingale Theory ( Anna Aksamit and Ben Goldys )
• MATH4512: Stochastic Analysis ( Ben Goldys )
• MATH4513: Topics in Financial Mathematics (Not offered in 2024) ( Marek Rutkowski )
• MATH4313: Functional Analysis ( James Parkinson )
• MATH4314: Representation Theory ( Andrew Mathas )
• MATH4311: Algebraic Topology ( Anne Thomas )
• MATH4312: Commutative Algebra ( Ruibin Zhang )
• STAT4026: Statistical Consulting ( Jennifer Chan , John Ormerod and Jean Yang )
• STAT4028: Probability and Mathematical Statistics ( Anna Aksamit and Michael Stewart )
• STAT4027: Advanced Statistical Modelling ( Jennifer Chan )
• STAT5611: Statistical Methodology ( Michael Stewart and Neville Weber )

Students are advised to check the courses offered in January at the AMSI Summer School and also courses available via the Advanced Collaborative Environment (ACE) . Taking any of these courses for credit requires permission to enrol in the unit AMSI4001.

More detailed information about courses and assessment can be obtained from the relevant handbook or program coordinator .

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© 2002-2024 The University of Sydney. ABN:  15 211 513 464. CRICOS number:  00026A. Phone:  +61 2 9351 2222. Authorised by:  Head, School of Mathematics and Statistics.

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Senior Thesis

This page is for Undergraduate Senior Theses.  For Ph.D. Theses, see here .

So that Math Department senior theses can more easily benefit other undergraduate, we would like to exhibit more senior theses online (while all theses are available through Harvard University Archives , it would be more convenient to have them online). It is absolutely voluntary, but if you decide to give us your permission, please send an electronic version of your thesis to cindy@math. The format can be in order of preference: DVI, PS, PDF. In the case of submitting a DVI format, make sure to include all EPS figures. You can also submit Latex or MS word source files.

If you are looking for information and advice from students and faculty about writing a senior thesis, look at this document . It was compiled from comments of students and faculty in preparation for, and during, an information session. Let Wes Cain ([email protected]) know if you have any questions not addressed in the document.

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Shirin Boroushaki

Nonlinear analysis and partial differential equations (pde).

During my PhD, my research primarily focused on the analysis of partial differential equations (PDEs) from a pure mathematical perspective. Specifically, my focus was on proving the existence of solutions to a specific class of PDEs known as Stochastic PDEs.To provide a bit of insight into the existence problem: Traditionally, one of the classic methods used to demonstrate that an elliptic (non-time-dependent) PDE has a solution is through the Euler-Lagrange variational method. In this method, we establish that the solution to an elliptic PDE can be found as the minimum of a functional. This concept resembles finding the optimum value of a function in calculus using derivatives. However, when dealing with PDEs whose solution changes over time, such as parabolic PDEs, and especially Stochastic PDEs, the Euler-Lagrange variational method does not readily apply. In my thesis, I introduced an alternative variational approach known as the self-dual method, which has been developed over the past two decades. This method enables us to obtain solutions to Stochastic PDEs by identifying them as minima of appropriately chosen self-dual functionals. The tools I applied throughout this research encompassed concepts from Calculus of Variations, Functional Analysis, Convex Analysis, and Stochastic Processes. As my mathematical journey has evolved, I've also discovered a passion for teaching math and math pedagogy. My goal is to inspire students' curiosity and courage to step into the journey of mathematical exploration, to share with them the inherent beauty of mathematics, and to help them grasp the concepts.

Richard Brewster

Discrete mathematics and theoretical computing science.

My research sits at the intersection of Discrete Mathematics and Theoretical Computing Science. I am particularly interested in algorithmic aspects of graph theory. Typically, the development of efficient algorithms is guided by the development of mathematical theory and vice versa. A rich area where this happens is graph homomorphisms. Here we see ideas for example from structural graph theory, graph colourings, category theory, and arc consistency checks (from AI). Other topics include graph coverings and packings, and combinatorial reconfiguration.

Fatma Mahmoud

Fuzzy topological applications in information systems.

Topological concepts exist not only in almost all branches of mathematics, but also in many real-life applications ranging from kids’ games to the applications in DNA and modern technology of image analysis. We believe that topological structures and its generalizations are important base for modification of knowledge extraction and processing. Recently, topology is concerned with the problem of ambiguities in the information, since the topological view of the boundary region is the clearest view to treat the area of uncertainty in knowledge, this line implied to the topological facts in many new applications such as structural analysis, in chemistry, physics, and biology. There is a study is done applying topological concepts in information systems. We can generalized this study to Fuzzy topological concepts.

Sean McGuinness

Combinatorics, graph theory and matroid theory.

Broadly speaking my research interests lie in the field of Combinatorics. My specific interests in this area are Graph Theory and Matroid Theory, the latter being more of recent interest to me. Matroid theory is a subject which lies at the crossroads of three subjects: Algebra, Geometry, and Combinatorics. All three contribute a rich diversity of ideas. My interest here has been more on the combinatorial side. I have worked on problems which involve extending known properties of graphs to matroids. I am also interested in various unsolved conjectures, for example, Rota’s basis conjecture and White’s conjecture. Much of my recent research focusses on the properties of bases in matroids. My research interests also touch on subjects tangential to matroid theory. For example, the Alon-Tarsi conjecture for Latin squares. I am also interested in using tools and concepts from Linear algebra, projective geometry and probability.

Combinatorics on Words

A word is a sequence of symbols taken from some finite alphabet. In the area of combinatorics on words, we are mostly interested in long words over small alphabets. Which patterns can be avoided, and which patterns must inevitably occur? We use tools from combinatorics, algebra, and computing to answer fundamental questions about words.

Applied Mathematics and Optimisation

The topics you might work on as an Honours student in my group revolve around applied analysis of partial differential equations, including but not limited to: weak solutions, numerical schemes, asymptotic expansions, fluid flow, mathematical and computational optimisation problems, special functions. Most projects done in my group require a variety of skills: analytical derivations and proofs, coding, visualisation of surfaces or functions. Depending on the project, you might do different amounts of each. To enjoy your work with me you would need to like calculus, algebra, differential equations and numerical analysis. You might not like coding, but you should be able to do it.

Saeed Rahmati

Algebraic topology and manifolds.

Topological manifolds are generalization of curves and surfaces to higher dimensions; they are topological spaces that “locally’’ look like a Euclidean space. Smooth manifolds are topological manifolds that we can do calculus on them. I’m interested in properties of manifolds that we can express using bridges between Topology and Algebra.

Statistical Analysis and Bioinformatics

My research areas primarily focus on statistical analysis and bioinformatics, with a specific emphasis on comparative genomics. I am passionate about exploring the evolutionary relationships and functional implications of genomic variations across different species. Through comparative genomics, I aim to unravel the underlying mechanisms that drive species diversity and adaptation. I utilize various statistical and computational approaches to analyze large-scale genomic datasets, integrating genomic, transcriptomic, and proteomic data to gain a comprehensive understanding of biological systems. I am particularly interested in studying comparative genomics by employing methods of mathematics and statistics and developing computational tools for analyzing large-scale biological datasets. By supervising Honours students, I hope to foster their passion for scientific inquiry and provide them with the necessary guidance to explore comparative genomics and its applications.

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Guidelines for writing a thesis

These guidelines are intended for students writing a thesis or project report for a  Third Year Project Course ,  Honours year  or  Postgraduate Coursework Project . Postgraduate research students should see  Information about Research Theses  for postgraduate research students.

Before you start your Honours or Project year, you should speak to members of staff about possible thesis topics. Find out who works in the areas that you are interested in and who you find it easy to talk mathematics with. If at all possible, settle on a topic and supervisor before the start of the first semester of your Honours or Project year.

Most students see their supervisor about once a week, although this is usually open to negotiation between the student and the supervisor. Even if you haven't done much between visits it is a good idea to have a regular chat so that your supervisor can keep track of how you are going. You can expect your supervisor to:

• Help you select - and modify - your topic.
• Direct you to useful references on your topic.
• Help explain difficult points.
• Provide feedback on the direction of your research.
• Read and comment on drafts of your thesis.
• Help prepare you for your talk.
• Give general course advice.

Your thesis or project report is an overview of what you have been studying in your Honours or Project year. Write it as if you were trying to explain the area of mathematics or statistics that you have been looking at to a fellow student.

• Include an introduction that explains what the project is all about, and what its contents are. (It is sometimes better to leave writing this part to the end!) For many reports, a conclusion or summary is appropriate.
• Your thesis should be a coherent, self-contained piece of work.
• Your writing should conform to the highest standards of English. Aim at clarity, precision and correct grammar. Start sentences with capital letters and end them with full-stops. Don't start sentences with a symbol.
• Take great care with bibliographic referencing. Wherever some material has an external source, this should be clear to the reader. Don't just write in the introduction: 'This report contains material from [1],[2] and [3]' - give the references for the material wherever it is used. Don't gratuitously pad your reference list with references that are not referred to in the text. Check current journals for acceptable referencing styles.
• Be careful not to plagiarise. What constitutes plagiarism is perhaps a little different in mathematics and statistics compared to some other subjects since there is a limit to how different you may be able to make a proof (at least in its basic structure). We do, however, expect the report to be written in your own words. A basic rule is: if you put a fact or an idea in your report which is not your own, the reader should be able to tell where you got this fact or idea.
• The University has  policies on academic honesty and plagiarism  which all students should familiarise themselves with.

Generally, mathematics reports and theses are almost always typed in LaTeX. If you are going to type it yourself, you should allow a certain amount of time to become familiar with this software. Indeed, starting to learn LaTeX well before you actually want to write is a very good idea.

You should not underestimate the time it takes to produce a polished document. You will almost certainly need several drafts. It is very difficult to concentrate on getting the mathematics, spelling, grammar, layout, etc., all correct at once. Try getting another student to proofread what you have written - from their different viewpoint they may pick up on lots of things that you can't see.

P R Halmos (1970) in  How to write mathematics, Enseignement Math.  ((2) 16, 123-152) has the following advice: "The basic problem in writing mathematics is the same as in writing biology, writing a novel, or writing directions for assembling a harpsichord: the problem is to communicate an idea. To do so, and to do it clearly:

• you must have something to say (i.e., some ideas), and you must have someone to say it to (i.e., an audience)
• you must organize what you want to say, and you must arrange it in the order you want it said in
• you must write it, rewrite it, and re-rewrite it several times
• and you must be willing to think hard about and work hard on mechanical details such as diction, notation, and punctuation.

That's all there is to it."

His other advice includes:

• Say something: "To have something to say is by far the most important ingredient of good exposition---so much so that if the idea is important enough, the work has a chance to be immortal even if it is confusingly misorganized and awkwardly expressed..... To get by one the first principle alone is, however, only rarely possible and never desirable."
• Audience: "The second principle of good writing is to write for someone. When you decide to write something, ask yourself who it is that you want to reach." Your broad audience will be fellow Masters and Honours students, who may not be experts in your thesis topic. "The author must anticipate and avoid the reader's difficulties. As he(/she) writes, he(/she) must keep trying to imagine what in the words being written may tend to mislead the reader, and what will set him(/her) right."
• Organise: "The main contribution that an expository writer can make is to organize and arrange the material so as to minimize the resistance and maximize the insight of the reader and keep him(/her) on the track with no unintended distractions". 
• Think about the alphabet: "Once you have some kind of plan of organization, an outline, which may not be a fine one but is the best you can do, you are almost ready to start writing. The only other thing I would recommend that you do first is to invest an hour or two of thought in the alphabet; you'll find it saves many headaches later. The letters that are used to denote the concepts you'll discuss are worthy of thought and careful design. A good, consistent notation can be a tremendous help".
• Write in spirals: "The best way to start writing, perhaps the only way, is to write on the spiral plan. According to the spiral plan the chapters get written in the order 1,2,1,2,3,1,2,3,4 etc. You think you know how to write Chapter 1, but after you've done it and gone on to Chapter 2, you'll realize that you could have done a better job on Chapter 2 if you had done Chapter 1 differently. There is no help for it but to go back, do Chapter 1 differently, do a better job on Chapter 2, and then dive into Chapter 3... Chapter 3 will show up the weaknesses of Chapters 1 and 2".
• Write good English: "Good English style implies correct grammar, correct choice of words, correct punctuation, and, perhaps above all, common sense."

More information on how to write mathematics:

• Lee, K. A guide to writing mathematics
• Lee, K. Some notes on writing mathematics 
• Jackson, M. Some notes on writing in mathematics
• Reiter, A. Writing a research paper in mathematics
• Honours thesis
• Postgraduate Coursework Project
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