Fall 2014 MATH 278 002 LEC

Topics in Analysis
Schedule: 
SectionDays/TimeLocationInstructorCCN
002 LECMWF 11-12P 740 EVANSHARRISON, V C54491
Units/CreditFinal Exam GroupEnrollment
4NONELimit:18 Enrolled:5 Waitlist:0 Avail Seats:13 [on 10/09/14]
Additional Information: 

Prerequisites: Consent of instructor.  

Syllabus:
NEW TOPOLOGICAL METHODS IN THE CALCULUS OF VARIATIONS

I. Differential topology from the discrete to the smooth continuum
         A.Dirac chains and infinitesimal calculus 
            1. Tensor, symmetric and antisymmetric algebras
            2. Koszul complex 
            3. Dual pairs
        B. Topological vector spaces     
            1. Topological spaces and continuous functions (basics only)
            2. Banach spaces 
            3. Inductive and projective limits, compact linking maps 
            4. Dual and predual spaces, weak and strong topologies         
        C. Differential forms, currents and chains 
            1. Schwartz and tempered distributions, de Rham currents, Whitney’s sharp and flat chains, differential chains 
            2. Operator algebras   
        D. Basic measure theory 
            1. Hausdorff measure and dimension 
            2. Radon measures 
            3. Whitney isomorphism theorem 
II. Applications 
        A. Calculus of variations 
            1. Models of surfaces
                * Brief overview of historical models: Douglas’s immersed disks, Federer and Fleming’s integral currents (embedded and orientable submanifolds), rectifiable currents, Reifenberg's surfaces, Almgren’s varifolds, Harrison’s dipole surfaces 
                * Spanning sets defined using homology (Reifenberg, Almgren) vs cohomology (Harrison & Pugh), invariance properties
            2. Existence of surfaces spanning a prescribed curve minimizing elliptic functionals 
            3. Higher dimension and codimension     
        B. Further topics might include: lower semicontinuity of Hausdorff measure, Reynolds’ transport theorem, continuous coproduct on differential chains, predual to wedge product on forms, quantum field theory, fluid flows. Some of these additional topics are established results, some are works in progress, and some are dreams!

Office:  829 Evans Hall

Office Hours: MWF 2-3

Required Text: A variety of references will be provided

Recommended Reading: Ask me for my H110 linear algebra text "Universal Linear Algebra for Students of Math and Physics" if you are not familiar with the tensor algebra and its universal property.

Grading: 

Homework: 

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