Rice
FIS

FIS image header
  •  
  •  
  •  
  •  
  •  
Download 
Scholarly Interest Report
         
Robert M. Hardt
Professor
W. L. Moody Professor of Mathematics
 
e-mail:hardt@rice.edu
 
  • Bachelor of Science Mathematics (1967) Massachusetts Institute of Technology, Cambridge, Massachusetts
  • Ph.D. Mathematics (1971) Brown University, Providence, RI
 
Primary Department
   Department of Mathematics
Picture
 
Websites
 Robert Hardt's homepage
 
Research Areas
 geometric measure theory, partial differential equations, calculus of variations, continuum mechanics, real algebraic geometry
 
Summary of Recent Research Proposal
 

This project lies in the area of geometric variational calculus, treating the behavior of singularities and energy concentration for various optimal or stationary functions, fields, or geometric structures subject to geometric or analytical constraints. Specified investigations focus on the relation between energy and topological obstruction in mappings between manifolds, liquid crystal materials, improper slicing of polynomial varieties, transport problems, and the regularity of relaxed energy minimizers. In various higher dimensional cases, energy concentration of limits of smooth mappings may occur along sets of infinite measure and is related to homotopically nontrivial mappings of spaces. This concentration behavior corresponding to any rational homotopy invariant of the target mapping can now be described. Singularities in liquid crystal materials will be also studied in several contexts. The theories of improper intersections of polynomial zero sets from algebraic geometry will be investigated to understand related behavior in analysis and partial differential equations. Combined transport systems such as occurs in postal routes, the circulatory system, and plant root systems can be described variationally with recently discovered geometric structures. Underlying many physical phenomena is a least energy principle whereby certain configurations or fields or geometric shape are distinguished by their property of having less energy or area than competing objects. The external constraints often lead to singularities, which are characterized by rapid changes of structure occurring in very small spatial regions. For example, one observes dislocation faults in solids under stress, domain walls in magnetized materials, liquid edges and corners in soap films, and point, curve, and surface defects in various liquid crystal materials. We deal with new mathematical structures and theories necessary to explain and predict such phenomena. In these problems, the theoretical studies of pure mathematics, the numerical computational studies of applied mathematics, and the phenomenological studies from physics all benefit each other and all have a crucial scientific role.

 

Geometric Calculus of Variations

 

Classically the Calculus of Variations has involved study of one dimensional problems involving finding paths minimizing functionals such as length or action integrals. The treatment of higher dimensional problems giving minimal surfaces, optimal shapes, or energy efficient vectorfields requires the development of suitable analytic and geometric machinery in the fields of Partial Differential Equations and Geometric Measure Theory. The general program involves formulating mathematically the appropriate minimizing model (e.g. in continuum mechanics) and then proving theorems on the existence of minimizing configurations, and then establishing properties such as the regularity, stability, and computability of the minimizing configuration.

 
Geometric Measure Theory and the Calculus of Variations
 

Geometric Measure Theory treats measure-theoretic properties of geometrically defined sets of various dimensions. The geometric calculus of variations treats k dimensional sets which minimize or maximize some functional involving their k dimensional size, such as area-minimizing soap films, length-minimizing transport paths, or critical space-times in general relativity. Such extremal sets may exhibit most places a regular, manifold-like behavior, but occasionally have special singularities, and a large goal is to understand their behavior.

 
Singularity Behavior in Some Geometric Variational Problems
 

Underlying many physical phenomena is a least energy principle whereby certain configurations or fields or geometric shapes are distinguished by their property of having less energy or area than other potential objects. Moreover, competing energies or constraints often occur, as in modelling plant shape, with numerous factors such as strength, economy of material,and fluid transport efficiency. The external constraints and minimizing properties also often lead to singularities, which are characterized by rapid changes of structure occurring in very small spatial regions. For example, one observes dislocation faults in solids under stress, domain walls in magnetized materials, liquid edges and corners in soap films, and microstructure complexity in crystalline materials. We deal with new mathematical structures and theories necessary to explain and predict such phenomena. With these problems, mathematical theory, numerical computational, and phenomenological studies all benefit each other and all have a crucial scientific role. This project lies in the area of geometric variational calculus, treating the behavior of singularities and energy concentration for various optimal or stationary functions, fields, measures, or geometric structures, possibly subject to constraints. Specified investigations focus on the relation between energy and topological obstruction in mappings to manifolds, the higher dimensional calculus of variations in metric spaces, higher order mapping problems, microstructure computation, combined transport-shape problems, and the regularity of relaxed energy minimizers. In various higher dimensional cases, energy concentration of limits of smooth mappings to a manifold may occur along sets of infinite measure producing new geometric objects and is related to the nonvanishing homotopy of the manifold. This concentration behavior corresponding to any nontorsion homotopy invariant can now be described, and bubbling related to torsion invariants is being investigated. The theories of improper intersections of polynomial zero sets from algebraic geometry will be investigated to understand related behavior in analysis. Models for plant shape have provided completely new variational problems involving cross-sectional area constraint and shape requirements for efficient fluid transport.

 
Teaching Areas
 calculus, analysis, geometry, topology, partial differential equations
 
Selected Publications
 Refereed articles
 

Onur Alper, Robert Hardt, Fanghua Lin "Defects of liquid crystals with variable degree of orientation." Calculus of Variations and Partial Differential Equations: 32 pages.Submitted

 
 

Quentin Funk and Robert Hardt "A multi-valued Platea Problem." : 18 pages.In Revision

 
 

Pedro F. dos Santos, Robert M. Hardt, James D. Lewis, and Paulo Lima-Filho "An Explicit Cycle Map for Motivic Cohomology of Real Varieties." In Progress

 
 

Quentin Funk and Robert Hardt "A Multi-valued Plateau Problem." In Progress

 
 

R. Hardt & T. Riviere "Sequential Weak Approximation for Maps of Finite Hessian Energy." Calculus of Variations and PDE, 54(no.3) (2015) : 2713-2749.

 
 

T. De Pauw and R. Hardt "Some Basic Theorems on Flat G Chains." Journal of Mathematical Analysis and Applications, 418 No.2 (2014) : 1047–1061.

 
 

R. Hardt & T. Riviere "Sequential Weak Approximation for Maps of Finite Hessian Energy." Submitted

 
 

T. DePauw & R. Hardt "Some Basic Theorems on Flat G Chains." Submitted

 
 

T. DePauw, R. Hardt, W.Pfeffer "Homology of Normal Chains and Cohomology of Charges." Submitted

 
 

Journal of Algebraic and Geometric Topology 11:5 (2011), 2477-2546.

 
 

Robert Hardt, Rolf Ryham "Some Explicit Coarea Flows." Submitted

 
 

American Journal of Mathematics, vol. 134 no.1 (2012), pp.1-69

 
 

D. Cox, R. Hardt, P. Kloucek "Convergence of Gibbs measures associated with simulated annealing." SIAM J. Math. Anal., 39, no.5 (2008) : 1472-1496.

 
 

R. Hardt and T. Riviere "Connecting rational homotopy type singularities." Acta Mathematica, 200, no.1 (2008) : 15-83.

 
 

Qing Han, Robert Hardt, and Fanghua Lin "Singular Sets of Higher Order Elliptic Equations." Communications in Partial Differential Equations, 28 (2003) : 2045-2063.

 
 

R. Hardt and T. Riviere "Connecting Topological Hopf Singularities." Annali Scuola Normale di Pisa, 2 (2003) : 287-344.

 
 

Thierry De Pauw and Robert Hardt "Size Minimization and Approximating Problems." Calculus of Variations and Partial Diferential Equations, 17 (2003) : 405-442.

 
 

Hardt, R., Thierry de Pauw "Size Minimization and Approximated Problems." Calculus of Variations and PDE (2002) : 37 pages.In Press

 
 

Hardt, R., Tristan Riviere "Connecting Topological Hopf Singularities." Annali Mathematica di Pisa (2003) Submitted

 
 Articles
 

Robert Hardt, Pascal Lambrechts, Victor Tourtchine and Ismar Volic "Real Homotopy Theory of Semi-Algebraic Sets." Submitted

 
 

D. Cox, R. Hardt, and P. Kloucek "Convergence of Gibbs Measures Associated with Simulated Annealing." SIAM Journal of Mathematical Analysis (2008) : 27 pages.In Press

 
 

R. Hardt and T. Riviere "Connecting Singularities of Arbitrary Rational Homotopy Type." : 55 pages.Submitted

 
 

Hardt, R., Q. Han and F.H. Lin "Singular sets of higher order equations."  (2002) : 22 pages.Submitted

 
 

Hardt, R., Tristan Riviere "Ensembles singuliers topologiques dans les espaces fonctionnels entre varietes.." Seminar des Equations Derivee Partielles (2001) : 23.

 
 Books
 

Thierry De Pauw, Robert Hardt, Washek Pfeffer "Homology of Normal Chains and Cohomology of Charges." Memoirs of the American Mathematics Society, 247(1172) (2017/January 17) : 128 pages.

 
 

Robert Hardt, Editor "Some Themes on Variations, A Rice Undergraduate Conference." 

 
 Editor of books
 

Hardt, R. "Six Themes on Variations, A Rice Undergraduate Conference."  (2003) : 200.In Revision

 
 Book reviews
 

Robert Hardt "Review of "Constant Mean Curvature Surfaces, Harmonic Maps , and Integrable Systems" by F. Helein." Bulletin of the American Mathematics Society, 40 (2002) : 121-123.

 
 Refereed conference papers
 

Pedro F. dos Santos, Robert M. Hardt, and Paulo Lima-Filho, "Relating Equivariant and Motivic Cohomology via Analytic Currents." Contemporary Mathematics, American Mathematics Society, 646 (2015) : 19-40.

 
 

R. Hardt "Plateau Problems in Metric Spaces and Related Homology and Cohomology Theories." Contemporary Mathematics, American Mathematics Society, 646 (2015) : 1-17.

 
 

T. DePauw and R.Hardt "Applications of Scans and Fractional Power Integrals." Progress in Nonlinear Differential Equations and Their Applications, 59 (2004) : 19-32.

 
 Other
 

Thierry de Pauw and Robert M. Hardt "Size Minimization and Approximated Problems." Journal fur die Reine und Angewandte Matematika (Submitted)

 
 

R. Hardt and D. Kinderlehrer "A regularity result in ferromagnetism, ." Communications in Partial Differential Equations , 25 (2000) : 1235-1258.

 
 

R. Hardt, M. & T. Hoffmann-Ostenhof, N.Nadirashvilli "Critical sets of smooth solutions to elliptic equations ." Journal of Differential Geometry, 51 (1999) : 359-373.

 
 

R. Hardt and D. Kinderlehrer "A regularity result in ferromagnetism,"." Communications in Partial Differential Equations (2000(accepted))

 
 

R. Hardt and M. Wolf "Harmonic extensions of quasi-conformal maps to hyperbolic space." Pitman Research Notes in Math., CRC Press, New York (1999)

 
 

R. Hardt, M. & T. Hoffmann-Ostenhof, N.Nadirashvilli "Critical sets of smooth solutions to elliptic equations." Journal of Differential Geometry, (2000 (accepted))

 
 

R. Hardt, Q. Han, and F.H. Lin "Singular sets of higher order equations."  (submitted)

 
 

R. Hardt and D. Kinderlehrer "A regularity result in ferromagnetism." communications in Partial Differential Equations (submitted)

 
 

R. Hardt, F.H. Lin and C.Y. Wang "The p-energy minimality of x/|x|." Communications in Analysis and Geometry, 6 (1998) : 141-152.

 
 

R. Hardt, M. & T. Hoffmann-Ostenhof, N.Nadirashvilli "Critical sets of smooth solutions to elliptic equations." Journal of Differential Geometry (accepted)

 
 

R. Hardt, Q. Han, and F.H. Lin "Geometric measure of singular sets of elliptic equations." Communications in Pure and Applied Mathematics, 51 (1998) : 1425-1443.

 
Presentations
 Conference Committee Member
 

Organizer.  "Singularities in Analysis, Geometry, and Topology: A Conference Honoring the Retirements of Reese Harvey and John Polking." Rice University, Houston, TX. (11/11/2005-11/13/2005)

 
 Invited Talks
 

Speaker.  "

A Linear Isoperimetric Inequality for Algebraic and Analytic Varieties

." Texas Geometry and Topology Conference, Texas A&M University, College Station, TX. (November 5, 2016)

 
 

Speaker.  "A Linear Isoperimetric Inequality for Algebraic and Analytic Varieties." Workshop on Nonlinear Partial Differential Equations and Applications, South China normal University, Xiamen, PR China. (June 16, 2016)

 
 

Speaker.  "A Multi-Valued Plateau Problem." Sun Yat Sen University, , Guangzhou University, PRChina. (June 21, 2016)

 
 

Speaker.  "Line Defects in Nematic Liquid Crystals." South China Normal University, Guangzhou, PR China. (June 22, 2016)

 
 

Speaker.  "Line Defects in Nematic Liquid Crystals with a Modified Ericksen Model." South China University of Science and Technology, Guangzhou, PRChina. (June 22, 2016)

 
 

speaker.  "Line Defects in a Modified Ericksen Model." International Workshop on Nonlinear PDE, NYU Shanghai, Shanghai, P.R.China. (June 14, 2016)

 
 

Speaker.  "Line Defects in a Modified Ericksen Model of Nematic Liquid Crystals.." Mathematics and Mechanics in the 22nd Century: Seven Decades and Counting.../IMA, Eugene Oregon. (October 24, 2015)

 
 

Speaker.  "Line Defects in a Modified Ericksen Model.." Mathematical Analysis and Computation of Nematic Liquid Crystals/AMS Meeting, Loyola University, Chicago. (October 3, 2015)

 
 

Speaker.  "Rectifiable and Flat Homology and cohomology Theories." Geometric Measure Theory and the Calculus of Variations/Institute Fourier, University of Grenoble, France. (July 1, 2015)

 
 

Speaker.  "Singularities With and Without Topology.." Institute for Computational and Experimental Research in Mathematics, Brown University, Providence. (March 17, 2015)

 
 

Speaker.  "Spaces of Flat and Normal Chains and Cochains.." Nonlinear Function Spaces in Mathematics and Physical Sciences/, University of Lyon, France. (December 15,16,17, 2015)

 
 

Speaker.  "

One Dimensional Rectifiable Varifolds and Some Applications, 

." Mathematics Department, Tianjin University, Tianjin, P. R.China. (June 11, 2014)

 
 

Speaker.  "

One Dimensional Rectifiable Varifolds and Some Applications

." Mathematics Department, NYU Shanghai, , East China Normal University, Shanghai, P.R.China. (June 16, 2014)

 
 

Speaker.  "

One Dimensional Rectifiable Varifolds and Some Applications

." Workshop on Uniqueness in Analysis, and Geometry, Massachusetts Institute of Technology. (December 8, 2014)

 
 

Speaker.  "

Plateau Problems in Singular Spaces

." Mathematics Department, Beijing Normal University, Beijing, P.R. China. (June 6, 2014)

 
 

Speaker.  "Homology of Normal Chains and Cohomology of Charges in a Metric Space, ." Chern Mathematics Institute, Nankai University, Tianjin, P. R.China. (June 10, 2014)

 
 

Speaker.  "Plateau Problems in Singular Metric Spaces, ." Institute Of Natural Sciences, Shanghai Jiao-Tong University, Shanghai, P.R.China. (June 18, 2014)

 
 

Speaker.  "

Plateau Problems in Metric Spaces

." 2013 Midwest Geometry Conference, Oklahoma State University, Stillwater, OK. (10/19/2013)

 
 

Speaker.  "Some Homology and Cohomolgy Theories for a Metric Space." Ohio River Analysis Meeting, University of Cincinnati. (March 9, 2013) With Thierry DePauw, Washek Pfeffer

 
 

Speaker.  " 

Some Homology and Cohomology Theories for a Metric Space." Yamabe Conference, University of Minnesota, Minneapolis. (October 5, 2012)

 
 

Speaker.  "Flat Chains and Cochains a Metric Space." Calculus of Variations Conference, Max Planck Institute, Oberwolfoch, Germany. (July 24, 2012)

 
 

Speaker.  "Plateau Problems in Metric Spaces." Mathematics Colloquium, University of Minnesota, Minneapolis. (October 4, 2012)

 
 

Speaker.  "The Plateau Problem in Metric Spaces." Geometric Measure Theory Conference, Albert Einstein Institute for Gravitation, Potsdam, Germany. (July 3, 2012)

 
 

Speaker.  "The Plateau Problem in Metric Spaces." Differential Geometry Seminar, University of Paris VII. (May 22, 2012)

 
 

Speaker.  "Rectifiable and Flat Chains in a Metric Space." Eidgenössische Technische Hochschule Zürich, Zurich, Switzerland. (March 16, 2011)

 
 

Speaker.  "Rectifiable and Flat Chains in a Metric Space." Mathematics Department, University of Verona, Verona< Italy. (March 2, 2011)

 
 

Speaker.  "Rectifiable and Flat Chains in a Metric Space." Mathematics Department, University of Padua, Padua, Italy. (March 3, 2011)

 
 

Speaker.  "Slicing and Calibrations." Herbert Federer Memorial Conference, Brown University, Providence, RI. (April 16, 2011)

 
 

Speaker.  "   Continuous Harmonic Maps and Relaxed Energy." Partial Differential Equations Conference, National Academy of Sciences, Beijing. (July 15, 2010)

 
 

speaker.  "A Mass Decreasing Flow for Rectifiable Chains in some Metric Spaces." Differential and topological problems in modern theoretical physics, La Scuola Internazionale Superiore di Studi Avanzati di Trieste, Italy. (April 28, 2010)

 
 

Speaker.  "Some Remarks on Total Variation Flow.." Washington State University Mathematics Department, Washington State University. (March 4, 2010) With Rolf Ryham

 
 

Speaker.  "A Mass Decreasing Flow in a Metric Space." Beijing Capitol Normal University, Beijing, P.R.C.. (July 19, 2010)

 
 

Speaker.  "Some Explicit Coarea Flows." Wu Xinmou Birthday Conference, National Academy of Sciences, Beijing. (July 16, 2010) With Rolf Ryham

 
 

Speaker.  "Functions of Bounded Variation." Shoemacher Lecture 2, University of Toledo. (March 3, 2009)

 
 

Speaker.  "Geometric Measure Theory in a Metric Space." Mathematics Department, Notre Dame University, South Bend, IN. (March 5) With Thierry De Pauw

 
 

Speaker.  "Modeling Soap Films and Soap Bubbles." Schoemacher Lecture 1, University of Toledo. (March 2)

 
 

Speaker.  "Rectifiable and Flat G Chains in a Metric Space." Geometry-Analysis Seminar, Rice University. (September 30) With Thierry De Pauw

 
 

Speaker.  "Rectifiable and Flat G Chains in a Metric Space." International Conference on Liquid Crystals and Superconductivity, Guanzhou University, Guanzhou, PR China. (June 29) With Thierry De Pauw

 
 

Speaker.  "Rectifiable and Flat G-Chains in a Metric Space.." Shoemacher Lecture 3, University of Toledo. (March 4, 2009) With Thierry De Pauw

 
 

Speaker.  "Regularity of Harmonic Maps I." East China Normal University, Shanghai, PR China. (June 22)

 
 

Speaker.  "Regularity of Harmonic Maps II." East China Normal University, Shanghai, PR China. (June 23)

 
 

Speaker.  "Regularity of Harmonic Maps III." East China Normal University, Shanghai, PR China. (June 24)

 
 

Speaker.  "Some Remarks on Imaging and Total Variation Flow.." Nanjing Mathematics Institute, Nanjing, PR China. (June 25) With Rolf Ryham

 
 

Speaker.  "Some Remarks on Total Variation Flow.." AMS Special Session on Partial Differential Equations, Wprchester Polytechnic Institute, Worchester, MA. (April 26)

 
 

Speaker.  "Rectifiable and Flat G-Chains and Scans in a Metric Space." Quasiconformal Mappings and Analysis on Metric Spaces/U.Mich., U. Michigan, Ann Arbor. (May, 2008) With T. De Pauw

 
 

"Some Mathematical Properties of Total Variation Flow." Institute of Pure and Applied Mathematics, UCLA conference center, Lake Arrowhead, CA. (December 10, 2008)

 
 

"Convergence to Gibbs Measures Absolutely Continuous with Respect to Hausdorff Measures on Manifolds." Institute for Pure and Applied Mathematics, UCLA. (May 4, 2007)

 
 

"Four Lectures on Integral Currents." East China Normal University, Shanghai. (May 21-24,2007)

 
 

"Four Lectures on Some Basic Results on Geometric Measure Theory in Metric Spaces." University Of Washington, Seattle. (August 13-16,2007)

 
 

"Slices, Rectifiability, and Scans." Institute for Pure and Applied Mathematics, UCLA. (May 2, 2007)

 
 

"Some Remarks on W^{2,2} Maps." Int'l Conference on Mathematical Theory of Superconductivity and Liquid Crystals, East China Normal University, Shanghai. (May 16, 2007)

 
 

"Some Remarks on W^{2,2} Maps." Geometrie et Equations Differetial Partielles, Universite de Marseilles. (June 28, 2007)

 
 

"Some Remarks on W^{2,2} Maps." Conference on Geometric Analysis, Korean Institute of Advanced Science, Seoul. (September 8, 2007)

 
 

Invited speaker.  "Bubbling Limits for Maps Between Manifolds." Colloquium/ University of Kentucky, Lexington, KY. (2/17/06)

 
 

Invited speaker.  "Bubbling Limits for Maps Between Manifolds." Princeton University, Princeton, New Jersey. (4/7/06)

 
 

Invited speaker.  "Bubbling Limits of Smooth Maps Corresponding to Rational Homotopy Classes." Universite Catholique de Louvain, Brussels, Belgium. (3/3/06)

 
 

Invited Speaker.  "Nodal and Singular Sets of a Solution of an Elliptic Equation." Universite Catholique de Louvain, Brussels, Belgium. (3/17/06)

 
 

Invited speaker.  "Rectifiable Chains in a Metric Space with Coefficients in a Group." Stanford University. (5/17/06)

 
 

Invited speaker.  "Rectifiable Chains in a Metric Space with Coefficients in a Group." University of Oklahoma, Norman, OK. (5/7/06)

 
 

Invited speaker.  "Some Variational Problems Involving W^{2,2} Mappings." University of California at Davis, Davis, CA. (November 21, 2006)

 
 

Invited speaker.  "Some Variational Problems Involving W^{2,2} Mappings." Georgia Institute of Technology, Atlanta, GA. (December 7, 2006)

 
 

Speaker.  "Some Variational Problems Involving W^{2,2} Mappings." Center for Nonlinear Analysis, Carnegie-Mellon U., Pittsburgh, PA. (Oct. 21, 2006)

 
 

Invited speaker.  "The Plateau Problem in Metric Spaces." Pennsylvania State University, State College, PA. (October 23, 2006)

 
 

Speaker.  "Bubbling Limits and Continuous Approximation Criteria for Maps between Manifolds." Notre Dame University, South Bend, IN. (4/7/2005)

 
 

Speaker.  "Bubbling Limits and Continuous Approximation Criteria for Maps between Manifolds." University of Illinois at Chicago, Chicago, IL. (4/8/2005)

 
 

Speaker.  "Convergence to Gibbs Measures Absolutely Continuous with Respect to Hausdorff Measures on Manifolds." PDE Conference, East China Normal University, Shanghai, China. (6/15/2005)

 
 

Speaker.  "Convergence to Gibbs Measures Absolutely Continuous with Respect to Hausdorff Measures on Manifolds." PDE Meeting, University of Florida, Gainesville, FL. (11/13/2005)

 
 

Speaker.  "Energy,Topology,and Bubbling Phenomena for Maps Between Manifolds ." Conference on Geometry and Analysis, Zhejiang Univ., Hangzhou. China. (6/24/2005)

 
 

Speaker.  "Some Mass Decreasing Flows Involving Fractional Powers of Densities." SIAM PDE Meeting, Houston, TX. (12/6/2004)

 
 

Speaker.  "Bubbling Limits and Continuous Approximation Criteria for Maps between Manifolds." University of Minnesota, Minneapolis, MN. (11/8/2004)

 
 

Speaker.  "Bubbling Limits and Continuous Approximation Criteria for Maps between Manifolds." University of Southern California, Los Angeles, CA. (9/16/2004)

 
 

Speaker.  "Isoperimetric Problems for Polyhedra." Pennsylvania State University, State College, PA. (4/23/2004)

 
 

Speaker.  "Isoperimetric Problems in Polyhedra." Harvey Mudd College, Claremont, CA. (9/17/2004)

 
 

Speaker.  "Some Mass Decreasing Flow Problems Involving Fractional Mass Densities." Society of .Industrial & Applied Math. PDE Meeting, Houston, TX. (12/8/2004)

 
 

Speaker.  "Surfaces Minimizing Integrals of Fractional Power Densities." Carnegie-Mellon University, Pittsburgh, PA. (4/26/2004)

 
 

Speaker.  "Surfaces Minimizing Integrals of Fractional Power Densities." Pennsylvania State University, State College, PA. (4/22/2004)

 
 

"A Sobolev Mapping Application of Scans." Institut de Mathematiques de Luminy, Marseilles. (September 18, 2003)

 
 

"A Variational Introduction to Scans." Institut de Mathematiques de Luminy, Marseilles. (September 17, 2003)

 
 

"Geometric Variational Problems with Densities." University College, London. (March 14, 2003)

 
 

"Modelling Singularities of Nematic Liquid Crystals." Harvey Mudd College, Claremont, California. (December 1, 2003)

 
 

"Plateau-type Variational Problems Using Scans." University Catholique de Louvaine, Brussels. (September 11, 2003)

 
 

"Plateau-type Variational Problems and Scans." University of Washington, Seattle. (July 7, 2003)

 
 

"Rectifiability of Energy Concentrations for Sobolev Maps." University College, London. (March 11, 2003)

 
 

"Scans for Mappings and Variational Problems." Purdue University, W. Lafayette, Indiana. (May 22, 2003)

 
 

"Scans in Metric Spaces Defined Over Various Groups." Institut de Mathematiques de Luminy, Marseilles. (September 19, 2003)

 
 

"Variational Problems ithe Densities." Banff Institute Research Station, Banff. (July 27, 2003)

 
 

"Applications of Scans." Three Cities Seminar, ETH, Zurich. (June, 2002)

 
 

"Applications of Scans and Fractional Power Integrands." Amer.Math.Soc.-Italian Joint Meeting, University of Pisa, University of Pisa. (June, 2002)

 
 

"Approximation of Sobolev Maps by Smooth Maps." David Gilbarg Memorial Conference, Stanford University. (April 2002)

 
 

"Fractional Power Integrands." Southeast Regional Geometry Seminar, Birmingham. (April 2002)

 
 

"Geometry Seminar." Stanford University, California. (March 2001)

 
 

"Size Minimization and Approximating Problems." Harmonic Maps and Morphisms, University of Brest, Brest, France. (July, 2002) With Thierry De Pauw

 
 

"Size minimizers." Midwest PDE Conference, University of Kentucky. (March 2002)

 
 

"Sobolev Maps and the Topology of Manifolds." Calculus of Variations, Oberwolfoch, Oberwolfoch, Germany. (June, 2002)

 
 

"Some Variational Problems Involving Fractional Power Integrands." Mathematics Department, Emory University, Atlanta, GA. (November 2002)

 
 Lectures
 

Speaker.  "

Some new uses of functions with finite total variation.

." Harold J. Gay Lecture, Worchester Polytechnic Institute, Worchester, MA. (Dec. 9, 2011)

 
 

Speaker.  "Flat Chains." Workshop on Geometric Measure Theory, Mathematical Sciences Research Institute. (July 18-21, 2011)

 
 

Lecturer.  "Rectifiable and Flat G Chains in a Metric Space." ERC Workshop on Geometric Measure Theory in Metric Spaces, Scuola Normale Superiore, Pisa, Italy. (Jan. 10-14, 2012)

 
 

"The Nature of Fractals and Fractals in Nature." Joseph Priestley Lecture, !st Unitarian Universalist Church, Houston. (December 19, 2007)

 
 Other
 

Conference co-organizer.  "Midwest Geometry Conference." University of Oklahoma, Norman, OK. (5/5/06-5/7/06)

 
 

Co-organizer.  "Undergraduate Conference on Algebraic Geometry." Mathematics Department, Rice University, Houston, TX. (2/10/06-2/12/06)

 
 

Mini-Conference Organizer.  "Some Variational Problems in Biology." SIAM PDE Meeting, Houston, TX. (12/7/2004-12/8/2005)

 
 

"Bubbling, Bubbles, and Scans." Texas Geometry and Topology Conference, Texas A&M. (November 2001)

 
 

"Connecting Topological Hopf Singularities." Geometry Seminar, Stanford University. (March, 2001)

 
 

"Connecting Topological Hoph Singularities." Hildebrandt Birthday Conference, Max Planck Institute, Leipzig, Germany. (June 2001)

 
 

"Isoperimetric Problems for Polyhedra." AMS Special Session on PDE, University of Tennessee. (October 2001)

 
 

"Limits of Finite Energy Maps." Harmonic Maps Conference, University of Rome, Italy. (October 2001)

 
 

"Limits of Maps from B4 to S2." Harmonic Maps and Morphisms, University of Marseille, France. (May 2001)

 
 

"Regularity and Singularity of Harmonic Maps." Harmonic Maps Conference, University of Rome, Italy. (October 2001)

 
 

"Size Minimization and Approximated Problems." Calculus of Variations, Kansas State University. (October 2001)

 
 

"Smooth Harmonic Maps and Their Limits." harmonic Maps Conference, University of Rome, Italy. (October 2001)

 
 

"Some Infinite Mass Limits." Ginzburg-Landau and Harmonic Maps, Cambridge University, England. (July 2001)

 
 

"Some Variational Problems with Infinite Mass Limits." AMS Special Session on Surface Geometry and Shape Perception, Stevens Institute of Technology. (April 2001)

 
 

"Limits of sequences of smooth maps to the sphere ." Colloquium, University of Houston. (November, 1999)

 
 

"Smooth maps to the sphere." seminar, New York University. (February, 2000)

 
 

"A 10 hour course in geometric measure theory." Advanced Mathematics Summer School, Institute of Theoretical and Computational Physics, Beijing. (July, 1999)

 
 

"Limits of sequences of smooth maps to the sphere." PDE Seminar, Fudan University, Shanghai. (July, 1999)

 
 

"Limits of sequences of smooth maps to the sphere." Mt. Baldy Conference on Analysis, Harvey Mudd College. (November, 1999)

 
 

"Limits of sequences of smooth maps to the sphere." Pacific Northwest Geometry Seminar, Oregon State University. (November, 1999)

 
 

"Limits of sequences of smooth maps to the sphere." American Mathematics Society Special Session, University of Texas. (October, 1999)

 
 

"Limits of sequences of smooth maps to the sphere." Analysis Seminar, Academy Sinica, Beijing. (July, 1999)

 
 

"New harmonic map constructions." PDE Seminar, Ecole Normale Supieure Cachan, Paris. (May, 1999)

 
 

"Nodal and Critical Sets of a Solution of an Elliptic Equation." PDE Seminar, Max Planck Institute, Leipzig. (June, 1999)

 
 

"Nodal and Critical Sets of a Solution of an Elliptic Equation." South China Conference on Applied Mathematics, Tongi University, Shanghai. (July, 1999)

 
 

"Nodal and Critical Sets of a Solution of an Elliptic Equation." Colloquium, University of Washington. (March, 1999)

 
 

"Some New Harmonic Maps from the 3-ball to the 2-sphere." Analysis Seminar, University of Washington. (March, 1999)

 
 

"Some new harmonic maps from the 3-ball to the 2-sphere." Analysis Seminar, Beijing University. (July, 1999)

 
 

"Some new harmonic maps from the 3-ball to the 2-sphere." American Mathematics Society Session on PDE, University of Florida. (March, 1999)

 
 

"Energy minimizing maps and area minimizing currents." Calculus of Variations, Oberwolfoch, Germany. (July, 1998)

 
 

"Minimizing biharmonic maps." Geometry Seminar, Ecole Norm. Sup. Cachan, Paris. (July, 1998)

 
 

"Models of Liquid Crystals." Applied Mathematics Seminar, Pennsylvania State University. (October (1998))

 
 

"Nodal and Critical Sets of a Solution of an Elliptic Equation." Mathematics Colloquium, Pennsylvania State University. (October (1998))

 
 

"The minimality of x/|x|." Conference on Geometric Measure Theory, University of Freiburg, Germany. (July, 1998)

 
 Seminar Speaker
 

Speaker.  "Rectifiable and Flat G-Chains in a Metric Space." Universite Catholique de Louvain, Brussels, Belgium. (May 10, 2010) With Thierry De Pauw

 
 

speaker.  "G-chains in a Metric Space." Eidgenossische Technische Hochschule, Zurich. (February, 2008)

 
 

speaker.  "G-chains in a Metric Space." Universite Catholique de Leuven. (Jan, 2008) With T. DePauw

 
 Workshops
 

Speker.  "Some Homology and Cohomology Theories for a Metric Space." Lawson Birthday Conference, Simons Center, State University of New York at Stony Brook. (October 27, 2012)

 
 

Organizer of Special Session.  "Geometric Variational Problems." Joint Amer. Math. Soc. Mexican Math. Soc. Meeting, Houston, TX. (5/14-15/2004)

 
 

Organizer of Mini-symposium.  "Some Problems in Mathematical Biology." Society of .Industrial & Applied Math. PDE Meeting, Houston, TX. (12/7/2004)

 
Editorial Positions
 Associate Editor, Advances in the Calculus of Variations. Walter de Gruter, Berlin and NY. (2015 - 2015)

 Associate Editor, University of Houston Mathematics Journal. University of Houston. (2015 - 2015)

 Associate Editor, University of Houston Mathematics Journal. University of Houston. (2009 - 2009)

 Associate Editor, Journal of Geometric Analysis. (2009 - 2009)

 Associate Editor, Advances in the Calculus of Variations. Walter de Gruter, Berlin and NY. (2009 - 2009)

 Associate Editor, Advances in the Calculus of Variations. Walter de Gruter, Berlin and NY. (2010 - 2010)

 Associate Editor, University of Houston Mathematics Journal. University of Houston. (2010 - 2010)

 Associate Editor, Advances in the Calculus of Variations. Walter de Gruter, Berlin and NY. (2011 - 2011)

 Associate Editor, Advances in the Calculus of Variations. Walter de Gruter, Berlin and NY. (2012 - 2012)

 Associate Editor, University of Houston Mathematics Journal. University of Houston. (2012 - 2012)

 Associate Editor, University of Houston Mathematics Journal. University of Houston. (2011 - 2011)

 Associate Editor, University of Houston Mathematics Journal. University of Houston. (2013 - 2013)

 Associate Editor, Advances in the Calculus of Variations. Walter de Gruter, Berlin and NY. (2013 - 2013)

 Associate Editor, University of Houston Mathematics Journal. University of Houston. (2014 - 2014)

 Associate Editor, Advances in the Calculus of Variations. Walter de Gruter, Berlin and NY. (2014 - 2014)

 Associate Editor, Advances in Calculus of Variations. Walter de Gruter, Berlin and NY. (2016 - 2016)

 Associate Editor, University of Houston Mathematics Journal. University of Houston. (2016 - 2016)

 Associate Editor, Houston Journal of Mathematics. University of Houston. (1999 - 1999)

 Associate Editor, University of Houston Mathematics Journal. University of Houston. (2008 - 2008)

 Associate Editor, Houston Journal of Mathematics . University of Houston. (2000 - 2000)

 Associate Editor, University of Houston Mathematics Journal. University of Houston. (2006 - 2006)

 Associate Editor, University of Houston Mathematics Journal. University of Houston. (2007 - 2007)

 Associate Editor, Houston Journal of Mathematics. University of Houston. (2001 - 2001)

 Associate Editor, University of Houston Mathematics Journal. University of Houston. (2005 - 2005)

 Associate Editor, Journal of Geometric Analysis. (2008 - 2008)

 Associate Editor, Journal of Geometric Analysis. (2006 - 2006)

 Associate Editor, Journal of Geometric Analysis. (2007 - 2007)

 Associate Editor, Journal of Geometric Analysis . Mathematica Josephina. (2000 - 2000)

 Associate Editor, Journal of Geometric Analysis. Mathematica Josephina. (1999 - 1999)

 Associate Editor, Journal of Geometric Analysis. Mathematica Josephina. (2001 - 2001)

 Associate Editor, Advances in Calculus of Variations. Walter de Gruter, Berlin and NY. (2007 - 2007)

 Associate Editor, Advances in the Calculus of Variations. Walter de Gruter, Berlin and NY. (2008 - 2008)

Supervised Theses & Dissertations
 Scott Berger, Ph.D. Edge Length Minimizing Polyhedra. (2002) (Thesis Director)

 Simon Morgan, Ph.D. Variational Problems in Singular Spaces. (2002) (Thesis Director)

 Zheng Meng, Ph.D. Plant shape and a Cross-Sectional Area Constraint. (2005) (Thesis Director)

 James Peterson, Ph.D. Coordinate Scans, Compactness Properties, and Area Minimization.. (2006) (Thesis Director)

 Yue Wu, Ph.D. Applications of Rauzy Induction on the Generic Ergodic Theory of Exchange Transformations.. (2006) (Thesis Committee Member)

 Soomin Kim, Ph.D. Limits of Minimal Surfaces with Increasing Genus. (2007) (Thesis Committee Member)

 Amanda Knecht, Ph.D. Weak Approximation for Degree Two del Pezzo Surfaces at Places of Bad Reduction. (2007) (Thesis Committee Member)

 Wei Zhu, Ph.D. Minimizing and flow problems for multiple-valued functions and maps.. (2007) (Thesis Director)

 Eric Samansky, Ph.D. Convergence of Gibbs Measures and the Behavior of Shrinking Tubular Neighborhoods of Fractals and Algebraic Sets.. (2007) (Thesis Director)

 Jer-Chin (Luke) Chuang, Ph.D. Transgressive Chains, Harmonic Cycles, and Subdivisions. (2007) (Thesis Committee Member)

 Matt McClelland, Ph.D. Deformation of Symmetric Scherk Type Minimal Surfaces. (2008) (Thesis Committee Member)

 Karoline Pershell, Ph.D. Some conditions for recognizing a 3-manifold group. (2009) (Thesis Committee Member)

 Ryan Dunning, Ph.D. Asymptotics Under Self-Intersection for Minimizers of Self-Avoiding Energies. (2009) (Thesis Director)

 Heather Hardway, Ph.D. Pattern Formation in Systems of Reaction Diffusion Equations Modeling Genetic Networks. (2009) (Thesis Committee Member)

 Casey Douglas, Ph.D. Perturbed Genus One Scherk Surfaces and their Limits. (2009) (Thesis Committee Member)

 Zheng Gan, Ph.D The spectral properties of limit periodic Schrodinger operators. (2011) (Thesis Committee Member)

 Thomas McGaffey, PhD Regularity and Nearness Theorems for Local Lie Groups. (2011) (Thesis Director)

 Ryan Scott, Ph.D Minimizing the mass of the codimension two skeleton for unit volume convex polyhedra. (2011) (Thesis Director)

 Colin Carroll, PhD Minimizers of a Vector-valued Coarea Functional. (2012) (Thesis Director)

 Evelyn Lamb, PhD The Hopf differential and harmonic maps between branched differential structures. (2012) (Thesis Committee Member)

 Colin Caroll, PhD Minimizers of a Vector-valued Coarea Functional. (2012) (Thesis Director)

 Mauricio Rivas, Ph.D UNCONSTRAINED VARIATIONAL PRINCIPLES AND MORSE INDICES FOR LINEAR ELLIPTIC EIGENPROBLEMS. (2013) (Thesis Committee Member)

 Letao Zhang, Ph.D Deformations of Hilbert Schemes of Points on K3 Surfaces and Representation Theory. (2014) (Thesis Committee Member)

 Paul Munger, Ph.D Spectral Regularity in Models of Aperiodic Order. (2014) (Thesis Committee Member)

 Darren Ong, Ph.D Spectral Characteristics of aperiodic, CMV and Schroedinger Operators. (2014) (Thesis Committee Member)

 Yingpei Wang, Ph.D On the approximation of the Dirichlet to Neumann map for high contrast two phase composites and its applications to domain decomposition methods. (2014) (Thesis Committee Member)

 Quentin Funk, Ph.D Rectifiable Homology of Semi-Algebraic Sets . (2015) (Thesis Director)

 Quentin Funk, PhD Rectifiable Homology of Semi-Algebraic Sets . (2015) (Thesis Director)

 Jake Fillman, Ph.D Spectral Analysis of One-Dimensional Operators. (2015) (Thesis Committee Member)

 Andy Huang, Ph.D Harmonic Maps of Punctured Surfaces into the Hyperbolic Plane. (2016) (Thesis Committee Member)

 Jorge Acosta, Phd The Asymptotics of Representations for Cyclic Opers. (2016) (Thesis Committee Member)

 Quentin Funk, Ph.D Two Variants on the Plateau Problem . (2016) (Thesis Director)

 Andy Huang, PhD Harmonic Maps of Punctured Surfaces into the Hyperbolic Plane. (2016) (Thesis Committee Member)

 Jorge Acosta, Ph.D Asymptotics of Representations for Cyclic Opers. (2016) (Thesis Committee Member)

 Quentin Funk, Ph.D Two Variants on the Plateau Problem . (2016) (Thesis Director)

 Carol Downes, Ph.D A Mass-minimizing Flow for Real-Valued Flat Chains with Applications to Transport Networks. (2017) (Thesis Director)

 Tam Do, Ph.D Stability of 1D flows?. (2017) (Thesis Committee Member)

 Mario Bencomo, Ph.D Discontinuous Galerkin and Finite Difference Methods for the Acoustic Equations with Smooth Coefficients. (2017) (Thesis Committee Member)

 Chao Li, Ph.D An Extension of First Variation to Noninjective Maps. (2017) (Thesis Director)

 Carol Downes, Ph.D Ramified Transport and Transport Flows with Applications. (2017) (Thesis Director)

 Mario Bencomo, PhD Discontinuous Galerkin and Finite Difference Methods for the Acoustic Equations with Smooth Coefficients. (2017) (Thesis Committee Member)

 Jianqiu Wu, Ph.D Minimal Mobius Bands in the 3 sphere and Harmonic Maps to Cones. (2017) (Thesis Director)

 Charles Ouyang, Ph.D Higgs Moduli Space?. (2018) (Thesis Committee Member)

 Xian Dai, Ph.D Higgs Moduli Space?. (2018) (Thesis Committee Member)

 Jianqiu Wu, Ph.D Non-orientable Minimal Surfaces in S^3 and Smooth Minimal Transport Networks. (2018) (Thesis Director)

 Emily Pierce, Ph.D Pinching Variations and Stationary Networks. (2019) (Thesis Director)

Awards, Prizes, & Fellowships
 Fellow of the American Mathematics Society, American Mathematics Society (Nov.4, 2014)

 AMS Centennial Fellowship Committee, American Mathematics Society,

 member, American Mathematics Society ((2000))

 Physical Sciences Review Committee, University of California at Irvine (March, 2000)

 Mathematics Department Review committee, Indiana University (January, 2000)

 Panel Review in Partial Differential Equations, National Science Foundation (February, 2000)

 member, American Mathematics society ((1999))

 Natural Sciences Review Committee, Brown University (November, 1999)

Residencies & Summer Festivals
 Residency
 Centro De Giorgi Visiting Scholar. 2 month sabbatical visit, Scuola Norm. Superior Pisa, . (Jan. 8-Mar. 1, 2011)
 Eidgenössische Technische Hochschule Visiting Scholar. 1 month sabbatical visit, Zurich, Switzerland. (March 10-April 10, 2012)
 Mathematical Sciences Research Institute. 1 month sabbatical visit, U California, Berkeley. (April 20-May 20, 2011)
 Visiting Scholar, Paris VII. (May1-May31, 2012)