Undergraduate research and mentoring

DOMATH Summer 2022

A group photo of Demetre's DoMath team taken outside on a sunny day in front of a brick building. On the left are two young men (Nico and Ben) smiling at the camera with folded arms. On the right are two young women (Hiba and Khiyali) smiling at the camera. In the middle is a professional young man (Demetre) smiling at the camera and holding a mask.

During the Summer of 2022, I organized a research program for 4 undergraduate Duke students.

Project leader: Professor Demetre Kazaras

Project manager: Kai Xu (Graduate student, Duke)

Team members: Hiba Benjeddou, Benjamin Goldstein, Khiyali Pillalamarri, Nicolas Salazar

Project summery: A minimal surface is a surface in 3-space which locally minimizes surface area. These structures are found in nature as membranes experiencing equal pressure from opposing sides, such as soap films spanning a wireframe. In our project we investigated and found new examples of 'minimal mod-3 surfaces'. These objects are made up of pieces of minimal surfaces meeting each other 3 at a time along 'singular curves' so that the angles between colliding faces are always 120 degrees. Not much is known about these minimal mod-3 surfaces -- our goal was to find new examples and investigate what properties the singular curves might have.

We first considered surfaces of revolution, obtained by rotating appropriate collections of catenary curves. Infinitely many examples were found, and their singular curves display a rich structure. Another class was constructed with a 'screw-motion symmetry' by carefully arranging helicoidal surfaces to intersect at 120 degrees. Our third approach was to design wire-frames with special symmetry so that the minimal-area surface spanning them could be extended to all of 3-space by rotation and translation. We also investigated the intrinsic geometry of the mod-3 surfaces themselves, understanding how distances behave on a particular example called the 'round mod-3 disc'.

Results will be described in a forthcoming manuscript.

Below are some images produced during our work.

An animation showing a collection of transparent, orange circular surfaces.

An animation showing a collection of thin blue curves intersecting each other at 120 degree angles. Revolving these curves in 3-space forms the orange surfaces in the previous animation.

An interesting mod-3 minimal surface with rotational symmetry

The above animations were created by Benjamin Goldstein.

This image shows three hexagon-like surfaces perfectly fit together. Each hexagon-like surface has parts which curl upwards and downwards which form a sort of hole when fit together. The surfaces are made of hundreds of small triangles of different shapes and colors ranging smoothly from orange to blue.

A Scherk-style mod-3 minimal surface with translational symmetry

Imagined by Hiba Benjeddou, graphics created by Demetre Kazaras

A remarkable embedded mod-3 minimal surface with screw-motion symmetry

Created by Khiyali Pillalamarri

Gromov-Haursdorff and Sormani Wenger Intrinsic Flat Convergence of Smocked Metric Spaces: Winter and Spring 2019

I am pictured walking behind several young mathematicians working at a large square table.

In late 2018, Professor Christina Sormani and I were working on a paper (An intrinsic flat limit with no geodesics) together and stumbled across a collection of problems which were simply too fun and approachable to keep to ourselves. We decided to organize a research program for undergraduate students from CUNY Lehman, Stony Brook University, and Rutgers built around these problems. I helped run this program for several weeks and helped students write up our results in a published paper. See the first link below for a description of these problems, which make reference to diagrams used in traditional sewing practices called "smocking patterns" (see figures to the right or below).

Results:

Paper by myself, students, and Christina Sormani "Smocked metric spaces and their tangent cones"

Follow up paper by Christina Sormani and subset of students: "SWIF convergence of smocked metric spaces"

The images above and below were produced by Christina Sormani. The images depict the stitching patterns which, when pulled tight and reduced to 0 length, produced "smocked metric spaces."

If you keep repeating this patter outwards, what will this shape converge to?

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