**Atlanta, GA**

All over campus this summer, undergraduates are working with Georgia Tech researchers. Many programs are in full swing, modeled after the Research Experiences for Undergraduates (REU) program of the National Science Foundation (NSF).

The School of Mathematics likely takes the prize for the most number of programs by one unit: six. By summer’s end, seven professors, three postdoctoral mentors, and five graduate students would have worked with 13 undergraduate students. The undergrads come from 11 colleges and universities, including three in Georgia: Agnes Scott College, Georgia Tech, and Spelman College.

Funding comes from various NSF grants and the School of Mathematics.

**Why REUs**

REU programs play the same role for research careers as high school sports do for the NFL and NBA, says School of Mathematics Professor Igor Belegradek. Talent presenting early must be nurtured and honed as soon as possible.

Belegradek organized the summer 2018 REUs with colleague Dan Margalit.

“We have a rich history of undergraduate research in mathematics, as you can see on our website,” Margalit says. “It’s a testament to our faculty’s intellectual creativity and dedication to undergraduate education.”

REUs have important benefits for students, faculty mentors, and the School of Mathematics.

They help bring students to the School’s graduate program. They enable members of underrepresented minorities get advanced training and positive experiences in math research.

REUs advance the research of faculty. “We give students problems that we are genuinely interested in,” Margalit. “They are integral to our research programs.”

REUs also provide mentoring experience to early-career researchers – graduate students and postdoctoral researchers – serving as mentors. “The training is valuable for them,” Margalit says. “It helps give them confidence in their own research and make them marketable for job searches.”

Undergraduates’ ability to penetrate difficult problems inspires Margalit. “They are fearless and creative, trying approaches that I might not think of,” he says. “They might not understand every bit of background that goes into a problem. But we, as mentors, can airlift them to the front lines of the problem.”

Undergraduates "are fearless and creative, trying approaches that I might not think of. They might not understand every bit of background that goes into a problem. But we, as mentors, can airlift them to the front lines of the problem."Dan Margalit

**Cutting-Edge Research**

Although Margalit’s program – on mapping class groups – has six students, other REUs have only one or two students. Three began as early as May 21; one will last until Aug. 10. In sessions lasting from five to seven weeks, the mathematicians will tackle problems in various cutting-edge areas.

Following are two examples of problems Georgia Tech undergrads will be confronting.

**Shadow Problem**

Mohammad Ghomi has been working with Georgia Tech undergraduate **Alexander Avery** since May 21. From Ghomi’s list of open problems in geometry of curves and surfaces, Avery chose the “shadow problem” for surfaces.

Ghomi explains the problem thus: Consider a convex object, such as a ball or an egg. When such object is illuminated from any direction, the dark region of the surface, called the shadow, forms a connected set. In other words, the shadow is one piece.

What about the converse? Suppose the shape of a surface is unknown. And suppose the shadow is one piece when illuminated from any direction. Does it follow that the surface is convex?

Ghomi published a solution in *Annals of Mathematics* in 2002. The answer is yes for surfaces similar to balls and eggs. But not for other shapes, such as donuts.

“Alex is working on the discrete version of this problem,” Ghomi says. Avery is looking at surfaces that are not smooth – like balls and eggs – but instead are composed of polygons glued along their edges. “Alex has been making good progress. It looks like the polyhedral case will be similar to the smooth case.”

**Legendrian Knots**

“In mathematics, knots can be thought of as pieces of string which are tied up and then have the ends glued together,” says Caitlin Leverson, one of the postdoctoral mentors. “An interesting problem is to decide whether two knots are the same or different.”

Legendrian knots satisfy additional conditions. Two Legendrian knots may look very different, but be the same. Invariants are methods of assigning values to knots so that two knots are assigned the same value if they are the same.

From May 29 to Aug. 10, Leverson will be working with two Georgia Tech fourth-year mathematics majors: **DeVon Ingram** and **Hunter Vallejos**. Their goal is to find Legendrian knots that are different yet are assigned the same value by the invariant.

Since his second year as a mathematics major, Ingram has done research with different professors, including outside the School of Mathematics. For example, he worked on computational complexity theory with Lance Fortnow, professor and chair, School of Computer Science.

Ingram appreciates the beauty of differential geometry and its relation to physics. He sees correspondence between knot invariants and topological quantum field theories. Because of these interests, “I am naturally drawn to a knot theory problem,” he says.

Vallejos has been doing research since he was in Oak Ridge High School, in Oak Ridge, Tennessee, just 10 miles from Oak Ridge National Laboratory (ORNL). One outcome of his stints at ORNL is a 2017 paper in the Journal of Economic Interaction and Coordination, of which Vallejos was first author.

“I love when algebra, geometry, and topology intersect,” Vallejos says. “Legendrian knot theory blends these three distinct fields, which makes it a rich subject to study.”

**Visiting Students**

Several of the undergraduate researchers this summer come from outside Georgia Tech. Among them are Johannes Hosle and Andrew Sack.

**Johannes Hosle** hails from South Bend, Indiana. He is a third-year math major in the University of California, Los Angeles. His major interests are analysis and number theory. Starting on June 18, he will work with Galyna Livshyts and Michael Lacey.

“The general area of my problem will be in harmonic analysis in convex geometry,” Hosle says. “My interest stems from a general interest in analysis. The types of problems in this branch of mathematics seem to resonate most with me.”

**Andrew Sack** hails from Gainesville, Florida. He is a fourth-year mathematics major from the University of Florida. A published author in the International Journal of Mathematics and Computer Science, he is one of two students who have been working with John Etnyre and Sudipta Kolay since May 30.

Etnyre also studies how to tell knots apart. In his approach, a knot is represented by a diagram of a loop on a paper. The loop can cross over itself as many times. “But each time the loop crosses over itself, you have to specify which of the two strands is on top of the other,” Etnyre says.

A coloring of a knot is a labeling of the strands by a method that has consistency at the crossings. The coloring can tell two knots apart. “The work is related to research trying to figure out how three-dimensional spaces can be put inside a five-dimensional space.”

“I’m interested in this research because, after taking two years of topology, I find it fascinating,” Sack says. “Previous research I’ve done centered on graph coloring. I can use some of the intuition I built around graph coloring to help better understand knot coloring.”

*This story was modified from a story appearing June 14th by Maureen Rohi.*

**Atlanta, GA**

The National Science Foundation (NSF) has awarded a Research Training Groups (RTG) grant to the Georgia Tech Geometry and Topology (GTGT) group. GTGT will use the $2.1 million grant over five years to train undergraduates, graduate students, and postdoctoral fellows. The GTGT project supports NSF’s long-range goal to increase the number of U.S. citizens, nationals, and permanent residents pursuing careers in mathematics.

School of Mathematics faculty members Igor Belegradek, John Etnyre, Stavros Garoufalidis, Mohammad Ghomi, Jennifer Hom, Thang Le, Dan Margalit, and Kirsten Wickelgren make up GTGT and are co-principal investigators of the grant.

**Why Study Topology and Geometry **

Etnyre answers this question. He explains:

“Topology is the study of spaces. They can be the space we live in or configurations of mechanical systems. Mathematicians also consider spaces of solutions to algebraic equations and partial differential equations, as well as even more abstract space.

“More specifically topology is the study of spaces where some notion of continuity makes sense. What are these spaces? How can we distinguish one space from another? What interesting properties do specific spaces have? These are the basics questions in topology, whose language pervades much of mathematics, science, and engineering.

“Geometry is, loosely speaking, the study of some kind of structure on a space. Riemannian geometry involves spaces on which you can measure lengths of vectors and the angles in between. Symplectic geometry allows one to study dynamical systems akin to classical mechanics on a space.

“Topology and geometry underlie a great deal of science and engineering. Whether trying to understand general relativity and the structure of the universe, design robust sensor networks, unravel DNA recombination, develop string theory, or countless other endeavors, the underlying language and ideas are likely to be that of geometry and topology.”

“Topology and geometry underlie a great deal of science and engineering. Whether trying to understand general relativity and the structure of the universe, design robust sensor networks, unravel DNA recombination, develop string theory, or countless other endeavors, the underlying language and ideas are likely to be that of geometry and topology.”

**Expected Outcomes **

Over its five-year run, the grant will enable the training of 60 undergraduate students, 22 graduate students, and 14 postdoctoral fellows. Supplementary funding from the College of Sciences will ensure three years of support for all postdoctoral fellows.

Etnyre says GTGT will leverage its access to Georgia Tech’s engineering programs to spark collaborations between engineers and mathematicians. Similarly, GTGT will use its proximity to institutions serving groups underrepresented in mathematics to help increase the representation of minorities and women in advanced mathematics.

Ultimately, Etnyre says, “we aim to develop students and postdoctoral fellows who are well-rounded scholars, accomplished teachers, and valuable members of the mathematics community.”

**Areas of Expertise**

The GTGT group is strong in various fields:

- Algebraic Topology: Kirsten Wickelgren
- Contact and Symplectic Topology: John Etnyre
- Geometric Group Theory: Igor Belegradek and Dan Margalit
- Global Riemannian and Differential Geometry: Igor Belegradek, John Etnyre, and Mohammad Ghomi
- Heegard-Floer Theory: John Etnyre and Jennifer Hom
- Low-Dimensional Topology: John Etnyre, Stavros Garoufalidis, Jennifer Hom, Thang Le, and Dan Margalit
- Quantum Topology: Stavros Garoufalidis and Thang Le
- Riemannian Geometry of Submaniforlds: Mohammad Ghomi

All these areas would benefit from the grant.

“We aim to develop students and postdoctoral fellows who are well-rounded scholars, accomplished teachers, and valuable members of the mathematics community.”

**Grant-Enabled Activities**

The grant enables the GTGT group to embark on several major activities:

- Expand the group by supporting graduate and postdoctoral fellowships
- Enhance educational opportunities for all students through new courses, expanded seminars and REU (Research Experiences for Undergraduates) opportunities, and a direct-reading program for undergraduates
- Firmly establish the annual Georgia Tech Topology Conference and the biennial Topology Students Workshop, continue the Southeastern Undergraduate Mathematics Workshop, and initiate the Georgia Tech Topology Summer School
- Strengthen professional development components of graduate and postdoctoral training
- Increase interaction with colleges and universities serving groups that are underrepresented in mathematics and expand outreach to precollege students
- Create a website to serve as repository of resources