I am a mathematics student and researcher currently completing my PhD at the University of Manchester. My work sits at the intersection of rigorous abstract algebra and modern computation, with a primary specialisation in the **Modular Representation Theory** of finite groups. My doctoral research focuses on computing the **Picard groups** of blocks. Specifically, investigating the derived symmetries of algebras with Dihedral, Semi-Dihedral, and Quaternion defect groups. By utilising the **GAP** (Groups, Algorithms, Programming) system to model high-dimensional algebraic structures, he tackles structural problems in Group Theory that are often resistant to purely theoretical approaches. His work contributes to foundational problems in the field, including the verification of local-global conjectures for groups of Lie type. Beyond pure mathematics, I bring a unique perspective on the application of mathematical logic to artificial intelligence. As a former Team Lead for Mathematics at **Mercor Intelligence**, I oversaw quality assurance for Large Language Model (LLM) training, bridging the gap between theoretical rigour and data-driven technology. An experienced educator, I have taught courses ranging from Linear Algebra and Calculus to Python programming across institutions in the UK and Pakistan. ### Research Expertise * **Modular Representation Theory:** Structural analysis of blocks of finite group algebras. * **Derived Categories:** Computation of Picard groups and auto-equivalences. * **Computational Algebra:** Design of algorithms in GAP and Python to solve algebraic problems. * **Groups of Lie Type:** Investigation of $PSL(2,q)$, $SL(2,q)$, and their block structures. ### Technical Skills * **Languages:** Python, GAP, LaTeX. * **Domains:** Computational Group Theory, Mathematical Logic, LLM Training & QA.

Junaid Ali Shah Syed

Ph.D. Candidate in Pure Mathematics | Researcher & Educator in Mathematics.

Research Expertise

Algebra and Representation theory.

Modular representation theory

Representation theory of Reflection Monoids, and Semigroups.

About

Education

University of Manchester

PhD in Pure Mathematics / 2026 (anticipated)

IBA Sukkur University

Master of Science in Mathematics / 2020

University of Sindh Jamshoro

Bachelor of Science in Mathematics / 2016

Experience

University of Manchester

PhD Researcher / September, 2022 — June, 2026

Determined the Picard group of the principal blocks of the group rings OP SL(2 q) and OP GL(2 q) when q = 1 3 (mod 4) with the Dihedral defect group D2n. Determined the Picard group of the principal block of the group rings OSL(2 q) when q = 1 3 (mod 4) with the Quaternion defect group Q2n. The significant results established are that the Picent of the principal block of OSL(2 q) when q = 1 3 (mod 4) with the Quaternion defect group Q2n is trivial. Using computational algebra systems (GAP) to verify and guide structural research.

Graduate Teaching Assistant / November, 2022 — December

Provided 3+ years of teaching and supervision for undergraduate mathematics courses, providing expert academic guidance and advanced problem-solving support. Conducted regular supervision sessions to foster student understanding, resulting in improved student performance in challenging concepts.

Lead Postgraduate Researcher (PGR) Representative / September, 2024 — September, 2025

Served as the primary liaison for 100+ PGRs. Successfully communicated critical problems and difficulties to the Faculty of Science and Engineering (FSE) and highlighted issues at the departmental level. Secured a significant increase in research and travel fund allocation and successfully resolved course registration issues for all PGRs. Advocated and implemented free daily departmental coffee for PGRs, improving networking among PGRs and Faculty members.

Sukkur IBA University

MS Researcher / February, 2020 — December, 2020

Conducted an extensive review of Complex Representations of Boolean Reflection Monoids of types An1 and Bn. Performed computations of modular representations of the Boolean Reflection Monoid of types An1 and Bn. Conducted a thorough review of Coxeter Arrangement Monoids as part of the project. Successfully computed complex representations of the Coxeter Arrangement Monoid of type An1.

Chorltons High School and College Manchester UK

Teaching Intern / July, 2024 — July, 2024

Observed and participated in classroom settings to understand classroom dynamics and develop effective methods for communicating complex mathematical concepts to younger learners.

National University of Modern Languages Pakistan

Lecturer in Mathematics / February, 2022 — July, 2022

Delivered undergraduate Mathematics courses to students in the Computer Science and Business Administration faculties, tailoring instruction to diverse applications.

Sukkur IBA University Pakistan

Visiting Lecturer in Mathematics / February, 2019 — February, 2020

Taught foundational mathematics courses to students in the preparatory/foundation semester, focusing on building essential analytical and computational skills for subsequent degree programs.

Sukkur IBA University (Mathematics Resource Centre) Pakistan

Subject Specialist in Mathematics / February, 2019 — June, 2021

Provided individualized academic support and tutoring services to undergraduate students, reinforcing core mathematical concepts.

Army Public College System Pano Aqil Pakistan

Lecturer in Mathematics / August, 2018 — December, 2018

Instructed advanced mathematics courses to college students.

Army Foundation Education System Pakistan

Subject Specialist in Mathematics / August, 2017 — August, 2018

Taught mathematics at the secondary level, emphasizing conceptual clarity and analytical problem-solving skills.

Mercor Intelligence Remote

Team Lead Mathematics Expert / August, 2025 — October, 2025

Rapidly promoted to Team Lead in three months, taking on the supervision of 3 Reviewers and 30 Writers. Designed team operations, optimized data management, and developed standardized training protocols to ensure high-quality task production. Applied mathematical rigor to enhance LLM accuracy; conducted final stage meta-reviews and reported quality benchmarks to the Strategic Project Lead.

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