/Mathematical skill stack
Massive Skills with Massive Impact
Mathematical skills transcend the formulaic and the computational. Potent in both breadth and depth, they position you to build solutions in any field. Our map of 70+ skills and abilities gives you a head start in defining your professional value.
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The Foundation of Your Success
Understanding your skill set is crucial for professional potency. While mathematical knowledge is undoubtedly important, it’s the underlying skills developed through mathematical study that are more likely to open doors in the non-academic job market. These skills — from systems thinking to intuitive technical communication — are highly transferable across numerous roles and industries, including throughout deep tech.
Mathematicians have significant breadth and depth in their skill sets — far beyond what most realize. While mathematical knowledge is important, its acquisition and utilization rest on skills that truly set mathematicians apart. We’ve collected more than 70 of those skills, honed by the experiential drivers of studying, teaching, and researching mathematics at its highest levels.
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Characteristics of this Map
This skill map was built using a dual approach: a deep, first-principles analysis of the mathematical experience, combined with a practical reverse-engineering of industry needs. The result bridges the gap between academic mathematical ability and real-world professional demands, providing a clear roadmap for mathematicians to articulate and leverage their diverse skill sets.
We carefully built this map to have the following characteristics:
Domain-Invariant
The skills developed through the study of mathematics are highly transferable, often more so than the specific knowledge gained. Such high transferability easily allows mathematicians to pivot into diverse career tracks, from analytics and data science to technical communication.
Future-Proof
The continued evolution of artificial intelligence and automation might obsolesce certain aspects of mathematical know-how, making it more important than ever to recognize and cultivate our uniquely human skills.
Business-Friendly
Effective communication is key in the job market. Rather than using dense mathematical jargon, we’ve described these skills in terms that are more likely to resonate with non-academic employers.
Skills > Knowledge
By shifting the focus from knowledge to skills, our goal is to help mathematicians recognize, articulate, and leverage their unique abilities.
Math-Centric
While some of these skills may certainly be cultivated in other disciplines, our map seeks to interpret each through a distinctly mathematical lens.
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Skill Categories Menu
Click on a category to learn more:
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Strategic & Operational Skills
Developed by teaching and working in academic environments, operational skills are the real-world, get-things-done abilities that often fly under the radar.
Hidden Professional Superpowers
This category contains a diverse toolkit that includes clear communication, resource management, project planning, leadership, mentorship, and more.
These skills, refined through years of academic experience, transfer into various industries across deep tech, enabling mathematicians to find roles everywhere from startups to large corporations.
Many mathematicians don’t realize how much of this skill set they’ve mastered. These aren’t just skills for the lecture hall or the research lab — they’re professional superpowers that position mathematicians to not only solve complex problems but also guide teams and drive initiatives forward.
Leadership & Management
Manage resources, people, and projects both formally and informally.
Resource Management
- Budgeting and allocating resources effectively, given financial constraints, to optimally achieve departmental goals
- Monitoring expenditures, and prioritizing tasks, purchases, or projects.
- Managing the procurement process, negotiating vendor contracts, and actively managing vendor relationships to ensure timely and cost-effective acquisition of goods and/or services.
- Organizing conferences, trips, or other events that utilize the resources of the department.
- Ordering materials, organizing a lab, or raising funds for student activities.
Personnel Management
- Supervising and coordinating the work of tutors, TAs, RAs, or other direct reports.
- Offering support, advice, and direction to new or junior faculty or support staff.
- Establishing performance expectations and conducting performance evaluations.
- Implementing strategies to enhance employee achievement and effectiveness.
- Resolving any conflicts or issues by effectively addressing disputes and escalating issues when necessary.
Project Management
- The ability to prioritize as well as adapt to changing priorities.
- Meeting deadlines and managing time well, even with multiple, competing projects.
- Developing project plans, defining project objectives, creating timelines, and establishing project milestones.
- Communicating risks of projects to relevant stakeholders.
- Planning at both the strategic and tactical levels.
Program Management
- Developing and implementing program-level strategic plans.
- Collaborating with multiple departments and/or stakeholders to ensure coordination of program activities and allocation of program resources.
- Developing goals, objectives, targets, and supporting policies and procedures for a program.
- Tracking and monitoring the success of program initiatives.
- Coordinating meetings with relevant stakeholders and working closely with leadership to establish program priorities.
Advising & Mentoring
- Sharing advice with and motivating students to do their best.
- Providing expertise and guidance based on knowledge and experience.
- Listening, guiding, and helping students to develop learning strategies that contribute to their long-term academic and professional success.
- Working with new or junior faculty to help them understand and strive toward departmental goals.
- Creating a culture of excellence and shared success.
Reporting & Presenting
Analyze, visualize, and communicate key research findings.
Reporting
- Collecting and organizing relevant and accurate information from a variety of sources, such as databases, surveys, and research papers.
- Analyzing and interpreting data to identify trends and key insights.
- Writing comprehensive and well-reasoned reports such as executive summaries, progress reports, and research summaries.
- Ensuring the accuracy, completeness, and reliability of information by rigorously fact-checking and proofreading documents.
- Using reporting tools and software to create professionally formatted documents utilizing visual aids where necessary.
Data Visualization
- Creating data stories and visualizations for both technical and non-technical audiences, using a variety of tools such as charts, tables, graphs, graphics, and related tools to share findings discovered during data analysis.
- Using tools such as R (ggplot), Python (matplotlib), Tableau, or Excel to create visually appealing and informative graphics to enhance audience understanding.
- Selecting appropriate data sources, cleaning the data, and ensuring data integrity for accurate visualizations.
- Choosing the most suitable visual representation based on consideration both of the nature of the data and the needs of the audience.
- Developing strong data analysis skills to identify relevant patterns that inform meaningful visual representations.
Formal & Informal Presentation
- Planning and managing presentation time effectively to ensure a smooth flow of information.
- Managing Q&A sessions by listening attentively, providing well-reasoned responses, and handling challenging or unexpected inquiries.
- Adjusting presentations given audience feedback to find the right pace and language to deliver effective communication.
- Giving formal presentations to leadership or presenting research at conferences.
- Delivering informal presentations to students or colleagues via lectures, summaries, or colloquia.
Measuring & Monitoring
Estimate and assess information to compare situations to stated criteria.
Defining Metrics
- Clearly understanding the objectives of the measurement process, including that which is to be measured and against what criteria.
- Selecting relevant metric techniques and aligning them with departmental and project goals.
- Clearly articulating and communicating success criteria.
- Identifying and developing relevant Key Performance Indicators (KPIs) to provide measurable and actionable insights.
- Using both quantitative and qualitative approaches to capture impactful information.
Assessing & Evaluating
- Using metrics to determine progress or other development toward stated goals.
- Designing and implementing data collection methods to gather relevant and accurate information for evaluation.
- Analyzing collected data using relevant quantitative and/or qualitative techniques to distill accurate conclusions.
- Applying established frameworks to guide the evaluation process.
- Adhering to rubrics and other guides to ensure fair and unbiased evaluation.
Monitoring
- Continuing to assess on a predetermined or as-needed basis.
- Establishing quality assurance mechanisms to ensure the accuracy and reliability of data collected for monitoring purposes.
- Using feedback to determine if metrics need to be updated or revised.
- Refining strategies and approaches using insights discovered in the monitoring process.
- Tracking any potential risks by identifying deviations from expected outcomes.
Communication
Share complex information with a variety of audiences in a multitude of formats.
Verbal Communication
- Expressing ideas, instructions, and information clearly and concisely.
- Engaging with colleagues and other stakeholders in exploratory conversations to deeply understand concepts and implications.
- Using active listening techniques, such as asking clarifying questions, to accelerate comprehension.
- Adapting communication style and language to suit different audiences and contexts to communicate technical concepts in an accessible and intuitive manner.
- Demonstrating confidence and poise when speaking in front of large audiences.
Written Communication
- Writing proofs and explanations that reflect and clearly articulate logical thought processes.
- Corresponding with students, professors, colleagues, advisors, committees, and leadership via email and other formats.
- Adapting writing style to suit the context and audience, and adjusting tone for different purposes such as formal, persuasive, conversational, and informative.
- Writing technical documents, such as research papers and reports, using precise language and organized structure.
- Carefully reviewing and revising written work for errors, coherence, and clarity.
Teaching
Curate and communicate technical and conceptual information, and evaluate technologies, strategies, and effectiveness.
Curriculum & Course Development
- Conducting needs assessments, literature reviews, surveys, interviews, and using data analysis to identify gaps in existing curricula, and developing informed recommendations and strategies for improvement.
- Ensuring the alignment of curricula with accreditation standards, and departmental, institutional, state, and national benchmarks, as well as industry standards.
- Collaborating with faculty members, subject matter experts (SMEs), and other stakeholders to gather requirements and input, reach consensus, and develop proposed curricula.
- Implementing best practices in adult learning techniques, using a variety of instructional approaches including group work, class and group discussion, lectures, and hybrid and online models to enhance student outcomes.
- Creating engaging learning materials, and evaluating and selecting informational and technological learning aids and resources.
Educational Technology
- Analyzing requirements from a variety of stakeholders to select technologies that meet the needs of all users, usually under cost or contractual constraints.
- Adept at integrating innovative teaching methodologies that use emerging technologies to enhance student engagement and facilitate active learning.
- Using Learning Management Systems (LMSs) and related technologies for assignments and assessments.
- Conducting thorough market surveys and assessing potential technologies for instructional use.
- Leveraging technology for efficient course organization, administration, and content delivery.
User Experience Research
- Conducting usability evaluations to assess the ease of use and efficacy of educational technologies.
- Using qualitative and quantitative methods to understand user needs, behaviors, and preferences.
- Conducting task analyses to identify user goals, motivations, and pain points.
- Collaborating with faculty and administrators to align user experience research to educational goals and curriculum design.
- Understanding how your audience will receive complex information and learning to anticipate their questions and misunderstandings.
Assessing & Evaluating
- Designing relevant and reliable assessment tools, such as exams, quizzes, projects, and rubrics to measure student learning outcomes and comprehension effectively.
- Collecting and analyzing assessment data using quantitative and qualitative methods, using statistical software and data visualization techniques to gain deeper understanding of results.
- Conducting course and program evaluations to assess their effectiveness, relevance, and alignment with institutional and departmental goals as well as accreditation requirements.
- Participating in the peer assessment and review processes, providing constructive feedback and contributing to a culture of professional growth.
- Utilizing assessment data and evidence to enable data-driven decision making processes regarding possible areas for improvement in the teaching and learning processes.
Interpersonal
Work with others to create, test, and implement solutions.
Working in Teams
- Listening to input from others and defending your ideas logically.
- Effectively collaborating with colleagues through active participation, sharing of ideas, and contributing to the achievement of team goals.
- Communicating effectively and engaging in open and constructive conversations.
- Adapting to changing team dynamics by demonstrating willingness to take on new responsibilities and generally remaining flexible.
- Effectively managing conflicts or disagreements by seeking mutually beneficial resolutions.
Working across an Organization
- Working with faculty in other departments to conduct joint research, launch educational initiatives, and/or form research programs.
- Aligning objectives and coordinating efforts across a variety of stakeholders.
- Participating in events or initiatives to expand your professional network.
- Demonstrating comfort with interacting with all levels of the organization.
- Understanding the interconnectedness of different departments by considering the broader impact of decisions and actions.
Working with Outside Partners
- Working with community organizations and local businesses to increase opportunities for students to engage in professional activities.
- Managing external relationships by working to foster trust and mutual understanding, monitoring performance, and addressing any issues diplomatically.
- Building relationships with external business partners, such as vendors, to advance departmental goals.
- Clearly conveying expectations, constraints, requirements, and feedback.
- Utilizing negotiation skills to establish mutually beneficial contracts or agreements.
Negotiation & Persuasion
- Writing effective grant, research, and other funding proposals.
- Utilizing effective persuasion techniques, such as logical reasoning and evidence-based arguments, to influence and convince others to take desired actions.
- Exerting influence tactfully in informal environments.
- Expressing thoughts and concerns effectively and respectfully.
- Advocating for your interests while respecting the needs of others.
Operational
Successfully navigate large, complex organizations and comply with standards and protocols.
Navigating Bureaucracy
- Understanding the policies, protocols, and regulations that exist at the departmental, institutional, state, and federal levels and their effect on your work processes.
- Using the proper channels of communication and escalation to handle sensitive situations.
- Demonstrating the ability to adapt to evolving administrative requirements and adjust strategies and approaches effectively.
- Competently interacting with different departments and understanding decision-making processes and stakeholders within large organizations.
- Establishing and maintaining positive working relationships with key stakeholders such as department heads, administrators, and support staff.
Compliance
- Exercising strict adherence to procedures involving data privacy, confidentiality, and any other legal or regulatory protocols at the departmental, institutional, state, and federal levels.
- Participating in all reporting processes necessary for compliance related to grants, funding, and other resource usage.
- Conducting risk assessments to identify potential compliance vulnerabilities and devising strategies to mitigate risks.
- Compiling and submitting required compliance reports to regulatory bodies.
- Maintaining accurate documentation and records of administrative processes, ensuring compliance and facilitating streamlined operations.
Enterprise Software
- Proficiency in using enterprise software applications such as ERP systems, project management tools, and collaboration platforms.
- Utilizing enterprise software systems to streamline administrative tasks.
- Identifying opportunities to optimize workflows and automate processes using software enterprise features, improving efficiency and reducing manual input.
- Using software and related tools for communication, HR tasks, professional training and certification, and scheduling.
- Customizing software functionality to align with departmental objectives.
Process Optimization
- Using process mapping techniques to analyze existing business processes and identify bottlenecks and opportunities for improvement.
- Utilizing data analysis techniques to collect and analyze data related to business processes in search of ways to increase efficiency and cost-effectiveness.
- Contributing to a culture of continuous improvement by establishing feedback loops and collecting relevant information.
- Identifying opportunities to leverage technology solutions, automated processes, and digital tools to streamline business processes.
- Planning and implementing optimization processes, gaining buy-in for changes from stakeholders by clearly communicating the benefits.
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Conceptual & Applied Skills
This category encompasses the abilities most commonly associated with mathematics: advanced modeling, adept use of computational software, and razor-sharp logical reasoning.
From Theory to Business Practice
But it goes beyond the expected, delving into the nuances of an array of thinking modes including systems thinking, first-principles analysis, and creativity. Crucially, this category balances both conceptual with the procedural, recognizing that many mathematicians excel in pure theory as much as in practical computation.
These skills offer mathematicians an advantage in both technical and nontechnical roles across various industries. In technical fields, the ability to create complex models, implement robust algorithms, and ensure reproducible results is invaluable for roles in data science, software engineering, and research & development. Meanwhile, the conceptual strengths — abstract reasoning and creative problem-solving — translate well into nontechnical roles in strategy, analysis, and management.
Reasoning Ability
Analyze and solve complex problems systematically.
Quantitative Reasoning
- The ability to work meaningfully and non-trivially with quantity, which includes understanding how to quantify phenomena, and knowing when quantification is even possible.
- Demonstrating comfort and proficiency in working with quantities, computing accurately, and understanding mathematical relationships in quantitative contexts.
- Analyzing numerical data using various statistical and machine learning methods to draw meaningful conclusions.
- Understanding fundamental statistical, approximation, and computational concepts and applying them appropriately in data analysis.
- Applying logical reasoning and critical thinking skills to interpret quantitative information and data, evaluate evidence, and draw substantiated conclusions.
Qualitative Reasoning
- Systematically coding and categorizing information, identifying recurrent concepts.
- Synthesizing qualitative data across multiple sources to identify common themes, and developing a comprehensive understanding of complex relationships and concepts.
- Understanding structural similarities in seemingly disparate entities.
- Performing logical operations on classes of objects.
- Applying interpretive skills to analyze qualitative data, generating a holistic understanding of multifactorial topics.
Logical Ability
- The ability to find substantive relationships among ideas within the framework of cause and effect.
- Applying formal and informal logical rules to draw conclusions.
- The ability to identify and maintain logical consistency in reasoning, avoiding contradictory statements or flawed conclusions.
- Adapting logical reasoning strategies to new or unfamiliar situations.
- Extensive experience with propositional logic, including the manipulation of logical operators to assess the truth value of compound statements.
Thinking Styles & Capacities
Utilize multiple modes of thinking for innovative problem-solving.
Critical Thinking
- Being deliberate and systematic in thought to form conclusions that can be tested and updated.
- The ability to think with a keen eye for detail and a high degree of precision about technical, complex material.
- Conducting objective analysis, evaluation, and interpretation of information to form well-reasoned conclusions.
Analytical Thinking
- Breaking down complex problems or situations into smaller components, analyzing them individually, and drawing meaningful insights.
- Identifying key principles, elements, and variables involved.
- Approaching problem-solving and decision-making in a systematic and logical manner, identifying connections, correlations, and causal relationships.
Systems Thinking
- Understanding complex systems by identifying and understanding their interconnected components and the feedback loops that exist among these components, and comprehending cause and effect in the aggregate.
- Recognizing when a system is more than the sum of its parts, and gaining a deep understanding of how various elements interact to influence a system’s behavior.
- Examining emergent properties that arise from interactions among system components, which are not present in individual components, and recognizing the dynamic, nonlinear nature of complex systems.
First-Principles Thinking
- Breaking down problems and assumptions into fundamental principles, foundational truths, and/or axioms, and then building logical conclusions based on those essential components.
- Critically examining assumptions, challenging their validity, and questioning and eliminating preconceived notions.
- Gaining scalability and transferability of solutions by identifying domain-invariant principles that can be applied across different situations.
Synthetic Thinking
- Integrating diverse ideas and information to develop comprehensive, logically sound understandings and solutions.
- The ability to assimilate disparate elements and reconcile them effectively.
- Engaging in multidimensional analysis, considering various factors, variables, and concepts and exploring the interconnectedness and interdependencies among such elements.
Strategic Thinking
- Analyzing the relationships and dynamics among various factors to identify leverage points and utilize them advantageously.
- Aligning iterative actions and decisions with broader goals.
- Adopting a long-term perspective with consideration of future outcomes.
Tactical Thinking
- Formulating short-term plans to accomplish specific objectives within a larger strategy.
- Remaining flexible and adaptable as tactics fail or partially succeed and updating them to reach long-term goals.
- Optimally allocating and utilizing available resources to maximize efficiency, and quickly identifying obstacles to maintain forward motion.
Lateral Thinking
- Approaching problems from unconventional perspectives to uncover and/or develop innovative solutions.
- Generating unexpected connections and exploring alternative pathways.
- Exiting your comfort zone and remaining open-minded and receptive to different ideas, diverse opinions, and unconventional solutions.
Conceptual Thinking
- Grasping and manipulating abstract concepts, understanding underlying principles, and developing conceptual frameworks.
- Drawing parallels among the abstract and the concrete using analogy and metaphor to relate the conceptual to the familiar.
- Distilling complex ideas by extracting core principles and features and their implications.
Associative Thinking
- Making connections between seemingly disconnected concepts, ideas, or domains.
- Using analogies, metaphors, and cross-disciplinary perspectives to gain unique or novel insights.
- Transferring knowledge and applying concepts from one context to another by understanding the generalizability of such ideas.
Divergent Thinking
- Generating a wide range of ideas or possibilities and exploring multiple possibilities.
- Using nonlinear, expansive modes of thought to produce unique or unconventional solutions.
- Thinking in an unconstrained manner, removing self-imposed limitations to explore unusual ideas without initial concern for practicality.
Metacognition
- Reflecting on your thinking processes and assessing your mental models.
- Recognizing personal cognitive biases, assumptions, and limitations in order to challenge and expand analytical ability.
- Monitoring and understanding your problem-solving strategies and approaches in order to improve cognitive performance.
Creativity
Rethink assumptions, employ tools inventively, and adapt to change.
Creativity
- Challenging assumptions and exploring multiple perspectives.
- Using tools in original ways to generate novel ideas, approaches, or solutions.
- Remaining flexible and experimenting with different strategies and tactics.
- Embracing uncertainty and taking calculated risks.
- Breaking away from conventional thinking patterns and looking for solutions that are not standard or clear.
Algorithmic, Computational, & Programming Skills
Translate abstract concepts into tangible processes.
Algorithm Design
- The ability to apply theory to develop actionable, step-by-step processes that yield solutions.
- Extracting essential information or characteristics from a problem and understanding the causality of core elements.
- Systematically breaking down complex problems into smaller units and modules.
- Translating mathematical processes into code or other digital algorithms by identifying the sequential steps required to reach a solution.
- Understanding algorithmic feasibility and time complexity, and optimizing resource usage appropriately.
Computational & Statistical Software
- Using statistical, mathematical, and analytical tools such as Python, R, Matlab, SAS, Tableau, and Excel to solve scientific, financial, and other applied problems.
- Working proficiency in statistical and analytical libraries which can be used for data visualization, modeling, data processing, and data analysis.
- Applying supervised and unsupervised learning algorithms to build predictive models and classifiers.
- Running simulations and solving optimization problems.
- Merging and analyzing data from various file formats, databases, or APIs.
Programming
- Writing reusable, readable, efficient, testable code in Python, R, C++, Java, Javascript, and knowledge of the syntax, data structures, control flow, and the libraries of such languages.
- Proficiency in identifying and resolving errors, bugs, and other issues using systematic debugging techniques, log analysis, and testing.
- Familiarity with version control systems and experience with branching, merging, and collaboration features to manage code changes and track revisions.
- Working knowledge of databases and their structures, as well as how to query them using SQL, GraphQL, Gremlin, etc.
- Understanding and implementing coding best practices such as documentation, appropriate naming conventions, indentation, and modularization.
Data Management & Analysis
Turn raw data into compelling statements.
Data Management
- Understanding data structures and architecture for collecting, organizing, and storing data.
- Ensuring data accuracy and integrity as well as adhering to all data privacy and confidentiality policies.
- Ensuring data accuracy and integrity as well as adhering to all data privacy and confidentiality policies.
- Building datasets from a multitude of sources.
- Documenting data sources by use of data dictionaries, metadata, and data lineages to ensure usability of data assets.
Data Manipulation
- Preprocessing and cleaning data by deduping, imputing missing values, handling anomalies and outliers, and correcting inconsistent formatting.
- Preparing data for analysis and modeling by normalizing, reshaping, aggregating, and converting data types.
- Ensuring data compatibility and consistency across different datasets.
- Building datasets for analysis by merging, joining, ordering, sorting, subsetting, and formatting dates and times.
- Applying statistical and machine learning techniques to interpret data and extract insights.
Data Visualization
- Creating visual representations of data such as charts, scatter plots, bubble charts, correlation matrices and heatmaps, tree maps, and network diagrams.
- Selecting appropriate visualization techniques to effectively convey insights into the underlying data.
- Interpreting and analyzing data visually.
- Considering the target audience and their needs, constructing relevant information hierarchies, and tailoring visualizations for audience comprehension.
- Proficiency with tools such as ggplot, Tableau, or Excel.
Data Analysis
- Performing exploratory data analysis using statistical, machine learning, and other techniques such as segmentation; clustering; and random forests and other classification techniques.
- Calculating measures of central tendency and measures of dispersion.
- Conducting hypothesis testing, constructing confidence intervals, conducting ANOVAs, chi-square tests, and other inferential statistical tests.
- Working knowledge of regression analysis (logistic, linear, nonlinear) and of time series methods such as moving averages, exponential smoothing, and ARIMAs.
- Experimentally engineering features and applying machine learning techniques, selecting appropriate techniques for different data types and research questions.
Mathematical & Statistical Modeling
Accurately capture and represent systems and processes.
Modeling
- Defining problems in a manner such that they can be translated into mathematical or statistical models by understanding the problem domain and entities.
- Constructing mathematical or statistical formulations that effectively represent essential components, variables, and their relationships within the system being modeled.
- Deeply understanding and evaluating assumptions and any potential risks or limitations associated with modeling.
- Formalizing problems in mathematically valid ways by understanding the domain of the problem and selecting the most relevant tools.
- Using advanced techniques such as Monte Carlo and other simulation techniques; statistical and stochastic techniques; time series methods such as ARIMAs, autocorrelation, and spectral analysis; and many, many others.
Analysis of Accuracy
- Assessing the accuracy and reliability of models by comparing their predictions or outputs with real-world observations or empirical data.
- Evaluating model performance and/or goodness-of-fit by employing techniques such as confusion matrices, logarithmic loss, area under the curve (AUC), precision, recall, sensitivity, specificity, f-score, regression metrics, receiver operating characteristic curve (ROC), bias, gain and lift charts, Gini coefficients, Kolmogorov Smirnov charts, and many, many more.
Model Life Cycle
- Understanding each step of the modeling process and its importance.
- Formulating the problem to be modeled in a mathematically accurate and tractable way, selecting mathematical and statistical tools, estimating and/or tuning parameters, iterating and optimizing, and building performant models.
- Documenting model development thoroughly, including explicitly stating model assumptions, methods, and parameters to ensure transparency and reproducibility.
- Interpreting results and outputs by translating complex mathematical concepts into meaningful, intuitive, and actionable (if possible) insights for stakeholders.
- Maintaining model applicability by optimizing, tuning, training on new data, and monitoring performance.
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Deep Problem-Solving Acumen
These skills embody the mathematician’s unique ability to work with unwavering focus for extended periods, often in solitude, driven by an insatiable curiosity to discover the immutable truths of the universe.
The Transformative Power of Mathematical Thinking
This category represents the heart and soul of mathematical thinking, encompassing a profound set of skills including intellectual humility, the ability to distill complex problems to their essence without losing critical information, and the capacity to impose structure on apparent chaos.
These skills reflect a depth of problem-solving ability that positions mathematicians to ask truly novel questions and skillfully pursue answers. While these abilities may not feature prominently in job posts, they are arguably the most vital for those seeking to push the boundaries of technology and human understanding.
With them, mathematicians are able to forge connections between seemingly disparate ideas, invent entirely new problem-solving methodologies, and innovate in ways that allow us to view various layers of spacetime — or even peer beyond it.
Research Skills
Identify, structure, and solve original problems.
Identify and Formulate Original Problems
- Proficiency in identifying research gaps or areas of inquiry that contribute to the advancement of mathematics.
- The ability to source and analyze existing literature, theories, and/or data to identify research questions or hypotheses that extend or challenge current thinking.
- Designing appropriate research methodologies, such qualitative, quantitative, mixed-methods, or theoretical approaches, and experimental design.
- Forming, testing, and updating hypotheses and/or conjectures based on data, information, and careful analysis.
- Crafting well-structured research proposals, including research objectives, methodology, and expected outcomes.
Continuous Learning
- Staying current with emerging research methodologies, data collection and analysis techniques, and analytical approaches.
- Demonstrating a genuine passion for exploring new ideas, acquiring new knowledge, and seeking innovative solutions.
- Keeping up-to-date with new analytical software and technology.
- The capacity and desire to teach yourself new things that enhance and expand your knowledge and abilities.
- Reading newly published papers in your field of expertise to remain at the frontier of advancements.
Independence
- The ability to manage yourself in terms of time, resources, information, and effort.
- Demonstrating independence of thought to uncover original questions and challenges to conventional or current thinking.
- The ability to architect effective self-directed learning plans and implement them successfully.
- Proficient in finding answers on your own with little to no guidance, no precedent, and no templates to follow.
- Intrinsic motivation to set ambitious goals with the ability to follow through with little assistance.
Sustained Focus
- The ability to work on complex problems over long timelines, maintaining a prolonged concentration.
- Maintaining motivation and perseverance despite encountering repeated challenges and setbacks.
- The ability to adhere to the initial goal of the endeavor in the face of distraction, balanced with the ability to pursue new paths when advantageous.
- Overcoming obstacles by systematically working through challenges and demonstrating resilience.
- Working consistently despite periods of boredom, frustration, or disengagement.
Tenacious & Resilient Learning
An unyielding commitment to intellectual depth, persisting through the challenges and solitude that accompany genuine mastery.
The Ability to Do the Unsexy Work
- A willingness to do work that others find mundane or undesirable, demonstrating a consistent and reliable work ethic decoupled from the recognition resulting from the work.
- Maintaining dedication through monotony, tedium, and setbacks, and sustaining professionalism and high standards irrespective of the nature of the work.
- Demonstrating a commitment to the completion of tasks even when they lack immediate reward, recognition, or appreciation.
- The ability to willingly and diligently tackle tasks or take on responsibilities that are neither glamorous nor necessarily prestigious, but are essential to the successful completion of a project.
- Embracing a growth mindset, recognizing any and all tasks that offer an opportunity for learning and self-improvement, and actively seeking such work.
The Ability to Learn Independently
- Proven ability to acquire new knowledge, skills, and expertise through self-directed learning, demonstrating autonomy and an unyielding drive for personal growth.
- Exhibiting resourcefulness in finding learning resources such as books, articles, and online courses and learning communities.
- The ability to critically evaluate and discern the quality, reliability, and relevance of information sources.
- Engaging in metacognitive practices to self-evaluate, identify areas for improvement and growth, and develop effective learning strategies.
- Setting clear goals to create personalized learning plans that are well-directed and adapted to the subject matter.
The Ability to Learn Thoroughly
- Proficient in acquiring knowledge and skills through comprehensive and in-depth learning approaches, emphasizing depth of understanding over speed of acquisition, demonstrating a commitment to thorough mastery of the subject matter.
- Developing a strong conceptual understanding by connecting new information to existing knowledge, recognizing the interconnections between different ideas.
- Integrating information from a multitude of sources and synthesizing it effectively to form a coherent and comprehensive understanding.
- Prioritizing depth of understanding over breadth of coverage, focusing on mastering core concepts and skills before exploring peripheral or tangential areas.
- Embracing a lifelong learning mindset, recognizing that learning is a continuous process, and actively seeking opportunities for ongoing development and improvement.
Systematic Solution-Building
Crafting clarity and structure out of complexity, balancing strict constraints with imaginative problem-solving approaches.
Problem Formulation & Architecting
- Deconstructing problems into their fundamental components and recognizing dependencies and relationships.
- Establishing clear boundaries and scope for a given problem, ensuring a focused approach and avoiding unnecessary complexity.
- Creating systematic frameworks or models to organize problem information, identifying key variables and unknowns.
- Prioritizing problem components based on their significance, urgency, or feasibility, and determining the appropriate sequence for addressing them.
- Thoroughly conducting information gathering processes and literature reviews to collect relevant data, results, and insights necessary for problem analysis and solving.
Creation of Clarity from Chaos
- Transforming complex and chaotic information into organized, structured, and understandable forms.
- Developing organizational frameworks and mental models to categorize and arrange information to form coherent and logical representations.
- Applying techniques to simplify complex concepts or information, distilling them into manageable parts, reducing complexity, and enhancing clarity.
- Synthesizing and integrating scattered or fragmented information and effectively managing stress and uncertainty to remain focused.
- Maintaining a solution-building mindset, embracing challenges as opportunities for growth, and leveraging chaos as a catalyst for new perspectives and innovative solutions.
Reverse-Engineering
- Adopting a reverse thinking approach, starting from the desired outcome or solution and working backward to unravel the necessary steps required to achieve it.
- Developing a conceptual understanding of the problem or system, recognizing key principles, processes, or mechanisms that underpin its functionality.
- Adopting an iterative approach to problem-solving, continually refining the understanding of the problem through feedback, analysis, and adjustments to the reverse-engineering process.
- Proficient in deconstructing complex problems or systems to understand their underlying processes, components, or principles.
- Reconstructing the problem or system by assembling the deconstructed elements, analyzing their interactions, and identifying potential areas for improvement.
Problem-Solving under Extreme Constraint
- Proficiency in recognizing and conveying the limitations, factors, or conditions that restrict or otherwise shape problem-solving processes.
- Evaluating the relevance and impact of constraints, determining their significance in defining the problem space.
- Extensive experience solving complex problems with highly restrictive or challenging constraints.
- Demonstrating perseverance and maintaining the motivation to explore multiple potential paths until a viable solution is found.
- Finding innovative ways to work within constrained systems and develop sound solutions.
Translation of Hard Problems into Tractable Problems
- Adopting an iterative and incremental approach to problem-solving, continuously refining problem translations and tractability strategies based on analysis of partial solutions.
- Employing simplification strategies, such as assuming ideal conditions, reducing dimensions, or applying approximation methods.
- Identifying and extracting essential components or principles from complex problems, abstracting away unnecessary details, and focusing on the core elements that impact the problem’s solvability.
- Developing effective problem representations to capture the problem’s structure and isolate potential solvable subproblems.
- Applying algorithmic thinking to devise step-by-step procedures that can be implemented to solve specific subproblems, sequentially working toward a solution of the overall problem.
Core Principle Discernment
The art of grasping the immutable, bringing clarity to the enduring aspects of complex problems.
Pursuit of the Fundamental
- Seeking and identifying essential concepts, principles, and underlying theories that form the foundation of complex entities.
- Developing a strong conceptual understanding of the subject matter, recognizing core principles and frameworks.
- Distilling complex information down into its essential elements, filtering out extraneous details, and maintaining focus on the fundamental concepts that drive the situation.
- Utilizing analytical and systems thinking to dismantle complex phenomena into constituent elements.
- Displaying a marked curiosity to question underlying reasons, assumptions, and principles that govern systems.
Definition Formation
- Demonstrating a strong grasp of fundamental concepts and their interrelationships.
- Using questioning and clarification techniques to resolve ambiguity and achieve precision and accuracy.
- Integrating diverse information to form concise definitions that capture the essence of complex concepts and phenomena.
- Organizing and structuring fundamental information effectively.
- Applying well-formed definitions to problem-solving processes, clarifying problem statements and identifying root causes.
Assumption Identification
- Proficiency in recognizing implicit assumptions that underlie arguments or analysis.
- Engaging in active listening to identify unstated presuppositions in others’ statements and asking probing questions to uncover and clarify hidden assumptions.
- Finding the foundation of arguments or lines of reasoning by examining the logical connections between statements in excruciating detail.
- Analyzing the broader context of information, data, circumstances, or situations, and considering factors that may influence underlying assumptions.
- Using systems thinking to identify interconnected assumptions within complex systems, recognizing how assumptions made in one area may impact other system components.
Identification of Invariance
- Recognizing the stable factors that have a consistent impact on the problem or phenomenon under consideration.
- The ability to identify fixed and unchanging variables and parameters.
- Designing experiments and investigations that isolate variables and systematically test their influence, identifying those that exhibit consistent behavior.
- Leveraging prior knowledge and experience to identify constants based on established principles.
- Considering the broader context and specific domain knowledge to determine which variables are likely to be constant, understanding the underlying factors that contribute to their stability.
Identification of Dependencies
- Investigating the underlying factors that drive changes in observable entities.
- Awareness of external factors or environmental conditions that influence the relationships among variables.
- Evaluating different scenarios to understand how future changes in an entity might impact others.
- Analyzing feedback loops to recognize how changes in one variable lead to reinforcing or balancing effects on another.
- Establishing correlations empirically via statistical or other techniques.
Identification of Constraints
- Adept at recognizing and articulating constraints to facilitate targeted solution development.
- Proficiency in establishing the scope of challenges through rigorous constraint identification and analysis.
- Expertise in leveraging understanding of constraints to guide strategic planning and execution.
- Competent in defining problem parameters to clarify potential avenues for innovation.
- Proficient in identifying and analyzing limiting factors to optimize solutions strategies.
Finding the Signal in the Noise
- Proficiency in identifying meaningful patterns or valuable information in noisy situations, demonstrating a keen ability to extract and interpret relevant signals amidst extraneous noise.
- Developing a knack for recognizing significant anomalies or patterns that stand out from background noise.
- Proficiency in data analysis techniques to identify trends that indicate meaningful signals.
- Exhibiting excruciating attention to detail to identify subtle patterns and trends.
- Explicitly utilizing noise reduction techniques such as filtering, smoothing, and reduction of dimensionality to minimize extraneous or irrelevant data and enhance the visibility of relevant signals.
Rigorous Inquiry
A relentless pursuit of truth, unshackled by ego, constantly challenging and refining one’s understanding in the face of the unknown.
Counterintuition
- Proficiency in grasping and comprehending results or findings that challenge or contradict common intuition.
- Demonstrated ability to think beyond obvious assumptions and embrace unconventional perspectives to understand complex phenomena.
- Using logical reasoning to evaluate conflicting or counterintuitive results, identifying potential explanations that reconcile discrepancies.
- Developing an aptitude for recognizing underlying relationships and structures in complex systems, enabling the identification of counterintuitive actualities or unexpected correlations.
- Cultivating deep curiosity and epistemic humility, and revising beliefs when confronted with logical and/or experimental evidence.
First Step Is to Disprove Yourself
- Maintaining a rigorous and skeptical mindset, demonstrating the ability to actively seek counterexamples, disprove conjectures, or identify weaknesses in reasoning as a crucial step in refining theories, strengthening arguments, and advancing knowledge.
- Actively searching for exceptions that disprove or challenge the validity of a conjecture, helping to refine theories or identify boundary cases.
- Welcoming critical feedback from colleagues to identify gaps, weaknesses, and alternative interpretations.
- Designing experiments or tests to explore the boundaries of a conjecture, seeking situations in which it fails or contradicts established results.
- Conscientiously challenging your own ideas to develop conjectures and models that can withstand scrutiny.
The Ability to Know when You Are Wrong
- Demonstrating an unyielding commitment to intellectual rigor, integrity, and the pursuit of truth.
- Actively cultivating intellectual humility as a core value, acknowledging the fallibility of your own ideas, theories, or beliefs, and being receptive to new evidence or alternative viewpoints.
- Prioritizing evidence-based thinking, valuing empirical data, rigorous experimentation, and logical reasoning to inform beliefs and drive effective inquiry.
- Embracing the principles of scientific integrity, taking responsibility for your own work, and demonstrating a commitment to rectifying and learning from mistakes.
- Demonstrating an unwavering commitment to the pursuit of knowledge and fostering a mindset of continuous improvement, rigorous inquiry, and advancement.
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