Research Supervision

 Julian Lander, Area-minimizing integral currents with boundaries invariant under polar actions, Trans. Amer. Math. Soc. 307 (1988) 419-429.

Benny Cheng, MIT, 1987. Proved new families of very symmetric cones to be area-minimizing, such as the cone over the unitary matrices Un (n >= 4), by extending the theory of coflat calibrations. Thesis, "Area-minimizing equivariant cones and coflat calibrations," MIT.

Benny Cheng, Area minimizing cone type surfaces and coflat calibrations, Indiana U. Math. J. 37 (1988) 505-535.

Gary Lawlor, Stanford, 1988. Proved the five-year-old angle conjecture on which pairs of m-planes are area-minimizing. Developed a curvature criterion for area minimization and classified all area-minimizing cones over products of spheres. Gave the first example of nonorientable area-minimizing cones. Thesis:

Gary Lawlor, A sufficient criterion for a cone to be area-minimizing, Memoirs of the AMS 91, No. 464 (1991) 1-111.

Mohamed Messaoudene, MIT, 1988. Analyzed the mass norm in the first nonclassical case, L³R⁶. Thesis:

Mohamed Messaoudene, The unit mass ball of three-vectors in R⁶, Linear Alg. Appl. 265 (1997) 247-298.
 

Janaki Abraham, 1984-1986. Constructed wire models for studying uniqueness and singular structure of soap films.

Jeff Abrahamson, 1984-1986. Classified boundary as well as interior singularities for length-minimizing flat chains modulo nu.

Jeff Abrahamson, Curves length minimizing modulo nu in Rⁿ, Michigan Math. J. 35 (1988), 285-290.

Maciej Zworski, 1984-1986. Answered structural questions of current research interest about the decomposition of normal currents as infinite convex combinations of integral currents. Work included positive results, counterexamples, and applications.

Maciej Zworski, Decomposition of normal currents, Proc. Am. Math. Soc.102 (1988), 831-839.

Donald Kane, 1984-1985. Answered questions about when a pair of 3-dimensional planes is area-minimizing modulo nu, and general questions about the theory of calibrations modulo nu, with computer and theoretical calculations.

Michael McCutchan, 1985-1986. Proved existence and classified singularities for size-minimizing curves. (The higher dimensional problem is an open question of current research interest.) Paper, "Size-minimizing curves."

Adam Levy, Energy-minimizing networks meet only in threes, J. Und. Math. 22 (1990) 53-59.

1988 Geometry Group, summer, 1988. (Adam Levy (leader), Manuel Alfaro, Mark Conger, Ken Hodges, Rajiv Kochar, Lisa Kuklinski, Zia Mahmood, Karen von Haam). Showed that for some piecewise smooth elliptic integrands in R², segments can meet in fours.

Adam Levy, Manuel Alfaro, Mark Conger, Ken Hodges, Rajiv Kochar, Lisa Kuklinski, Zia Mahmood, and Karen von Haam, Segments can meet in fours in energy-minimizing networks, J. Und. Math. 22 (1990) 9 - 20.

Adam Levy, Manuel Alfaro, Mark Conger, Ken Hodges, Rajiv Kochar, Lisa Kuklinski, Zia Mahmood, and Karen von Haam, The structure of singularities in Phi-minimizing networks in R², Pacific J. Math.149 (1991) 201-210.

Mark Conger, 1988-1989. Proved that for the class of elliptic integrands in Rⁿ, there is a uniform bound c(n) on the number of segments that can meet in a minimizing network. Showed by example that in R³ that bound is at least 6. Thesis,"Energy-minimizing networks in Rⁿ."

Geometry Group, January 1989 (Mark Conger (associate advisor), Marcus Christian, Dylan Cooper, James Goodell). Studied length-minimizing networks in R³ and proved that the conjectured double "Y" structure is the length-minimizing network connecting the vertices of a regular tetrahedron. Paper,"Length-minimizing networks in R³: the tetrahedron and new examples."

1989 Geometry Group, part of NSF REU site at Williams, summer, 1989 (Josh Sher (group leader), Manuel Alfaro, Tessa Campbell, Andres Soto). Proved that shortest directed networks in the plane can meet in fours but not in sevens. Paper, "Length-minimizing directed networks can meet in fours."

Manuel Alfaro, 1989-1990. Proved the existence of shortest directednetworks in the plane. Thesis:

Manuel Alfaro, Existence of shortest directed networks in R², Pacific J. Math. 167 (1995) 201-214.

1990 Geometry Group, part of NSF REU site at Williams, summer, 1990 (Manuel Alfaro (group leader), Jeffrey Brock (associate group leader), Joel Foisy, Nickelous Hodges, Jason Zimba). Proved results on planar compound soap bubbles, included in Foisy's Honors thesis, "Soap bubble clusters in R² and R³," 1991 and the following publication:

Joel Foisy, Manuel Alfaro, Jeffrey Brock, Nickelous Hodges, Jason Zimba. The standard double soap bubble in R² uniquely minimizes perimeter, Pacific J. Math. 159 (1993), 47-59. Featured in the 1994 AMS What's Happening in the Mathematical Sciences.

Scott Berger (Princeton), 1990-1991. Searched for polyhedron of unit volume of least total edge length and proved related results. Junior paper, "The search for boundary-minimizing polytopes in two and three dimensions."

1991 Geometry Group, part of NSF REU site at Williams, summer, 1991 (Thomas Colthurst, Christopher Cox, Joel Foisy (group leader), Hugh Howards, Kathryn Kollett, Holly Lowy, Stephen Root). Studied planar compound bubbles and networks minimizing length plus the number of Steiner points.

Thomas Colthurst, Christopher Cox, Joel Foisy, Hugh Howards, Kathryn Kollett, Holly Lowy, and Stephen Root, Networks minimizing length plus the number of Steiner points, in Ding-Zhu Du and Panos M. Pardalos, ed., Network Optimization Problems: Algorithms, Complexity and Applications, World Scientific, 1993, pp. 23-36.

Christopher Cox, 1991-92. Proved results on the existence and structure of networks minimizing a cost depending on length and capacity. Thesis:

Christopher Cox, Flow-dependent networks: existence and behavior at Steiner points, Networks 31 (1998) 149-156.

Hugh Howards, 1991-1992. Proved examples of isoperimetric regions in surfaces. Thesis, "Soap bubbles on surfaces"; results included in following publication:

Hugh Howards, Michael Hutchings, and Frank Morgan, The isoperimetric problem on surfaces, Am. Math. Monthly 106 (1999) 430-439.

1992 Geometry Group, part of NSF REU site at Williams, summer, 1992 (Christopher Cox, group leader, Lisa Harrison, Michael Hutchings, Susan Kim, Janette Light, Andrew Mauer, Meg Tilton). Proved new results on planar soap bubbles. Paper, "The standard triple bubble type is the least-perimeter way to enclose three connected areas,"revised as the following publication:

Christopher Cox, Lisa Harrison, Michael Hutchings, Susan Kim, Janette Light, Andrew Mauer, and Meg Tilton, The shortest enclosure of three connected areas in R², Real Analysis Exchange 20 (1994/95) 313--335.

Susan Kim, 1992-93. Studied flow-dependent networks (see Cox above) for indistinguishable products. Paper, "Source and sink networks: no stochastic Steiner points in cost minimizers."

1993 Geometry Group, part of NSF REU site at Williams, summer, 1993 (Christopher French, Kristen Albrethsen, Heather Curnutt, Scott Greenleaf, Christopher Kollett). Proved least-energy way to enclose and separate two regions of prescribed area in the plane, where energy is given by the Manhattan metric, thus generalizing the square Wulff crystal. Results of paper, "The planar double Wulff crystal" appeared in the following publication:

Frank Morgan, Christopher French, and Scott Greenleaf, Wulff clusters in R², J. Geom. Anal. 8 (1998) 97-115.

Joseph Masters, 1994. Proved that the standard double bubble is the least-area way to enclose and separate two regions of prescribed area on the sphere.

Joseph Masters, The perimeter-minimizing enclosure of two areas in S², Real Analysis Exchange 22 (1996/7) 645-654.

1995 Geometry Group, part of NSF REU site at Williams, summer, 1995 (Megan Barber, Jennifer Tice, Brian Wecht (group leader)). Provided an improved theory of double salt crystals, taking into account the reduced cost of the interior wall. Also studied geodesics and geodesic nets on polyhedra, and shapes of double bubbles of immiscible fluids. Papers on "Geodesics and geodesic nets on regular polyhedra" and "Immiscible fluids" continued in publications below with publications of 1998 and 2000 Geometry Groups.

Brian Wecht, Megan Barber, and Jennifer Tice, Double salt crystals, Acta Crystallographica, Sect. A (Jan. 2000) 92-95.

1996 Geometry Group, part of NSF REU site at Williams, summer, 1996 (Alexei Erchak, Ted Melnick, Ramona Nicholson). Studied geodesic nets on polyhedra and double clusters of immiscible fluids. Papers on "Double clusters of immiscible fluids" and "Geodesic nets on polyhedra" continued in publications of 1998 and 2000 Geometry Groups.

Frank Morgan, Ted Melnick, and Ramona Nicholson, The soap bubble geometry contest, The Mathematics Teacher 90 (December, 1997) 746-750.

Brian Elieson, 1996-97. Studied networks with various weights, as in cluster interfaces. Thesis: "Cost minimizing networks separating immiscible fluids in R²."

1997 Geometry Group, part of NSF REU site at Williams, summer, 1997 (David Futer, David McMath, Brian Munson, Sang Pahk). Studied how the structure of clusters of immiscible fluids may change when additional fluids are added. Generalized work of Lawlor and Morgan by proving an equivalent point-placing "calibration" condition for the case of adding one fluid to n fluids or adding two fluids to three fluids. Proposed a counterexample for the case of adding two fluids to four fluids. Paper, "Cost-minimizing networks among immiscible fluids in R³." Continued in Munson undergraduate thesis, "Cost-minimizing networks and polyhedral cones," Univ. of Oregon, 1998, which won him the Aaron Novick Senior Fellowship at Oregon, and in joint paper with 1998 Geometry Group (below).

1998 Geometry Group, part of NSF REU site at Williams, summer, 1998 (Andrei Gnepp, Ting Fai Ng, Cara Yoder). Studied the isoperimetric problem on polyhedra and continued the work on immiscible fluids. Papers, "Isoperimetric domains on polyhedra and singular surfaces," continued in paper with 2000 Geometry Group below; a supplemental paper, "Two counterexamples on immiscible fluids;" and the following publication:

David Futer, Andrew Gnepp, David McMath, Brian Munson, Ting Fai Ng, Sang Pahk, and Cara Yoder (Geometry Groups 1997 and 1998), Cost-minimizing networks among immiscible fluids in R², Pacific J. Math. 196 (2000) 395-414.

1999 Geometry Group, part of NSF REU site at Williams, summer, 1999 (Cory Heilmann, Yvonne Lai, Ben Reichardt, Anita Spielman). Generalized the recent proof of the Double Bubble Conjecture from R³ to R⁴ and certain higher dimensional cases. Papers, "Component bounds for area-minimizing double bubbles" and the following publication:

Ben Reichardt, Cory Heilmann, Yuan Lai, and Anita Spielman, Proof of the Double Bubble Conjecture in R⁴ and certain higher dimensional cases, Pac. J. Math. 208 (2003) 347-366.

2000 Geometry Group, part of NSF REU site at Williams, summer, 2000 (Andrew Cotton, David Freeman, John Spivack). Studied isoperimetric problem on singular surfaces such as cylindrical can and proved double bubble conjecture for equal volumes in S³ and H³. Papers, "The isoperimetric problem on singular surfaces" and "The isoperimetric problem on double discs," subsumed in following publication:

Andrew Cotton, David Freeman, Andrei Gnepp, Ting Fai Ng, John Spivack, and Cara Yoder (Geometry Groups 1998 and 2000), The isoperimetric problem on some singular surfaces, J. Austral. Math. Soc. 78 (2005), 167-197..

Andrew Cotton and David Freeman, The double bubble problem in spherical and hyperbolic space, Intern. J. Math. Math. Sci. 32 (2002) 641-699.

2001 Geometry Group, part of NSF REU site at Williams, summer, 2001 (Joe Corneli, Paul Holt, Nicholas Leger, Eric Schoenfeld). Five papers, including "The double bubble on the cylinder" and "The double bubble on the flat 2-torus" (revised with 2002 Geometry Group publication below), "Partial results on double bubbles in S³ and H³."

Joe Corneli, Paul Holt, Nicholas Leger, and Eric Schoenfeld, Student Chapter Members at Board of Govenors Meeting, MAA FOCUS, November, 2001.

Frank Morgan, Paul Holt, Joseph Corneli, Nicholas Leger, and Eric Schoenfeld, Mathematicians on Michael Feldman's "Whad'Ya Know?" (with Morgan), MAA FOCUS, November, 2001.

Clay Mathematics Institute Summer Course in Geometric Measure Theory and the Proof of the Double Bubble Conjecture, MSRI, summer, 2001.

Miguel Carrión Álvarez, Joseph Corneli, Genevieve Walsh, and Shabnam Beheshti, Double bubbles in the three-torus, Exp. Math. 12 (2003), 79-89; ArXiv.

Eric Katerman, fall 2001, senior colloquium, “Doughnuts, basketballs, and inner tubes: an introduction to DeRham cohomology theory.”

Eric Katerman, De Rham cohomology of the rectangular torus, Furman University Electronic Journal of Undergraduate Mathematics 11 (2006).

2002 Geometry Group, part of NSF REU site at Williams, summer, 2002 (Eric Schoenfeld (associate advisor), Tracy Borawski, George Lee, Robert Lopez).

Joseph Corneli, Paul Holt, George Lee, Nicholas Leger, Eric Schoenfeld, and Benjamin Steinhurst, The double bubble problem on the flat two-torus, Trans. Amer. Math. Soc. 356 (2004) 3769-3820; ArXiv.

Robert Lopez and Tracy Borawski Baker, The double bubble problem on the cone, NY J. Math. 12 (2006), 157-167. http://nyjm.albany.edu:8000/j/2006/12-9.html

2003 Geometry Group, Williams, summer, 2003 (Joe Corneli, Neil Hoffman, Stephen Moseley).

Joseph Corneli, Neil Hoffman, Paul Holt, George Lee, Nicholas Leger, Stephen Moseley, Eric Schoenfeld, Double bubbles in S³ and H³, J. Geom. Anal. 17 (2007), 189-212.

Neil Hoffman, 2003-04. Double bubbles in S³, H³, and Gaussian space.

Jonathan Lovett, 2004-05. Rotating linkages in a normed plane.

Jack Cook, Jonathan Lovett, and Frank Morgan, Rotation in a normed plane, Amer. Math. Monthly 114 (Aug.-Sept., 2007), 628-632.

2004 Geometry Group, part of NSF REU site at Williams, summer, 2004 (Ivan Corwin, Stephanie Hurder, Vojislav Sesum, Ya Xu). Double bubbles in Gaussian space.

Joe Corneli, Ivan Corwin, Stephanie Hurder, Vojislav Sesum, Ya Xu, Elizabeth Adams, Diana Davis, Michelle Lee, Regina Visocchi, Double bubbles in Gauss space and spheres, Houston J. Math., to appear.

Ivan Corwin, Neil Hoffman, Stephanie Hurder, Vojislav Sesum, Ya Xu, Differential geometry of manifolds with density, Rose-Hulman Und. Math. J. 7 (1) (2006). http://www.rose-hulman.edu/mathjournal/v7n1.php

2005 Geometry Group, part of NSF REU site at Williams, summer, 2005 (Elizabeth Adams, Diana Davis, Michelle Lee, Regina Visocchi). Isoperimetric regions in Gauss sectors.

Elizabeth Adams, Ivan Corwin, Diana Davis, Michelle Lee, Regina Visocchi, Isoperimetric regions in Gauss sectors, Rose-Hulman Und. Math. J. 8 (1) (2007). http://www.rose-hulman.edu/mathjournal/v8n1.php

Michelle Lee, 2005-06. Isoperimetric regions in locally Euclidean manifolds and in manifolds with density, Honors thesis, Williams College, 2006.

Michelle Lee, Isoperimetric regions in surfaces and in surfaces with density, Rose-Hulman Und. Math. J. 7 (2) (2006). http://www.rose-hulman.edu/mathjournal/v7n2.php

Vojislav Sesum, 2005-06. The honeycomb problem on hyperbolic surfaces, Honors thesis, Williams College, 2006.

2006 Geometry Group, part of NSF REU site at Williams, summer, 2006 (Colin Carroll, Adam Jacob, Conor Quinn, Robin Scott Walters), (1) The honeycomb problem on hyperbolic surfaces and (2) planes with density.

Colin Carroll, Adam Jacob, Conor Quinn, Robin Walters, On generalizing the honeycomb theorem to compact hyperbolic manifolds and the sphere, SMALL Geometry Group, Williams College, 2006.

Colin Carroll, Adam Jacob, Conor Quinn, Robin Walters, The isoperimetric problem on planes with density, Bull. Austral. Math. Soc., to appear.

Conor Quinn, 2006-07. Area-minimizing partitions of the sphere, Honors thesis, Williams College, 2007.

Conor Quinn, Area-minimizing partitions of the sphere, Rose-Hulman Und. Math. J. 8 (2) (2007). http://www.rose-hulman.edu/mathjournal/v8n2.php

Nicholas D. Brubaker, Stephen Carter, Sean M. Evans, Daniel E. Kravatz, Sherry Linn, Stephen W. Peurifoy, Ryan Walker (Millersville University undergraduate workshop, April 2007), Existence and stability of the conjectured
surface area minimizing double bubbles in the three torus, Math Horizons, April, 2008.

2007 Geometry Group, part of NSF REU site at Williams, summer, 2007 (Max Engelstein, Anthony Marcuccio, Quinn Maurmann, Taryn Pritchard), Surface partitions and density problems.

Quinn Maurmann, Max Engelstein, Anthony Marcuccio, and Taryn Pritchard, Asymptotics of perimeter-minimizing partitions, Canadian Math. Bull., to appear.

Max Engelstein, Anthony Marcuccio, Quinn Maurmann, and Taryn Pritchard, Isoperimetric problems on the sphere and on surfaces with density, to appear.

Quinn Maurmann and Frank Morgan, Isoperimetric comparison theorems for manifolds with density, preprint (2008).

Matthew Simonson, 2007-08. Isoperimetric problems on non-Euclidean surfaces.

2008 Geometry Group, part of NSF REU site at Williams, summer, 2008 (Jonathan Dahlberg, Alexander Dubbs, Hung Tran, Edward Newkirk), Isoperimetric problems and manifolds with density.