Modern Mathematics for Technology, AI & Finance

This 10-lecture program introduces students to the foundational ideas, structures, and modern applications of mathematics. Beginning with high-school level concepts, the course expands into major fields of modern mathematics — including logic, analysis, algebra, geometry, topology, matrix theory, and financial mathematics.

  • ✔Build mathematical thinking and logic

  • ✔Understand the why behind formulas and theories

  • ✔Connect math to real-world systems

  • ✔Explore how mathematics drives today’s technology and finance

Principal Team

All instructors are graduates of the Peking University Department of Mathematics.

Yimin Yang, Ph.D

Professor of Mathematics, Duke University

Dr. Yimin Yang is a co-founder of Loyal Trust Bank, where he has served as Acting CEO, Board Director, and Chief Credit Officer. He previously built and led risk management divisions at PNC Bank and Truist Bank. Dr. Yang earned his B.S. in Mathematics from Peking University, M.S. in Computer Networking from Carnegie Mellon University, and Ph.D. in Mathematics from the University of Chicago. Before entering the financial industry, he was a professor at the University of Minnesota.

Topics include applications in large language models (AI/LLMs), neural networks, black-hole physics and game theory, blockchain encryption, bond pricing, stock-risk analysis, and portfolio management. Content may adjust to support student level and interest.

Why This Program Matters


  • ✔ Bridges high-school math to university-level thinking


  • ✔ Builds foundations for STEM, Computer Science, Engineering, Economics & Finance


  • ✔ Reveals how mathematical concepts shape technology and global markets


  • ✔ Strengthens academic readiness for college applications and interviews

Lecture Roadmap

  • Counting Infinity

    Lecture One (Counting Infinity): Foundations of Mathematical Logic: Counting and Comparing the Infinite. How did primitive people, who didn't understand numbers, deal with the concept of infinity? We know there are infinitely many rational numbers, but why are there still fewer of them than irrational numbers?

  • Unknown Numbers by Newton

    Lecture Two (Unknown Numbers by Newton): Foundation of Analysis (I): Testing two same numbers is one of the foundations of calculus. Mathematicians finally understand what real numbers are. How to correct Newton’s misinterpretation of mathematics? What is the Squeeze/Pinching Theorem for functions?

  • Zero vs Infinitely Small

    Lecture Three (Zero vs. Infinitely Small): Foundation of Analysis (II).  What is zero divided by zero? Why is Newton considered the father of all algorithms in the world? How do we add infinitely many zeros? One formula reveals all the mysteries of calculus.

  • Symbolic Rules

    Lecture Four (Symbolic Calculations and Rules): Foundation of Algebra & Statistics (I): Symbolic manipulation in mathematics, the factorization, and extrema of quadratic polynomials, and how they are used in regression statistical analysis.

  • Transformation of Objects

    Lecture Five (Transformation of Objects): Foundation of Geometric and Topology.  The Mathematical Principles Behind Monkey King Sun Wukong's Fiery Eyes: Euler Invariants. How can one discern the transformations of a cup that has come to life? How did Stephen Hawking penetrate black holes? How did the Nobel laureate from "A Beautiful Mind" use game theory to study the Nash equilibrium of criminals? It turns out these two approaches use the same method.

  • A World Built From Rules

    Lecture Six (When a World Determined only by Mathematical Rules): Foundation of Algebra (II): Why were Newton's
    mathematical calculations correct, even though his mathematical theories were incorrect? How do you find a formula that does not exist? Do parallel lines really intersect? How did a Young genius change the entire world of mathematics? How are geometric curves used in cryptography and blockchain?

  • Large Data Arranged

    Lecture Seven (Large Data Arranged): Foundation of Matrix(I). How can large amounts of data be stored? How can the relationships between data points be maintained? Matrix arithmetic. How can matrices be used to study bank credit risk? How can Markov process matrices be used to analyze DNA?

  • Big Data Transformed

    Lecture Eight (Transformation of big data): Foundation of Matrix (II). Transformations in Spaces of Different Dimensions: Dimensionality reduction and high-dimensional embedding in big data science. The relationship between neural networks and DVDs. Diagonalization of symmetric matrices, The Foundation of Large Language Models. How are probability matrices used to generate text?

  • Bond Investment Mathematics

    Lecture Nine (Bond Investment Mathematics): Foundation of Financial Mathematics (I). How are random variables and symmetric correlation matrices used to predict changes in interest rates? The basics of bond and fixed-income calculations: PCA analysis. How is probability used in lotteries? How can playing cards be shuffled to achieve maximum randomness?

  • Stock Risk Management

    Lecture Ten (Stock Risk Management Basics): Foundation of Financial Mathematics (II): Does pollen only become active after getting drunk? The principle of diversity in stock market analysis and the minimum risk portfolio. How to win (multiple) Nobel Prizes with the simplest quadratic polynomial? Why should everyone buy only the same stock? What stock is this? Are option securities and quantum mechanics simulations the same computation??
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