1. Under the Radar: The Ubiquity of Mathematics and Statistics in University Education
Mark Green, UCLA

2. The Big Questions
The role of Math has changed and expanded; what does this mean for the role of Math Depts?
What is our vision for Quantitative Education for the 21st Century?
Are we meeting the needs of our students?
What should WE, as faculty, do to adapt to this new environment?
What information would be helpful in answering these questions?

3. Quantitative Education for the 21st Century
Do we have a vision as a profession about our educational mission?
Such a vision should be nuanced: One size does not fit all
The goal is coherence without uniformity
An education that provides the foundation for a lifetime—relevant but not trendy
How do we build capacity to carry out this vision?
What changes do we need to make in our role as faculty to adapt to what such a vision asks of us?

4. A Few Questions
What information do we need that we do not already have?
What student populations are we trying to serve?
What mathematical content and skills will prepare our students for the world they will live in?
Can we shorten the time from discovery to entering the curriculum?
What departments should we partner with?
How do we move from individual-centered course innovation to systemic change?
How to we scale up our successes?
Where will the resources come from?
How best do we forge a consensus for change?

5. The “Unreasonable Effectiveness of Mathematics”
Prime Numbers-> Secure Internet Commerce
Operators on Hilbert Space-> Quantum Mechanics
Quaternions->Satellite Tracking, Video Games
Eigenvectors-> Google’s PageRank
Stochastic Processes-> Black-Scholes
Integral Geometry-> MRI and PET scans
Connections-> Gauge Fields

6. The “Unreasonable Effectiveness of Applications”
Electromagnetics -> Hodge Theory
Nuclear Physics -> Random Matrix Theory
Geosciences -> Chaotic Dynamical Systems
Superconductivity -> Ginzburg-Landau Equation
String Theory -> Gromov-Witten Invariants
Condensed Matter -> Complex Systems
Epidemiology -> Interacting Particle Systems
Deep Learning -> ??
Mathematics evolves. This is one of the ways new “mathematical species” come into existence.

7. My Subject: The State of Play at UCLA
Who requires what?
Who teaches what?
What are the major trends?

8. Majors That Require Math + Stat
All Life Science majors
Computational and Systems Biology
Environmental Science
Neuroscience
Physiological Science
Psychobiology
Cognitive Science
Economics
Business Economics
Anthropology (BS)
Human Biology and Society (BS)
Applied Math (but not Math!)

9. Majors Requiring Math But Not Stat
All Physical Science
All Engineering
Earth and Environmental Science
Climate Science
Math (tracks except Applied Math)

10. Majors Requiring Stat But Not Math
Psychology
Sociology
Political Science
Public Affairs
International Development Studies
Communications
Human Biology and Society (BA)
Anthropology (BA)

11. Easy to Remember Summary
Math + Stat: Life Sciences
Math: Physical Sciences and Engineering
Stat: Social Sciences
Neither: Humanities, Arts, Ethnic and Gender Studies
Key Takeaway: We have an enormous responsibility!

12. Some Trends New majors are proliferating (135 total at UCLA)
Majors which are not computational are birthing majors that are: e.g. Psychology  Neuroscience, Cognitive Science, Psychobiology; Anthropology and Human Biology and Society have both BA and BS tracks
Minors are proliferating (94 total at UCLA), often so students have computational credentials, e.g Digital Humanities minor
Lots of Math and Stat courses are being taught in other departments; often these courses marry disciplinary knowledge and quantitative methods
Data Science!

13. Into The Weeds With Mark
I want to talk about specific subjects, what courses they offer (not necessarily required) and the trends they illustrate
I will try to give an idea of WHY these various subjects need the topics that they do

14. Ecology and Evolutionary Biology
C119A: Mathematical and Computational Modeling in Ecology
C119B: Modeling in Ecological Research
133: Elements of Theoretical and Computational Biology (Intro to core Math ideas and models necessary to…)
C171: Practical Computing for Evolutionary Biologists and Ecologists
C172: Advanced Statistics in Ecology and Evolutionary Biology
M178: Computational Systems Biology: Modeling and Simulation of Biological Systems

15. Why Population models like dp/dt = r p – a p^2
Lotke-Volterra predator (y)-prey (x) model
dx/dt – ax – bxy; dy/dt = cxy-ey; a,b,c,e>0
Modeling effect of ecological diversity on resistance to invasive species (systems of ODE)
Modeling extinction, neutral evolution (probabilistic models)
Modeling speciation and the role of habitat (spatial statistics)
Island Biogeography (power laws, extinction models, spatial statistics,..)
Statistics of monitoring population distributions
predator-prey

16. Economics 41: Statistics for Economists
97: Economic Toolkit (“essential math and programming skills,” e.g. partial derivatives)
106G: Introduction to Game Theory
106GL: Introduction to Game Theory Lab
140: Inequality: Mathematical and Econometric Approach
141: Topics in Microecon: Mathematical Finance
142: Topics in Microecon: Probabilistic Microeconomics
143: Advanced Econometrics
144: Economic Forecasting (“theory and applications of time-series methods)
145: Topics in Microecon: Mathematical Economics
147: Financial Econometrics (reviews probability and statistics)

17. MS in Business Analytics
400: Mathematics and Statistics for Analytics
403: Optimization
406: Prescriptive Models and Data Analytics
434: Advanced Workshop on Machine Learning
436: Fraud Analytics

18. MS in Financial Engineering
402: Econometrics
403: Stochastic Calculus
405: Computational Methods in Finance

19. Why
Gathering and modeling economic data, modern portfolio theory (statistics, linear algebra)
Growth models (ODE’s)
Forecasting (Time-series, machine learning)
First welfare theorem, benefits of markets (optimization)
Equilibrium models (fixed-point theorems)
Modeling markets (stochastic processes, Bachelier, Black-Scholes)
Markets with incomplete or asymmetric information (game theory)
Income and wealth distribution, firm size distribution (PDE’s, mean field games)
Complex investment vehicles (algorithms and computation)

20. Sociology 111: Social Networks
112: Introduction to Mathematical Sociology
113: Statistical and Computational Methods for Social Research
M118: Simulating Society: Exploring Artificial Communities
191E: Undergraduate Seminar: Population Growth Models

21. Sociology Graduate Courses
208A-B: Social Network Methods
208C: Machine Learning for Social Scientists
210A-B-C: Intermediate Statistical Methods
212A-B: Quantitative Data Analysis
212C: Study Design and Other Issues in Quantitative Data Analysis
M213A: Introduction to Demographic Methods
M213B: Applied Event History Analysis (mostly different models)
M213C: Population Models and Dynamics
281: Selected Problems in Mathematical Sociology

22. Political Science 6: Introduction to Data Analysis
6R: Intro to Data Analysis, Research Version
170A: Studies in Statistical Analysis of Political Data
171A: Applied Formal Models: Collective Action and Social Movements (incl game theory, social networks,..)
Seminar of Politics of Algorithms

23. Why Analysis of survey data (linear algebra, multivariate statistics)
Social networks, science of science (graph theory, network science, probabilistic models, preferential attachment)
Emergent social phenomena (probabilistic models, Schelling’s model, collective action model)
Prediction (machine learning)
Coalitions and competition, emergence of cooperation (game theory, theory of repeated games)
Social choice theory (Arrow Impossibility Thm, Gibbard-Satterthwaite Thm)

24. Chemistry and Biochemistry
C100: Genomics and Computational Biology
110B: Intro to Statistical Mechanics and Kinetics
C122: Mathematical Methods in Chemistry (linear alg, complex analysis, ODE’s, PDE’s)
C126A: Computational Methods for Chemistry
C145: Theoretical and Computational Organic Chemistry
CM160A: Introduction to Bioinformatics
CM160B: Algorithms in Bioinformatics
C176: Group Theory and Applications in Inorganic Chem
M186: Stochastic Processes in Biochemical Systems

25. Why Chemical reaction dynamics, metabolic networks (ODE’s)
Sequence alignment across species (edit distance, sequence matching algorithms)
Genetic sequencing from DNA fragments (Smith-Waterman algorithm, dynamic programming)
3-dimensional protein structure (Fourier analysis, machine learning)
Molecular orbital theory (periodic table, representation theory)
Quantum molecular structure (Fourier analysis, numerical analysis)
Stochastic behavior of organisms (probabilistic models, stochastic processes)
Bioinformatics (Bayesian statistics, Markov chains, hidden Markov models, machine learning)

26. Mechanical and Aerospace Engineering
82: Mathematics of Engineering
103: Elementary Fluid Mechanics
107: Intro to Modeling and Analysis of Dynamical Systems
150A: Intermediate Fluid Mechanics (includes Navier-Stokes)
155: Intermediate Dynamics (Lagrangian mechanics, Euler eqn)
M168: Introduction to Finite Element Methods
169A: Introduction to Mechanical Vibrations
171A: Introduction to Feedback and Control Systems

27. Mechanical and Aerospace Engineering (continued)
C175A: Probability and Stochastic Processes in Dynamical Systems
181A: Complex Analysis and Integral Transforms
182B: Mathematics of Engineering
182C: Numerical Methods for Engineering Applications
184: Introduction to Geometry Modeling (conics, Bezier curves, ..)

28. Why This is rather typical engineering math
A new trend is using machine learning as a short cut to computation

29. Computer Science
112: Modeling Uncertainty in Information Systems (random variables, Bayes’ Thm, Markov chains,..)
CM121: Introduction to Bioinformatics
CM122: Algorithms for Bioinformatics
CM124L Computational Genetics
145: Introduction to Data Mining
M146: Introduction to Machine Learning
161: Fundamentals of Artificial Intelligence
168: Computational Methods for Medical Imaging

30. Computer Science (continued)
170: Mathematical Modeling Methods for CS (numerical and symbolic computation, matrix algebra, statistics, optimization, spectral analysis)
180: Introduction to Algorithms and Complexity
M182: Systems Biomodeling and Simulation Basics
183: Introduction to Cryptography
M184, M185, CM186 as before

31. Why
Search (PageRank algorithm, eigenvectors, stochastic matrices, Perron-Frobenius Theorem)
Imaging (PDE’s, heat equation, wavelets, numerical analysis, compressed sensing, machine learning)
Cryptography (number theory, lattice theory)
Algorithms (graph theory, complexity theory)
Dimensionality reduction (linear algebra, singular value decomposition, Johnson-Lindenstrauss Theorem)

32. UCLA Medical School
Now has a Department of Computational Medicine, chaired by a professor from CS

33. Why
Epidemiology (SIR model, ODE’s, spatial statistics, evolutionary models)
Individualized medicine (clustering, machine learning)
Physiological modeling (ODE’s, PDE’s, agent-based models)
Modeling of infections, modeling of drug delivery, cancer modeling (ODE’s, PDE’s)
Genetic diseases (phylogenetic methods, parsimony, the , linkage disequilibrium, probabilistic methods)
Brain mapping (vector fields, differential geometry, compressed sensing)

34. UCLA School of Law The Dean, Jennifer Mnookin, trained at MIT
Has PULSE (Program on Understanding Law, Science and Evidence)
Two professors have a grant to study policy and governance issues in AI (I am part of a group that periodically has lunch to discuss this)
Has a class on “Disruptive Technologies” (I was a guest lecturer)

35. Why Forensic science (many mathematical techniques)
Algorithms and the law (explainability, transparency)
Regulatory science (agent-based models)
Theory of evidence, Jury selection (probability, machine learning)

36. Digital Humanities Minor
“Places project-based learning at the heart of the curriculum”
“Students use tools and methodologies such as three-dimensional visualization, data mining, network analysis, and digital mapping”
“Students have the opportunity to make significant contributions in fields ranging from archaeology and architecture to history and literature.”

37. Mathematics Tracks at UCLA
Applied Mathematics (32%)
Mathematics/Applied Science (1%)
Mathematics/Economics (19%)
Financial/Actuarial Mathematics (24%)
Mathematics for Teaching (1%)
Mathematics of Computation (12%)
Mathematics: Data Theory (rolling out this Fall)

38. Mathematics Major Learning Outcomes
Strong mathematical content knowledge of single and multivariate differential and integral calculus, and differential equations
Ability to synthesize material, solve problems, and think abstractly
Familiarity with linear algebra, techniques of proof, and foundations of real analysis
Ability to perform basic computer programming, especially in C++

39. Mathematics: Data Theory Learning Outcomes
Understanding of mathematical and statistical bases of most common methods of data science
Ability to explain in writing, with examples, how concepts of statistics and mathematics together solve real-world problems involving data
Skillfully manage data
Development, comparison, and testing of data-driven models to solve problems
Understanding and explanation of variability when fitting and interpreting models of real-world systems
Carrying out of reproducible data analysis using accepted practices of research community
Written and verbal communication of findings of analyses
Identification of areas of active research in data science
Insightfully address problems concerning ethics of data use and storage, including data privacy and security

40. Mathematics: Data Theory Learning Outcomes (continued)
Demonstrated mastery of concepts and skills of machine learning, modeling and supervised learning, dimension reduction and unsupervised learning, and deep learning
Demonstrated familiarity with numerous software tools used in statistical and data science work and research
Demonstrated knowledge of mathematical foundations, including pure and applied linear algebra, basic analysis, probability, and optimization theory
Study and evaluation of proofs of mathematical and statistical results employed in data theory
Work effectively in a team on a data science problem
Demonstrated eligibility for graduate study in applied mathematical science or statistical science

41. What You Should Do Be curious Be proactive
Be willing to go outside your comfort zone
Get plugged into existing efforts

42. What We Should Do as a Profession
This is the topic for the discussion period
”Nothing” is not the correct answer
Time is not our friend

43. THANKS MAY THE FORCE BE WITH YOU!
and

44. Psychology 100A: Psychological Statistics
119S: Neural Basis of Learning and Computing with Neurons (“how neural networks perform”)
142H: Advanced Statistical Methods in Psych (honors)
M144: Measurement and its Applications
186B: Cognitive Science Lab: Neural Networks

45. Computational and Systems Biology
M175: Stochastic Processes in Biochemical Systems
M186: Computational Systems Biology: Modeling and Simulation in Biological Systems

46. Bioengineering
C175: Machine Learning and Data-Driven Modeling (probability, hidden Markov models, dimensionality reduction, clustering)
M182: Systems Biomodeling and Simulation Basics (mostly first order ODE’s)
M184: Intro to Computational and Systems Biology
CM186: Computational Systems Biology: Modeling and Simulation of Biological Systems

47. Microbio., Immunol. and Mol. Gen. Molecular, Cell and Dev. Biol.
103BL Advanced Research Analysis in Virology (use bioinformatics and Math modeling software on datasets)
104BL, 109BL, 150BL, 187BL: Advanced Research Analysis in Virology/Developmental Biology/Plant-Microbe Ecology/Genomic Biology
110L: Integrative Approach to Discovery in Molecular, Cell and developmental Biology (included rigorous quantification and bioinformatics techniques)

48. Neuroscience
M135: Dynamical Systems Modeling of Physiological Processes
C172: Neuroimaging and Brain Mapping

49. Physics 105A-B: Analytic Mechanics
: Mathematical Methods of Physics (matrices, operators, Fourier series and integrals, complex variable, Riemann surfaces)
160: Numerical Analysis Techniques and Particle Simulations
180N: Computational Physics and Astronomy Lab

50. Atmospheric and Oceanic Sc iences
M120: Introduction to Fluid Mechanics
180: Numerical Methods in Atmospheric Science
C182: Data Analysis in Atmospheric and Oceanic Sciences (principal component analysis, time-series, clustering, model validation)

51. Earth, Planetary and Space Science
171: Advanced Computing in Geosciences (hypothesis testing, incomplete data, formal modeling, probabilistic testing of models)

52. Geography 166: Environmental Modeling
169: Introduction to Remote Sensing (introduction to digital image processing)
M171: Introduction to Spatial Statistics
172: Remote Sensing: Digital Image Processing and Analysis

53. Electrical and Computer Engineering
113: Digital Signal Processing (e.g. Fourier transform)
114: Speech and Image Processing Systems Design
131A: Probability and Statistics
132A: Introduction to Communications Systems (includes basic probability, hypothesis testing,..)
133A: Applied Numerical Computing
133B: Simulation, Optimization and Data Analysis
134: Graph theory in Engineering
141: Principles of Feedback Control (e.g. ODE’s)
142: Linear Systems: State Space Approach
C143A: Neural Signal Processing and Machine Learning
M146: Introduction to Machine Learning

54. Linguistics 180: Mathematical Structures in Language
185A-B: Computational Linguistics

55. Public Affairs 60: Using Data to Learn About Society
70: Information, Evidence and Persuasion
115: Using Quantitative Methods to Understand Social problems and their Potential Solutions

56. Society and Genetics
105A: Ways of Knowing in Life and Human Sciences (DNA sequencing, bioinformatics, statistics,..)