Instructor: Tianyu Zhang
Time: Winter 2022, December.
Discussion Time: TBD
Functional Analysis, Measure Theory, Real Analysis, Convex Analysis.
Analogues of many of the structures in real linear algebra have been studied under the heading of “convexity”. For example, the notions of subspace, affine sets(known as affine manifolds), system of linear equations, linear functionals, correspond respectively to the more general notions of convex cone, convex set, system of linear(or convex or concave) inequalities, positively homogeneous convex(or concave) functionals.
Strangely missing from this list is anything corresponding to “linear transformation”. Certain “convex transformations”, to be sure, have arisen naturally out of economic theory.
The concept of “monotone processes”, to which this seminar is devoted, does fill the gap corresponding to “linear transformation”, at least in a small way. It allows us to study convex analogues of all the topics mentioned above.
Generally speaking, monotone processes generalize not “linear transformation” but “positive linear transformation”. To the last part of this paper, in particular, is best viewed as an extension of the Perron-Frobenius theory of positive matrices rather than of ordinary eigenvalue theory. There is, however, a more general possible concept of “convex process”(see this in the last parts of the Convex Analysis) which does not have these limitations of positivity and monoticity.
Reference: R. Tyrrell Rockafellar, Monotone Processes of Convex and Concave Type, Memories of the American Mathematical Society, Number 77.
Lecture 1: Definition of Monotone Processes
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Lecture 2: Adjoints and Kuhn-Tucker Functions
Lecture 3: Inverse and Polars
Lecture 4: Monotone Convex Programs
Lecture 5: Combinatorial Operations
Lecture 6: Sub-eigenvalues and Growth Rates
Lecture 7: Eigensets
Lecture 8: Behavior in the Limit