phd in abstract algebra

• Schools & departments

Algebra PhD

Awards: PhD

Study modes: Full-time, Part-time

Funding opportunities

Programme website: Algebra

Upcoming Introduction to Postgraduate Study and Research events

Join us online on the 19th June or 26th June to learn more about studying and researching at Edinburgh.

Choose your event and register

Research profile

Our research group has strong links with the Geometry and Topology group, as well as Mathematical Physics. You’ll find these invaluable as opportunities to discuss your work and expand your thinking.

The School of Mathematics is a vibrant community of more than 100 academic and related staff supervising over 100 PhD students.

Working within one of the largest mathematics groups in the UK, you’ll be completing your degree in an environment that hums with a busy graduate school life, and you’ll have the chance to make your mark in seminars, workshops, clubs and outings.

The Hodge Institute is the home of algebra, geometry, number theory and topology research groups in the School of Mathematics.

• The Hodge Institute

Our interests include:

• non-commutative ring theory
• non-commutative algebraic geometry
• the geometry of algebraic numbers
• Lie-theoretic representation theory
• quantum algebra
• category theory

While we offer a large community of researchers under one roof, we believe in encouraging you to gain as broad a perspective as possible. The best way to do this is to involve yourself in the international dialogue on your research area, through attending conferences and symposia, and visiting your peers in centres of research worldwide.

Throughout your studies, you’ll be given opportunities to travel to events and institutions that will allow you to gain this perspective and open up new areas of investigation.

You can find out more on the Maxwell Institute Graduate School site.

Training and support

Mathematics has connections stretching across all the scientific disciplines and beyond. Our School is one of the country’s largest mathematics research communities in its own right, but you will also benefit from Edinburgh’s high-level collaborations, both regional and international.

Research students will have a primary and secondary supervisor and the opportunity to network with a large and varied peer group. You will be carrying out your research in the company of eminent figures and be exposed to a steady stream of distinguished researchers from all over the world.

Our status as one of the most prestigious schools in the UK for mathematics attracts highly respected staff. Many of our 100+ current academics are leaders in their fields and have been recognised with international awards.

Researchers are encouraged to travel and participate in conferences and seminars. You’ll also be in the right place in Edinburgh to meet distinguished researchers from all over the world who are attracted to conferences held at the School and the various collaborative centres based here.

You’ll find opportunities for networking that could have far-reaching effects on your career in mathematics.

You will enjoy excellent facilities, ranging from one of the world’s major supercomputing hubs to generous library provision for research at the leading level, including the new Noreen and Kenneth Murray Library at King’s Buildings.

Students have access to more than 1,400 computers in suites distributed across the University’s sites, many of which are open 24 hours a day. In addition, if you are a research student, you will have your own desk with desktop computer.

We provide all our mathematics postgraduates with access to software packages such as Maple, Matlab and Mathematica. Research students are allocated parallel computing time on ‘Eddie’ – the Edinburgh Compute and Data Facility. It is also possible to arrange use of the BlueGene/Q supercomputer facility if your research requires it.

Advice on applications

For advice on applications, see

• Hodge Institute page

We particularly encourage women and other underrepresented groups in mathematics to apply, and work with groups such as the Piscopia Initiative to improve the representation and inclusion of women and minorities in mathematics.

Entry requirements

These entry requirements are for the 2024/25 academic year and requirements for future academic years may differ. Entry requirements for the 2025/26 academic year will be published on 1 Oct 2024.

A UK first class honours degree, or its international equivalent, in an appropriate subject; or a UK 2:1 honours degree plus a UK masters degree, or their international equivalents; or relevant qualifications and experience.

International qualifications

Check whether your international qualifications meet our general entry requirements:

• Entry requirements by country
• English language requirements

Regardless of your nationality or country of residence, you must demonstrate a level of English language competency at a level that will enable you to succeed in your studies.

English language tests

We accept the following English language qualifications at the grades specified:

• IELTS Academic: total 6.5 with at least 6.0 in each component. We do not accept IELTS One Skill Retake to meet our English language requirements.
• TOEFL-iBT (including Home Edition): total 92 with at least 20 in each component. We do not accept TOEFL MyBest Score to meet our English language requirements.
• C1 Advanced ( CAE ) / C2 Proficiency ( CPE ): total 176 with at least 169 in each component.
• Trinity ISE : ISE II with distinctions in all four components.
• PTE Academic: total 62 with at least 59 in each component.

Your English language qualification must be no more than three and a half years old from the start date of the programme you are applying to study, unless you are using IELTS , TOEFL, Trinity ISE or PTE , in which case it must be no more than two years old.

Degrees taught and assessed in English

We also accept an undergraduate or postgraduate degree that has been taught and assessed in English in a majority English speaking country, as defined by UK Visas and Immigration:

• UKVI list of majority English speaking countries

We also accept a degree that has been taught and assessed in English from a university on our list of approved universities in non-majority English speaking countries (non-MESC).

• Approved universities in non-MESC

If you are not a national of a majority English speaking country, then your degree must be no more than five years old* at the beginning of your programme of study. (*Revised 05 March 2024 to extend degree validity to five years.)

Find out more about our language requirements:

• Academic Technology Approval Scheme

If you are not an EU , EEA or Swiss national, you may need an Academic Technology Approval Scheme clearance certificate in order to study this programme.

Fees and costs

Tuition fees, scholarships and funding, featured funding.

• School of Mathematics funding opportunities
• Research scholarships for international students

UK government postgraduate loans

If you live in the UK, you may be able to apply for a postgraduate loan from one of the UK's governments.

The type and amount of financial support you are eligible for will depend on:

• your programme
• the duration of your studies
• your residency status.

Programmes studied on a part-time intermittent basis are not eligible.

• UK government and other external funding

Other funding opportunities

Search for scholarships and funding opportunities:

• Search for funding

Further information

• Graduate School Administrator
• Phone: +44 (0)131 650 5085
• Contact: [email protected]
• School of Mathematics
• James Clerk Maxwell Building
• Peter Guthrie Tait Road
• The King's Buildings Campus
• Programme: Algebra
• School: Mathematics
• College: Science & Engineering

Select your programme and preferred start date to begin your application.

PhD Algebra and Number Theory - 3 Years (Full-time)

Phd algebra and number theory - 6 years (part-time), application deadlines.

We strongly recommend you submit your completed application as early as possible, particularly if you are also applying for funding or will require a visa. We may consider late applications if we have places available. All applications received by 22 January 2024 will receive full consideration for funding. Later applications will be considered until all positions are filled.

• How to apply

You must submit two references with your application.

Find out more about the general application process for postgraduate programmes:

PhD in Mathematical Sciences

The Doctor of Philosophy (PhD) program in Mathematical Sciences represents achievement of a broad knowledge of the various branches of mathematics, of the ability to communicate mathematics in both written and oral form, and of a demonstrated creative ability in a particular branch of mathematics.

The program prepares students for careers in academia and as professional mathematicians. Students have the opportunity to work with some of the world's leading experts in a variety of research areas. The Department has particular strength in the interrelated fields of analysis , partial differential equations , and probability . Students interested in more applied directions can work with experts in  mathematical bioscience , computational finance , and  cryptography ..

Admission Requirements

Students applying for the program should have or be expecting to obtain a bachelor’s degree either in mathematics or a related field.

The following background is required:

• Multivariable calculus at the level of MATH 2063
• Ordinary differential equations at the level of MATH 2073 or MATH 2074
• Linear algebra at the level of MATH 2076
• Experience with reading and constructing mathematical proofs at the level of MATH 3001

The following background is recommended:

• Analysis at the level of MATH 3002 or MATH 5101/6001
• Abstract linear algebra at the level of MATH 5103/6003  
• Additional advanced courses in pure or applied mathematics

An official general GRE score is required for admission.  This requirement is waived for applicants with at least 6 graduate credits in a relevant field with a cumulative GPA of 3.2 or higher.  The requirement is also waived for UC undergraduates with a degree in a relevant field and a cumulative GPA of 3.5 or higher.  A quantitative score of 160 or higher is recommended.

The English proficiency requirement is met for applicants with degrees earned in English from accredited universities and colleges in  the US or other English-speaking countries .

Financial Support

Most of our PhD students receive full financial support via a teaching or research assistantship, and most are supported through their entire UC career.  Travel support is available for students to attend or present their work at conferences.  

All applicants for the PhD program are automatically reviewed for graduate assistantship eligibility at the time of application. 

• Financial aid opportunities for students in the Mathematical Sciences Department
• Tuition and fees for graduate and professional students

Application Instructions

Applicants will need to meet the minimum requirements to be considered for the program. Completed applications will be reviewed beginning February 1 . We will continue to receive applications until all positions are filled.

All application materials from international students requiring a US visa must be received prior to April 1 (but sooner is better) in order to allow time for the necessary paperwork to be processed. The visa application process can often take 90 days or more to complete.

How to apply: 

1. Create  an online application

2. Include these documents in your application:

• Three letters of recommendation. The application system will automatically send an email to each of the recommenders with a link to submit their letters. 
• Unofficial copy of transcript (official transcript will be required if you are admitted to the program). 
• GRE general test score 
• Statement of purpose/cover letter 
• English Proficiency for international students

3. Pay the application fee

UC’s CEEB college code is 1833, as established by The College Board . CEEB codes are used to ensure that test scores are sent to the correct institution. 

• More information about submitting your application materials
• FAQs for the admission process

Program Description

The credit-hour requirement includes a minimum of 90 graduate credits beyond the bachelor's degree or a minimum of 60 credits beyond a master's degree, including 7 hours in dissertation research, with a GPA of 3.3 or higher. 

All incoming PhD students are required to take the  qualifying exam before the beginning of their first semester. Students who do not pass this exam at the PhD level are placed in the appropriate 6000 - level courses. The Mathematics Qualifying Exam is based on the two-semester sequence Advanced Calculus MATH6001-6002 and the one semester course Abstract Linear Algebra MATH6003. 

All PhD students must pass four preliminary examinations . Each Preliminary Exam is offered twice a year. Examinations based on a course given during Fall Semester are offered after the end of Spring Semester and at the beginning of the following Fall Semester. Examinations based on a course given during the Spring Semester are offered at the beginning of the Fall Semester and at the beginning of the following Spring Semester.

After the preliminary examinations, an advanced examination in the area of examination of the student is required. An advanced exam may either be a written exam, a presentation or a series of presentations. The exam will be administered by a committee. Generally, this committee will form the students’ dissertation committee.

Visit the curriculum guide to learn about the required courses. More details concerning the requirements of the program are explained in the Mathematical Sciences Department’s Graduate Handbook . See the course descriptions for information on the content.

About Cincinnati

Cincinnati is a big city with a small-town feel. The cost of living is low, but the quality of life is high.  Forbes named Cincinnati the #5 most affordable city and the #9 best city for raising a family. Cincinnati has ranked  the best place to live in Ohio by U.S. News & World Report, also the fourth-best city in the country for parks . UC is home to over 10,500 graduate students, 20% of which are international students.

• Why Cincinnati  
• Estimated living expenses  (for international students)

For further information, please contact the Graduate Program Director, Dr. Robert Buckingham:

• Email: [email protected]
• Phone: 513-556-4085

See the full list of our graduate programs

Mathematics, PHD

On this page:, at a glance: program details.

• Location: Tempe campus
• Second Language Requirement: No

Program Description

Degree Awarded: PHD Mathematics

The PhD program in mathematics is intended for students with exceptional mathematical ability. The program emphasizes a solid mathematical foundation and promotes innovative scholarship in mathematics and its many related disciplines.

The School of Mathematical and Statistical Sciences has very active research groups in analysis, number theory, geometry and discrete mathematics.

Degree Requirements

84 credit hours, a written comprehensive exam, a prospectus and a dissertation

Required Core (3 credit hours) MAT 501 Geometry and Topology of Manifolds I (3) or MAT 516 Graph Theory I (3) or MAT 543 Abstract Algebra I (3) or MAT 570 Real Analysis I (3)

Other Requirements (3 credit hours) MAT 591 Seminar (3)

Electives (24-39 credit hours)

Research (27-42 credit hours) MAT 792 Research

Culminating Experience (12 credit hours) MAT 799 Dissertation (12)

Additional Curriculum Information Electives are to be chosen from math or related area courses approved by the student's supervisory committee.

Students must pass:

• two qualifying examinations
• a written comprehensive examination
• an oral dissertation prospectus defense

Students should see the department website for examination information.

Each student must write a dissertation and defend it orally in front of five dissertation committee members.

Admission Requirements

Applicants must fulfill the requirements of both the Graduate College and The College of Liberal Arts and Sciences.

Applicants are eligible to apply to the program if they have earned a bachelor's or master's degree in mathematics or a closely related area from a regionally accredited institution.

Applicants must have a minimum cumulative GPA of 3.00 (scale is 4.00 = "A") in the last 60 hours of their first bachelor's degree program or a minimum cumulative GPA of 3.00 (scale is 4.00 = "A") in an applicable master's degree program.

All applicants must submit:

• graduate admission application and application fee
• official transcripts
• statement of education and career goals
• three letters of recommendation
• proof of English proficiency

Additional Application Information An applicant whose native language is not English must provide proof of English proficiency regardless of their current residency.

Additional eligibility requirements include competitiveness in an applicant pool as evidenced by coursework in linear algebra (equivalent to ASU course MAT 342 or MAT 343) and advanced calculus (equivalent to ASU course MAT 371), and it is desirable that applicants have scientific programming skills.

Next Steps to attend ASU

Learn about our programs, apply to a program, visit our campus, application deadlines, learning outcomes.

• Address an original research question in mathematics.
• Able to complete original research in theoretical mathematics.
• Apply advanced mathematical skills in coursework and research.

Career Opportunities

Graduates of the doctoral program in mathematics possess sophisticated mathematical skills required for careers in many different sectors, including education, industry and government. Potential career opportunities include:

• faculty-track academic
• finance and investment analyst
• mathematician
• mathematics professor, instructor or researcher
• operations research analyst
• statistician

Program Contact Information

If you have questions related to admission, please click here to request information and an admission specialist will reach out to you directly. For questions regarding faculty or courses, please use the contact information below.

• [email protected]
• 480/965-3951

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Basic Abstract Algebra

An introduction to abstract algebra, brief talk on ideological and political teaching in abstract algebra, how mathematicians assign homework problems in abstract algebra courses, an invitation to abstract algebra, verifying non-isomorphism of groups.

The concept of isomorphism is central to group theory, indeed to all of abstract algebra. Two groups {G, *} and {H, ο}are said to be isomorphic to each other if there exists a set bijection α from G onto H, such that $$\left( {a\;*\;b} \right)\alpha = \left( a \right)\alpha \; \circ \;(b)\alpha $$ for all a, b ∈ G. This can be illustrated by what is usually known as a commutative diagram:

On the Exponential Diophantine Equation (132m) + (6r + 1)n = z2

Nowadays, mathematicians are very interested in discovering new and advanced methods for determining the solution of Diophantine equations. Diophantine equations are those equations that have more unknowns than equations. Diophantine equations appear in astronomy, cryptography, abstract algebra, coordinate geometry and trigonometry. Congruence theory plays an important role in finding the solution of some special type Diophantine equations. The absence of any generalized method, which can handle each Diophantine equation, is challenging for researchers. In the present paper, the authors have discussed the existence of the solution of exponential Diophantine equation  (132m) + (6r + 1)n = Z2, where m, n, r, z are whole numbers. Results of the present paper show that the exponential Diophantine equation (132m) + (6r + 1)n = Z2, where m, n, r, z are whole numbers, has no solution in the whole number.

Abstract Algebra

Ring hypothesis is one of the pieces of the theoretical polynomial math that has been thoroughly used in pictures. Nevertheless, ring hypothesis has not been associated with picture division. In this paper, we propose another rundown of similarity among pictures using rings and the entropy work. This new record was associated as another stopping standard to the Mean Shift Iterative Calculation with the goal to accomplish a predominant division. An examination on the execution of the calculation with this new ending standard is finished. In spite of the fact that ring hypothesis and class hypothesis from the start sought after assorted direction it turned out during the 1970s – that the investigation of functor groupings furthermore reveals new plots for module hypothesis.

(m, n)-Ideals in Semigoups Based on Int-Soft Sets

Algebraic structures play a prominent role in mathematics with wide ranging applications in many disciplines such as theoretical physics, computer sciences, control engineering, information sciences, coding theory, and topological spaces. This provides sufficient motivation to researchers to review various concepts and results from the realm of abstract algebra in the broader framework of fuzzy setting. In this paper, we introduce the notions of int-soft m , n -ideals, int-soft m , 0 -ideals, and int-soft 0 , n -ideals of semigroups by generalizing the concept of int-soft bi-ideals, int-soft right ideals, and int-soft left ideals in semigroups. In addition, some of the properties of int-soft m , n -ideal, int-soft m , 0 -ideal, and int-soft 0 , n -ideal are studied. Also, characterizations of various types of semigroups such as m , n -regular semigroups, m , 0 -regular semigroups, and 0 , n -regular semigroups in terms of their int-soft m , n -ideals, int-soft m , 0 -ideals, and int-soft 0 , n -ideals are provided.

A transition to abstract algebra

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Abstract Algebra

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Abstract algebra is a broad field of mathematics, concerned with algebraic structures such as groups , rings , vector spaces , and algebras . On the 12-hour clock, \(9+4=1\), rather than 13 as in usual arithmetic

Roughly speaking, abstract algebra is the study of what happens when certain properties of number systems are abstracted out; for instance, altering the definitions of the basic arithmetic operations result in a structure known as a ring, so long as the operations are consistent.

For example, the 12-hour clock is an example of such an object, where the arithmetic operations are redefined to use modular arithmetic (with modulus 12). An even further level of abstraction--where only one operation is considered--allows the clock to be understood as a group. In either case, the abstraction is useful because many properties can be understood without needing to consider the specific structure at hand, which is especially important when considering the relationship(s) between structures; the concept of a group isomorphism is an example.

Levels of Abstraction in Abstract Algebra

Group theory.

• Ring Theory

Other Applications of Abstract Algebra

It is possible to abstract away practically all of the properties found in the "usual" number systems, the tradeoff being that the resulting object--known as a magma (which consists of a set and a binary operation, that need not satisfy any properties other than closure)--is simply too general to be interesting. On the other extreme, it is possible to abstract out practically no properties, which allows for many results to be found, but the resulting object (the usual number systems) is too specific to solve more general problems.

Most of abstract algebra is dedicated to objects that have a reasonable balance between generality and structure, most notably groups and rings (discussed in more detail below) in which most of the basic properties of arithmetic are maintained, but their specifics are left free. Still, some higher levels of abstraction are occasionally useful; quasigroups , for instance, are related to Latin squares , and monoids are often used in computer science and are simple examples of categories .

Main article: Group theory The possible moves on a Rubik's cube form a (very large) group . Group theory is useful as an abstract notion of symmetry , which makes it applicable to a wide range of areas: the relationship between the roots of a polynomial (as in Galois theory ) and the solution methods to the Rubik's cube are both prominent examples.

Informally, a group is a set equipped with a binary operation \(\circ\), so that operating on any two elements of the group also produces an element of the group. For example, the integers form a group under addition, and the nonzero real numbers form a group under multiplication. The \(\circ\) operation needs to satisfy a number of properties analogous to the ones it satisfies for these "normal" number systems: it should be associative (which essentially means that the order of operations doesn't matter), and there should be an identity element (0 in the first example above, and 1 in the second). More formally, a group is a set equipped with an operation \(\cdot\) such that the following axioms hold; note that \(\cdot\) does not necessarily refer to multiplication; rather, it should be viewed as a function on two variables (indeed, \(\cdot\) can even refer to addition):

Group Axioms 1) Associativity. For any \(x, y, z \in G \), we have \( (x \cdot y) \cdot z = x \cdot (y \cdot z) \). 2) Identity. There exists an \( e \in G \), such that \( e \cdot x = x \cdot e = x \) for any \(x \in G \). We say that \(e\) is an identity element of \(G\). 3) Inverse. For any \(x \in G\), there exists a \(y \in G\) such that \(x \cdot y = e = y \cdot x \). We say that \(y\) is an inverse of \(x\).

It is also worth noting the closure axiom for emphasis, as it is important to verify closure when working with subgroups (groups contained entirely within another):

4) Closure. For any \(x, y \in G \), \(x*y \) is also in \(G\).

Additional examples of groups include

• \(\mathbb{Z}_n\), the set of integers \(\{0, 1, \ldots, n-1\}\) with the operation addition modulo \(n\)
• \(S_n\), the set of permutations of \(\{1, 2, \ldots, n\}\) with the operation of composition .

\(S_3\) is worth special note as an example of a group that is not commutative , meaning that \(a \cdot b = b \cdot a\) does not generally hold. Formally speaking, \(S_3\) is nonabelian (an abelian group is one in which the operation is commutative). When the operation is not clear from context, groups are written in the form \((\text{set}, \text{op})\); e.g. the nonzero reals equipped with multiplication can be written as \((\mathbb{R}^*, \cdot)\).

Much of group theory (and abstract algebra in general) is centered around the concept of a group homomorphism , which essentially means a mapping from one group to another that preserves the structure of the group. In other words, the mapping of the product of two elements should be the same as the product of the two mappings; intuitively speaking, the product of two elements should not change under the mapping. Formally, a homomorphism is a function \(\phi: G \rightarrow H\) such that

\[\phi(g_1) \cdot_H \phi(g_2) = \phi(g_1 \cdot_G g_2),\]

where \(\cdot_H\) is the operation on \(H\) and \(\cdot_G\) is the operation on \(G\). For example, \(\phi(g) = g \pmod n\) is an example of a group homomorphism from \(\mathbb{Z}\) to \(\mathbb{Z}_n\). The concept of potentially differing operations is necessary; for example, \(\phi(g)=e^g\) is an example of a group homomorphism from \((\mathbb{R},+)\) to \((\mathbb{R}^{*},\cdot)\).

Main article: Ring theory

Rings are one of the lowest level of abstraction, essentially obtained by overwriting the addition and multiplication functions simultaneously (compared to groups, which uses only one operation). Thus a ring is--in some sense--a combination of multiple groups, as a ring can be viewed as a group over either one of its operations. This means that the analysis of groups is also applicable to rings, but rings have additional properties to work with (the tradeoff being that rings are less general and require more conditions).

The definition of a ring is similar to that of a group, with the extra condition that the distributive law holds as well:

A ring is a set \( R \) together with two operations \( + \) and \( \cdot \) satisfying the following properties (ring axioms): (1) \( R \) is an abelian group under addition. That is, \( R\) is closed under addition, there is an additive identity (called \( 0 \)), every element \(a\in R\) has an additive inverse \(-a\in R \), and addition is associative and commutative. (2) \( R \) is closed under multiplication, and multiplication is associative: \(\forall a,b\in R\quad a.b\in R\\ \forall a,b,c\in R\quad a\cdot ( b\cdot c ) =( a\cdot b ) \cdot c.\) (3) Multiplication distributes over addition: \(\forall a,b,c\in R\\ a\cdot \left( b+c \right) =a\cdot b+a\cdot c\quad \text{ and }\quad \left( b+c \right) \cdot a=b\cdot a+c\cdot a.\) A ring is usually denoted by \(\left( R,+,. \right) \) and often it is written only as \(R\) when the operations are understood.

For example, the integers \(\mathbb{Z}\) form a ring, as do the integers modulo \(n\) \((\)denoted by \(\mathbb{Z}_n).\) Less obviously, the square matrices of a given size also form a ring; this ring is noncommutative. Commutative ring theory, or commutative algebra, is much better understood than noncommutative rings are.

As in groups, a ring homomorphism can be defined as a mapping preserving the structure of both operations.

Rings are used extensively in algebraic number theory , where " integers " are reimagined as slightly different objects (for example, Gaussian integers ), and the effect on concepts such as prime factorization is analyzed. Of particular interest is the fundamental theorem of arithmetic , which involves the concept of unique factorization; in other rings, this may not hold, such as

\[6 = 2 \cdot 3 = \big(1+\sqrt{-5}\big)\big(1-\sqrt{-5}\big).\]

Theory developed in this field solves problems ranging from sum of squares theorems to Fermat's last theorem .

Abstract algebra also has heavy application in physics and computer science through the analysis of vector spaces . For example, the Fourier transform and differential geometry both have vector spaces as their underlying structures; in fact, the Poincare conjecture is (roughly speaking) a statement about whether the fundamental group of a manifold determines if the manifold is a sphere.

Related to vector spaces are modules , which are essentially identical to vector spaces but defined over a ring rather than over a field (and are thus more general). Modules are heavily related to representation theory , which views the elements of a group as linear transformations of a vector space; this is desirable to make an abstract object (a group) somewhat more concrete, in the sense that the group is better understood by translating it into a well-understood object in linear algebra (as matrices can be viewed as linear transformations, and vice versa).

The relationships between various algebraic structures are formalized using category theory .

• Group Theory Introduction

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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.

Chairs: George Bergman and Tony Feng

Algebra Faculty, Courses, Dissertations

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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Abstract Algebra

• Choi, Evelyn, Shannon Fan, Jadon Geathers, and Aoden Teo Masa Toshi. “On the Partition Function of Six Vertex Lattice Models Through Interpolation and Demazure Operators” .
• Mahoney, Dylan, and Bryan Park. “On the Weierstrass p_Function As an Exponential Map” .
• Hao, Hanson. “Investigations on Automorphism Groups of Quantum Stabilizer Codes” .
• Hao, Hanson, Eli Navaro, and Henri Stern. “Irreducibility and Galois Groups of Random Polynomials” .
• Ido, Yuzu, Ian Ruohoniemi, and Matthew Stevens. “Integer Forms and Elliptic Curves” .
• Greicius, Quinn. “Symmetric Homomorphisms of Abelian Varieties” .
• Singh, Jaydeep. “Topics in Lattice Gauge Theory: Behavior of Wilson Loops in the Thermodynamic Limit” .
• Bitz, Jared, and Cory Griffith. “A New Bound and New Techniques for the Erdos–Ginzburg–Ziv Constant for Finite Abelian Groups” .
• Tu, Leslie. “Schubert Calculus on the Grassmanian” .

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Abstract Algebra

By helpzwouldbenice May 26, 2009 in PhD in Economics

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Helpzwouldbenice.

I just finished my junior year at a top-10 LAC, and I was thinking of taking Abstract Algebra this coming fall in place of Real Analysis for three big reasons:

(a) Real Analysis is notoriously hard at my school (as I'm sure it is everywhere else; but people much smarter than me have told me it was extremely hard), and I'm not sure I would do well in it

(b) I took an Intro. to Analysis (basic proofs) class this past semester, and found I liked the Ordered Fields/Metric Spaces etc. part more than the Limits/Continuity/Connectedness part, so I think I might actually like the material in Abstract Algebra, and thus be motivated to perform well

© While I know that many on here and elsewhere have said that Abstract Algebra is pretty much irrelevant to economics, my feeling is that doing well in Abstract Algebra is a strong signal that one is comfortable with numbers and generally 'good' at math. My point here is that if the admissions process is a signalling game, and if I guess I'll do better in Abstract Alg than in Real Analysis, then it would be better to take the less relevant class in favor of the better signal...

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One of my recommendations came from an Abstract Algebra professor.

The main reason adcoms want to see math courses is to make sure you'll be comfortable with proofs. Some theorems from analysis are used here and there, but these theorems are extremely basic (i.e. Weierstrass).

If you think you'll enjoy Abstract Algebra, go for it. I wouldn't shy away from real analysis just because you think it's hard (grad micro is harder), but analysis is not required for admission. Most people in my class at Minnesota have not taken analysis before this year.

"I don't want to do it because I might not succeed"

Maybe you shouldn't be going for a PhD if this is how you feel.

It takes a lot of tools to be a great economist. One of the most important of those tools is a massive ego. Phrases like "people much smarter than me" signals to me that you don't think you're the smartest person within a 100 feet when you walk into a room. I don't see how that attitude is going to fly in grad school.

I sincerely wonder if you indeed think so. Or you are just being funny again?

zurich_econ

Torontonian.

"I don't want to do it because I might not succeed"   Maybe you shouldn't be going for a PhD if this is how you feel.

Maybe you shouldn't be that rude.

Nah, jeeves' comment is completely valid. My advisor has actually told me the very same thing. In response, I guess I'd say that my focus regarding grad school right now is just getting in. With more mathematical exposure and a certain context for the theorems I'm trying to prove in grad school, perhaps I may succeed there in classes much harder than Real Analysis. But honestly, I won't know if I don't get in. And if I want to go to as good a grad school as possible, I need to send as good signals as possible.

Anticipating a response to the above response, I know that grad school is much harder than undergrad, and the fact that I'm not willing to take the hardest possible math classes does seem to make me a subpar candidate. But I do like the research process and the idea of creative thinking, and I feel like there's more to grad school than proving theorems--- I'd guess that not every (great) economist out there is a mathematical wizard (reading Steven Levitt's bio provides a basis for this statement).

Not trying to be rude. It was completely candid.

That's a bad reason not to do analysis, especially if you want to do something like a PhD. If you want to minimize the possibility of failure as your objective function, then a PhD is not an optimal strategy. I'm not devaluing the OP at all . It is a perfectly valid preference to be averse to the possibility of failure, it's just that that preference isn't well suited to this line of work.

If you don't want to do analysis because you think its importance is overstated on this board (it is), or because you value working the algebra side of your brain or because you'd rather spend your time on research, then by all means, do it.

But for goodness sakes, don't forgo it just because there is a possibility you won't do well.

I do like the research process and the idea of creative thinking, and I feel like there's more to grad school than proving theorems--- I'd guess that not every (great) economist out there is a mathematical wizard (reading Steven Levitt's bio provides a basis for this statement).

This is a much better reason not to take analysis... but if this is your reason, there's no sense taking abstract algebra- instead write a killer thesis or something like that...

Not trying to be rude. It was completely candid. That's a bad reason not to do analysis, especially if you want to do something like a PhD. If you want to minimize the possibility of failure as your objective function, then a PhD is not an optimal strategy.

I think you make a reasonable point, jeeves -- that is, going into a PhD you have to be willing to work as hard as you can and still know things might not work out. Working through adversity is obviously going to be a big part of the next 5-6 years.

However, the OP is at a point where taking real analysis is not just a matter of obtaining knowledge to be applied toward a future goal -- it's a pretty significant signal as well. The OP could be considering the future benefit of learning RA + possible benefit of doing well vs. the cost of time and effort + possible cost of a poor signal to adcoms, in which case a sufficiently high probability of a poor performanace could make it worth foregoing.

Basically, my point is that I'm not sure it's an explicit fear of failure, but more a strong distaste for the consequences of said failure. At the same time, the OP could indeed do very well in this class. I think a lot of us have the habit of underestimating our ability and this can be a liability -- Harry Lime's comment has a large grain of truth to it.

OneMoreEcon

Not trying to be rude. It was completely candid.   That's a bad reason not to do analysis, especially if you want to do something like a PhD. If you want to minimize the possibility of failure as your objective function, then a PhD is not an optimal strategy. I'm not devaluing the OP at all . It is a perfectly valid preference to be averse to the possibility of failure, it's just that that preference isn't well suited to this line of work.   If you don't want to do analysis because you think its importance is overstated on this board (it is), or because you value working the algebra side of your brain or because you'd rather spend your time on research, then by all means, do it.   But for goodness sakes, don't forgo it just because there is a possibility you won't do well.

jeeves, you're getting confused. ;) OP didn't mention "aversion to failure" as maximizing his utility... he mentioned a significant probability of getting a poor grade (possibly due to a notoriously difficult prof, though this will be private information to the applicant) and sending a weak signal (and quite possibly an incorrect signal, depending on the prof's standards).

Assuming it's possible, OP's best course of action is probably to take Abstract Algebra for credit and audit Real Analysis. AA is more likely to send a better signal (weaker than equal grade in RA, better than worse grade in RA) to ad coms, while auditing helps prepare the OP for the limited amount of RA needed in 1st year micro.

As an anecdote, I went to what's regarded as a mediocre undergrad institution. RA was dual-listed as both an undergrad and a grad course. Some grad students took it as remedial coursework or a refresher if they've been out of school for a while. Two profs usually taught the course. I had the normal one (regarded as normal RA hard... not easy, but not impossible). The other prof is pissed off that students these days can get a masters in math without writing a thesis (i.e. coursework only), so he intentionally fails grades insanely harsh and fails most of the students.

You probably learn a lot more from the latter and write better proofs. But you also send a terrible signal with that 'C' you got in his class.

treblekicker

Economics 101.

obviously real is a much stronger signal..

I wouldn't be so sure. Abstract algebra is often considered to be the hardest undergraduate math class, and getting an A in it at a serious institution is as good a signal as it gets.

I think most students at my school consider real analysis the hardest undergraduate math course (at least the ones that I have talked to) followed by abstract algebra.

hubrishedge

This was the case at my undergrad school; I'm not sure that abstract algebra was the hardest undergrad math class, but it definitely made real analysis look like child's play.

Since you already have one RA class on your transcript, I think these classes will be pretty close substitutes for signaling. People on this form really overemphasize RA.

That said, abstract algebra will have absolutely no direct application to anything you ever do in economics. If you are maximizing over the objective function "future research ability", taking AA is not a solution :-).

pookie bear

This was the case at my undergrad school; I'm not sure that abstract algebra was the hardest undergrad math class, but it definitely made real analysis look like child's play.   Since you already have one RA class on your transcript, I think these classes will be pretty close substitutes for signaling. People on this form really overemphasize RA.   That said, abstract algebra will have absolutely no direct application to anything you ever do in economics. If you are maximizing over the object function "future research ability", taking AA is not a solution :-).

Just to add another data point, I felt that Abstract Algebra (B+) was a bit harder than Real Analysis (A) for me. Topology and Differential Geometry get the reputation as the hardest math classes at my school, but then again aren't entirely necessary for applying to grad school.

Topology and Differential Geometry get the reputation as the hardest math classes at my school, but then again aren't entirely necessary for applying to grad school.

The hardest undergraduate math class is the one you are currently taking. 6 months later it all seems trivial.

Real Analysis gets over emphasized on this board. Many students enter PhD programs without real analysis; your analysis class should be fine.

So true. I think the pound-for-pound toughest math struggle I had in my life was learning how to factor.

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New article examines the impact of mindfulness in treating alcohol use disorder

Published June 4, 2024

Kyler S. Knapp

Paul R. Stasiewicz

Clara M. Bradizza

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New research from the University at Buffalo School of Social Work suggests that assigning formal daily mindfulness homework may be beneficial for clients during treatment for alcohol use disorder.

Kyler S. Knapp, PhD, a postdoctoral fellow in the School of Social Work, led the study, titled “Daily mindfulness homework completion is associated with reduced drinking during a mindfulness-enriched emotion regulation treatment for alcohol use disorder.” The paper appears in the June issue of Addictive Behaviors .

Knapp's co-authors were alumnus Braden K. Linn, PhD ’18, MSW ’12, assistant professor at Pennsylvania State University, as well as Paul R. Stasiewicz, professor and Janet B. Wattles Endowed Chair, and Clara M. Bradizza, professor, both from the UB School of Social Work.

Homework is widely used in cognitive behavioral therapy (CBT) for alcohol use disorder (AUD). Theoretically, homework helps clients generalize skills acquired during treatment to their daily lives. However, clinical trials methodology has typically employed pre- and post-treatment assessments, which has made evaluating the contribution of homework to behavior change a challenge.

The current study leveraged daily diary data from a clinical trial of CBT for AUD to parse within- and between-person associations of mindfulness homework practice and alcohol consumption. Adults seeking treatment for AUD (N = 97) completed 12 treatment sessions and 84 consecutive daily smartphone surveys. Each day, participants reported on the frequency of prior day’s formal and informal mindfulness homework practice, the duration of prior day’s formal mindfulness practice, and prior day’s alcohol consumption. Multilevel models tested within- and between-person associations of mindfulness homework practice with the odds of drinking and heavy drinking, accounting for prior day’s alcohol use.

Results revealed that greater-than-usual frequency and duration of formal daily mindfulness homework practice, but not informal mindfulness practice, were associated with lower odds of a drinking day. Further, greater-than-usual duration of formal daily mindfulness homework practice, but not frequency of mindfulness practice (formal or informal), was associated with lower odds of a heavy drinking day. Results suggest that formal daily mindfulness homework practice may be beneficial and extend the literature by demonstrating that it coincides with reduced drinking and heavy drinking odds within-persons during AUD treatment. Apprising AUD clients of the potential value of homework may help boost its uptake.

This research contributes to one of the  Grand Challenges for Social Work tackling our nation's toughest social problems: Close the health gap.

Do you have questions or comments for the Office of the Provost? Let us know your thoughts and we’ll be happy to get back to you.

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• 6/4/24 New article examines the impact of mindfulness in treating alcohol use disorder