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How to Reach Me: If you cannot find me in my office, I must be in the Cyberspace! My academic activities (advising, authoring, review & editorial, ...) are Web-based, and so, for whatever purpose, communicating with me via e-mail is always highly recommended. My office e-mail: lih AT math DOT wsu DOT edu. My home e-mail (preferred): haijun DOT li AT gmail DOT com.
What Am I Doing? My current research has something to do with ``stochastic dependence'', which occurs in many natural, engineered and social systems, and is of paramount importance in understanding system structural behaviors. I view dependence patterns as emergent properties of system components' interactions over time, and thus investigation of underlying stochastic processes is often decisive in the study of stochastic dependence.
My View on Applications: The study of stochastic dependence requires domain knowledge, and the application domain of my research is reliability & risk. Historically, reliability theory was originally developed to help nineteenth century maritime insurance and life insurance companies compute profitable rates to charge their customers, and today reliability modeling and analysis are often viewed as analysis of technology and operational risk. But my view is that reliability theory has become a generic tool that is useful in the other fields such as financial risk, actuarial science, survival analysis, catastrophe modeling, environmental impact assessment.... My current application domain is financial risk, due in great part to public availability of financial data, which provide rich sources for statistical analysis of dependence models.
My Two-Cents on Theoretical Research: I like six famous words from David Hilbert: ``We must know, we shall know.'' Now ``standing on the shoulders of (these) giants'', we should not be merely satisfied with only experimental results (physical, computational, or statistical). We should make every effort in quest for fundamental theories. Here are some grand theoretical challenges (from a recent US grant proposal solicitation):
Mathematical Challenge: The Mathematics of the Brain
Develop a mathematical theory to build a functional model of the brain that is mathematically consistent and predictive rather than merely biologically inspired.
Mathematical Challenge: The Dynamics of Networks
Develop the high-dimensional mathematics needed to accurately model and predict behavior in large-scale distributed networks that evolve over time occurring in engineering, biology, and the social sciences.
Mathematical Challenge: Capture and Harness Stochasticity in Nature
Address Mumford’s call for new mathematics for the 21st century. Develop methods that capture persistence in stochastic environments.
Mathematical Challenge: Computational Duality
Duality in mathematics has been a profound tool for theoretical understanding. Can it be extended to develop principled computational techniques where duality and geometry are the basis for novel algorithms?
Mathematical Challenge: Beyond Convex Optimization
Can linear algebra be replaced by algebraic geometry in a systematic way?
Mathematical Challenge: Optimal Nanostructures
Develop new mathematics for constructing optimal globally symmetric structures by following simple local rules via the process of nanoscale self-assembly.
Mathematical Challenge: Creating a Game Theory that Scales
What new scalable mathematics is needed to replace the traditional Partial Differential Equations (PDE) approach to differential games?
Mathematical Challenge: An Information Theory for Virus Evolution
Can Shannon’s theory shed light on this fundamental area of biology?
Mathematical Challenge : What are the Symmetries and Action Principles for Biology?
Extend our understanding of symmetries and action principles in biology along the lines of classical thermodynamics, to include important biological concepts such as robustness, modularity, evolvability, and variability.
Mathematical Challenge: Computation at Scale
How can we develop asymptotics for a world with massively many degrees of freedom?