## PhD Students

**Julian Lander**, MIT, 1984. Gave the first positive results in general codimension on when a minimizing surface inherits the symmetries of the boundary. Thesis:

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, Mem. AMS 91, No. 464 (1991) 1-111.

**Mohamed Messaoudene**, MIT, 1988. Analyzed the mass norm in the first nonclassical case, Λ^{3}**R**^{6}. Thesis:

Mohamed Messaoudene, The unit mass ball of three-vectors in **R**^{6}, Linear Alg. Appl. 265** **(1997) 247-298.

## Undergraduate Research at MIT

**Roger Bassini**, 1981-1982. Studied singularities in networks minimizing integrands more general than length. Paper, “Length-minimizing networks for three points in **R**^{2}.”

**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 **R*** ^{n}*, 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.”

## Undergraduate Research at Williams

**Adam Levy**, 1987-1988. Proved that in networks minimizing for general smooth elliptic integrands in **R**^{2}, segments meet only in threes. Thesis published as follows:

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 **R**^{2}, 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 **R**^{2}, Pacific J. Math.149 (1991) 201-210.

**Mark Conger**, 1988-1989. Proved that for the class of elliptic integrands in **R*** ^{n}*, there is a uniform bound c(

*n*) on the number of segments that can meet in a minimizing network. Showed by example that in

**R**

^{3}that bound is at least 6. Thesis,”Energy-minimizing networks in

**R**

*.”*

^{n}**Geometry Group**, January 1989 (Mark Conger (associate advisor), Marcus Christian, Dylan Cooper, James Goodell). Studied length-minimizing networks in **R**^{3} 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 **R**^{3}: 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. Thesis, Existence of shortest directed networks in **R**^{2}., also explained that such networks can meet in sixes but not in sevens.

Manuel Alfaro, Existence of shortest directed networks in **R**^{2}, Pacific J. Math. 167 (1995), 201-214. Generalized to **R*** ^{n}* by Konrad J. Swanepoel, On the existence of shortest directed networks, J. Combin. Math. Combin. Comput. 33 (2000), 97–102.

**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—Zimba would later be lead writer on the Common Core mathematics standards). Proved results on planar compound soap bubbles, included in Foisy’s Honors thesis, “Soap bubble clusters in **R**^{2} and **R**^{3},” 1991 and the following publication:

Joel Foisy, Manuel Alfaro, Jeffrey Brock, Nickelous Hodges, Jason Zimba. The standard double soap bubble in **R**^{2} 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 **R**^{2}, 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 **R**^{2}, 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 **S**^{2}, 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 **R**^{2}.”

**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 **R**^{3}.” 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 **R**^{2}, 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 **R**^{3} to **R**^{4} 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 **R**^{4} and certain higher dimensional cases, Pacific 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 **S**^{3} and **H**^{3}. 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. In Theorem 2.4 they note solution to double bubble problem in hyperbolic plane and for immiscible fluids as well.

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 **S**^{3} and **H**^{3}.”

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?” 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, De Rham cohomology of the rectangular torus, Furman University Electronic Journal of Undergraduate Mathematics 11 (2006). Based on senior colloquium, fall, 2001.

**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, New York 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 **S**^{3} and **H**^{3}, J. Geom. Anal. 17 (2007), 189-212. Full version with computer code on arXiv at http://arxiv.org/abs/0811.3413.

**Neil Hoffman,** 2003-04. Double bubbles in **S**^{3}, **H**^{3}, and Gaussian space. Won 2003 MathFest CUR best overall student talk award.

**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. Stephanie won 2004 MathFest student talk award.

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. 34 (2008) 181-204.

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. Diana won 2005 MathFest student talk award.

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 spaces, 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. Walters got NSF post-doc after Chicago PhD.

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. 78 (2008), 177-197. *Note:* Proposition A.1 was obtained earlier by Abresch and Langer.

**Conor Quinn,** 2006-07. Least-perimeter partitions of the sphere, senior honors thesis, Williams College, 2007. Won 2006 MathFest student talk award.

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

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. Max won 2007 MathFest student talk award.

Quinn Maurmann, Max Engelstein, Anthony Marcuccio, and Taryn Pritchard, Asymptotics of perimeter-minimizing partitions, Canadian Math. Bull. 53 (2010), 516-525.

Max Engelstein, Anthony Marcuccio, Quinn Maurmann, and Taryn Pritchard, Isoperimetric problems on the sphere and on surfaces with density, New York J. Math. **15 **(2009), 97–123. http://www.emis.de/journals/NYJM/j/2009/15-5.html

*Note:* Isoperimetric Corollary 4.9 on the halfplane with density* *y* ^{a}* was obtained earlier by Carla Maderna and Sandro Salsa, “Sharp estimates for solutions to a certain type of singular elliptic boundary value problems in two dimensions,” Applicable Analysis 12 (1981), 307-321, Theorem 1.

Quinn Maurmann and Frank Morgan, Isoperimetric comparison theorems for manifolds with density, Calc. Var. PDE 36 (2009), 1-5.

Max Engelstein, The least-perimeter partition of a sphere into four equal areas, Disc. Comp. Geom. 44 (2010), 645-653; arXiv.org.

**Matthew Simonson,** 2007-08. The isoperimetric problem on Euclidean, spherical, and hyperbolic surfaces, J. Korean Math. Soc. 48 (2011), 1285-1325.

**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. Jon won 2008 MathFest student talk award.

Jonathan Dahlberg, Alexander Dubbs, Edward Newkirk, Hung Tran, Isoperimetric regions in the plane with density r^{p}, New York J. Math. 16 (2010), 31-51. http://nyjm.albany.edu/j/2010/16-4.html

**Edward Newkirk,** 2008-09. Least-perimeter partitions of the sphere, senior honors thesis, Williams College, 2009.

Edward Newkirk, Minimal connected partitions of the sphere, Rose-Hulman Und. Math. J. 11(2) (2010). http://www.rose-hulman.edu/mathjournal/v11n2.php

**Bret Thacher,** 2009. Extensions of extremal graph theory to grids, senior honors thesis, Williams College, 2009.

Bret Thacher, Extensions of extremal graph theory to grids, Rose-Hulman Und. Math. J. 10 (2) (2009). http://www.rose-hulman.edu/mathjournal/v10n2.php

**Deividas Seferis,** 2009. Isoperimetric regions on a weighted 2-dimensional lattice, senior honors thesis, Williams College, 2009.

Deividas Seferis, Isoperimetric regions on a weighted 2-dimensional lattice, Rose-Hulman Und. Math. J. 10 (2) (2009). http://www.rose-hulman.edu/mathjournal/v10n2.php

**2009 Geometry Group,** part of Williams NSF REU site, at Granada, Spain, summer, 2009 (Alexander Díaz, Nate Harman, Sean Howe, David Thompson), “Isoperimetric problems in sectors with density” (see my blog post), “Sphere partitions problem.” Sean won University of Arizona spring 2010 Excellence in Undergraduate Research Award, NSF graduate fellowship, and NSF postdoc. Alex won 2010 SACNAS poster award and received 2015 Ford Fellowship Honorable Mention. Nat won an NSF postdoc.

Alexander Díaz, Nate Harman, Sean Howe, David Thompson, Isoperimetric problems in sectors with density, Adv. Geom. 12 (2012), 589–619; arXiv.org (2010); see blog posts 1 and 2.

Frank Morgan, Sean Howe, Nate Harman, Steiner and Schwarz symmetrization in warped products and fiber bundles with density, Revista Mat. Iberoamericana 27 (2011), 909-918; arXiv.org (2009); see blog post.

Nicholas Neumann-Chun, An undergraduate’s experience at the joint mathematics meetings, MAA FOCUS, February/March 2010, 13.

**2010 Geometry Group**, part of Williams NSF REU site, summer, 2010 (Yifei Li, Michael Mara, Isamar Rosa Plata, and Elena Wikner), “Tiling with penalties and isoperimetry with density,” Rose-Hulman Und. Math. J. 13 (1) (2012), http://www.rose-hulman.edu/mathjournal/v13n1.php. Yifei won 2010 MathFest outstanding student talk award. Isa received a 2013 NSF Graduate Fellowship in Civil Engineering.

**2011 Geometry Group**, part of Williams NSF REU site, summer, 2011 (Ping Ngai “Brian” Chung, Miguel A. Fernandez, Niralee Shah, Luis Sordo Vieira, with Elena Wikner), “Perimeter-minimizing pentagonal tilings and isoperimetry with density.” Miguel won 2011 MathFest outstanding student talk award and the 2012 MAA annual joint meetings outstanding poster award. Luis won the Michigan spring 2012 MAA outstanding undergraduate presentation award and a 2013 NSF Graduate Research Fellowship. Brian received an Outstanding Presentation Award at the 2013 joint math meetings in San Diego.

Ping Ngai Chung, Miguel A. Fernandez, Yifei Li, Michael Mara, Frank Morgan, Isamar Rosa Plata, Niralee Shah, Luis Sordo Vieira, Elena Wikner, Isoperimetric pentagonal tilings, Notices Amer. Math. Soc. 59 (May, 2012), 632-640.

Ping Ngai Chung, Miguel A. Fernandez, Niralee Shah, Luis Sordo Vieira, Are circles isoperimetric in the plane with density e^{r}? Rose-Hulman Und. Math. J. 16(1) (2015).

Ping Ngai Chung, Miguel A. Fernandez, Niralee Shah, Luis Sordo Vieira, Elena Wikner, Perimeter-minimizing pentagonal tilings, Involve 7 (2014), 453-478.

**2012 Geometry Group**, part of Williams NSF REU site, summer, 2012 (Whan Ghang, Zane Martin, Steven Waruhiu). Isoperimetric problems (Geometry Group report), Williams College, 2012. (Log-Sobolev inequality on interval; perimeter-minimizing tilings of plane and **R**^{3}.)

Whan Ghang, Zane Martin, Steven Waruhiu, The sharp log-Sobolev inequality on a compact interval, Involve 7-2 (2014), 181-186, arXiv.org.

Whan Ghang, Zane Martin, Steven Waruhiu, Surface-area-minimizing *n*-hedral tiles, arXiv.org (2013). Revision with Paul Gallagher, David Hu, Maggie Miller, Byron Perpetua, Rose-Hulman Und. Math. J. 15(1) (2014).

Whan Ghang, Zane Martin, Steven Waruhui, Perimeter-minimizing tilings by convex and nonconvex pentagons, Rose-Hulman Und. Math. J. 16(1) (2015).

Roshan Sharma, The Weierstrass Representation always gives a minimal surface, Rose-Hulman Und. Math. J. 14(1) (2014). Write-up of Williams College senior colloquium talk, October 31, 2012.

Tejesh Pradham, A note on the wallet paradox, Morgan blog (2013). Note on Williams College senior colloquium talk, November 12, 2012.

**Zane Martin,** 2013. Perimeter-minimizing tilings by convex and nonconvex pentagons, senior honors thesis, Williams College, 2013; revised for publication with Ghang and Waruhiu (above). Honorable Mention, NSF Graduate Fellowship Program.

**2013 Geometry Group**, part of Williams NSF REU site, summer, 2013 (Paul Gallagher, David Hu, Zane Martin, Maggie Miller, Byron Perpetua). “Isoperimetric Problems.” Miller gave an “outstanding presentation” on the “Convex Region Isoperimetric Conjecture at the MAA student poster session at the 2014 Joint Mathematics Meetings in Baltimore.

Frank Morgan, Paul Gallagher, and Maggie Miller, Working up a Lather—Bubbles and Foam, Parts 3 and 4, Mathematical Moment interview by Michael Breen.

Paul Gallagher, David Hu, Zane Martin, Maggie Miller, and Byron Perpetua, Isoperimetric problem on the plane with density e^{–1/r}, Rose-Hulman Und. Math. J. 15(2) (2014).

**2014 Geometry Group**. Wyatt Boyer, Bryan Christopher Brown, Alyssa Loving, Sarah Tammen, with Gregory Chambers. Isoperimetric problems (densities, convex bodies, hyperbolic double bubbles) and optimal transportation with constraint.

Wyatt Boyer, Bryan Brown, Gregory Chambers, Alyssa Loving, and Sarah Tammen, Isoperimetric regions in **R*** ^{n}* with density r

^{p}, Anal. Geom. Metr. Spaces 4 (2016), 236–265. Tammen named Schafer Prize Runner-up for this work and received NSF graduate fellowship.

Wyatt Boyer, Bryan Brown, Alyssa Loving, Frank Morgan, and Sarah Tammen, Double bubbles in hyperbolic surfaces, *Involve* 11 (2018), 207-217.

Wyatt Boyer, Bryan Brown, Alyssa Loving, and Sarah Tammen, Optimal transportation with constant constraint, *Involve ***12**-1 (2019), 1-12.

**2015 Geometry Group**. John Berry, Matthew Dannenberg, Jason Liang, Yingyi Zeng. Isoperimetric problems (double Gaussian, convex bodies, tilings), relaxed packings.

John Berry, Eliot Bongiovanni, Wyatt Boyer, Bryan Brown, Matthew Dannenberg, Paul Gallagher, David Hu, Jason Liang, Alyssa Loving, Zane Martin, Maggie Miller, Byron Perpetua, Sarah Tammen, and Yingyi Zeng, The convex body isoperimetric conjecture in **R**^{2}, Rose-Hulman Und. Math. J., 2018.

John Berry, Matthew Dannenberg, Jason Liang, Yingyi Zeng, The convex body isoperimetric conjecture in **R**^{2} and polygons, preprint.

John Berry, Matthew Dannenberg, Jason Liang, Yingyi Zeng, The isoperimetric problem for the sum of two Gaussian densities, Involve **11**:4 (2018), 549–567; arXiv.org (2016).

John Berry, Matthew Dannenberg, Jason Liang, Yingyi Zeng, Symmetries of Cairo-Prismatic tilings, Rose-Hulman Und. Math. J. 17 (2016), http://scholar.rose-hulman.edu/rhumj/vol17/iss2/3.

John Berry, Matthew Dannenberg, Jason Liang, Yingyi Zeng, Relaxed disc packings, http://sites.williams.edu/Morgan/2015/08/14/relaxed-disk-packings/

**2016 Geometry Group**. Leonardo Di Giosia, Jay Habib, Lea Kenigsberg, Dylanger Pittman, Weitao Zhu. Lea awarded 2017 Schafer Prize honorable mention and NSF graduate fellowship.

Leonardo Di Giosia, Jay Habib, Lea Kenigsberg, Dylanger Pittman, Weitao Zhu, Balls isoperimetric in **R*** ^{n}* with volume and perimeter densities r

^{m}and r

^{k}, proof incomplete, version 2 arxiv.org (2019).

Leonardo Di Giosia, Jay Habib, Lea Kenigsberg, Dylanger Pittman, Weitao Zhu, Isoperimetry in surfaces of revolution with density, in preparation (2016).

Leonardo Di Giosia, Jay Habib, Lea Kenigsberg, Dylanger Pittman, Weitao Zhu, Least-perimeter tiling in the hyperbolic plane, preprint (2016).

Leonardo Di Giosia, Jay Habib, Lea Kenigsberg, Dylanger Pittman, Weitao Zhu, The log convex density conjecture in hyperbolic space, Rose-Hulman Und. Math. J. 18 (2017), http://scholar.rose-hulman.edu/rhumj/vol18/iss1/9/.

Leonardo Di Giosia, Jay Habib, Lea Kenigsberg, Dylanger Pittman, Dustin Ping, Weitao Zhu, Report on perimeter-minimizing *n*-hedral tiles, Williams College, 2016.

**2017 Geometry Group**. Eliot Bongiovanni, Alejandro Diaz, Arjun Kakkar, Nat Sothanaphan. Eliot won NSF graduate fellowship.

Eliot Bongiovanni, Alejandro Diaz, Arjun Kakkar, Nat Sothanaphan, The least-area tetrahedral tile of space, arxiv.org/abs/1709.04139, submitted Geometriae Dedicata (June 2018).

Eliot Bongiovanni, Alejandro Diaz, Leonardo Di Giosia, Jay Habib, Arjun Kakkar, Lea Kenigsberg, Dustin Ping, Dylanger Pittman, Nat Sothanaphan, Weitao Zhu, Double bubbles on the line with log-convex density, Anal. Geom. Metr. Spaces 6 (2018), 64–88. See blog post Bubbles inside bubbles.

Eliot Bongiovanni, Alejandro Diaz, Arjun Kakkar, Nat Sothanaphan, Isoperimetry in surfaces of revolution with density, Missouri J. Math. 30 (2018), 150-165; arxiv.org/abs/1709.06040.

“Outstanding Posters” (judged top 15%) for the MAA Student Poster Session at the Joint Mathematics Meetings 2018:

Double Bubbles on the Real Line with Log-convex Density by Eliot Bongiovanni and Nat Sothanaphan

The Least-Area Tetrahedral Tile of Space by Alejandro Diaz and Arjun Kakkar

Nat Sothanaphan, 1D Triple Bubble Problem with Log-Convex Density, submitted 1 March 2019, arxiv.org/abs/1805.08377

Nat Sothanaphan, Double Bubbles on the Line with Log-convex Density *f* with (log *f*)’ Bounded, Missouri J. Math. 30 (2018), 166-175; arxiv.org/abs/1807.02661.

## Beyond Williams

**2019 IAR** high school student research group. Ruiyu Roy Huang, Yiheng Willa Pan, Xinkai Oliver Qian, Mulei Lily Xu, Lu Sarah Wang [TA], Junfei Jefferson Zhou. “Isoperimetric regions on the line with density |x|^{p},” submitted 17 February 2019.

**2019 Geometry Group** in SUMRY at Yale. Jack Hirsch, Kevin Li , Jackson Petty, Christopher Xue. Research on isoperimetric problems.