Among the Ph.D. students who worked or are working with me in this area are Nik Chatzikonstantinou (current Ph.D. student at UR), Bochen Liu (postdoctoral researcher, Chinese University in Hong-Kong), Steven Senger (Assistant Professor at the University of Missouri), Belmiro da Silva (current Ph.D. student at UR), Krystal Taylor (Assistant Professor, Ohio State), Brianna Vick (current Ph.D. student at UR).

Another option is a set of similar problems in vector spaces over finite fields or in modules over commutative rings. While there is some overlap between the techniques involved in the two disciplines, the problems in the finite field setting have a non-trivial arithmetic component which frequently gives the problems a distinct flavor. In this setting we ask how large a subset of the d-dimensional vector space over a finite field needs to be to ensure that it contains vertices of an equilateral simplex or another geometric structure. Another key question is the sum-product question which asks for the smallest possible size of minimum of the sum set and the product set of a subset of a given finite field.

Among the Ph.D. students who worked with me in this area Esen Aksoy (postdoctoral research at the University of Ankara), Philipp Birklbauer (current Ph.D. student at UR), Jeremy Chapman (Associate Professor at the Lyons College), David Covert (Assistant Professor at the University of Missouri-St. Louis), Doowon Koh (Associate Professor at Chungbuk University), Alex McDonald (current Ph.D. student at UR), Brendan Murphy (postdoctoral researcher at the University of Bristol), Steven Senger (Assistant Professor at Missouri State University), Le Anh Vinh (Professor at the University of Hanoi).

In the discrete setting, the key problem is the Erdos distance conjecture, which asks for the minimal number of the pairwise distances determined by the elements of a finite subset of a d-dimensional vector space over the real numbers. This conjecture was resolved in two dimension in 2011 by Larry Guth and Nets Katz, but the higher dimensional case remains wide open. The sum-product phenomenon described above is very much alive and well in this setting as well.

Among the Ph.D. students who worked or are working with me in this area are Steven Senger (Associate Professor at Missouri State), Jonathan Passant (current Ph.D. student at UR) and Firdavs Rakhmonov (current Ph.D. student at UR).

Last, but not least, there is also an option of working with me on problems in applied data science. While I have not yet advised a student in this area, I have several interesting problems in clustering and dimension detection of large data sets that I would like to explore with a student in the near future.

The complete list of my current and former Ph.D. students can be found in my CV. You can find links to all of my publicans, both recent and not so recent, as well as the list of the invited lectures I have given over the years in my CV as well.

On Falconer distance set problem in the plane, with Guth, Ou and Wang, submitted for publication

On Gabor orthogonal bases and convexity, with Mayeli, published in Discrete Analysis

Rigidity, graphs and Hausdorff dimension, with Chatzikonstantinou, Mkrtchyan and Pakianathan,

Equilateral triangles in subsets of Euclidean space of large Hausdorff dimension, with Bochen Liu, accepted for publication in Israel Math Journal

Finite chains inside thin subsets of Euclidean space, with Bennett and Taylor, published in Analysis and PDE

A group theoretic viewpoint on the Erdos-Falconer problems and the Mattila integral, with Greenleaf, Liu and Palsson, published in Revista Iberoamericana

There is a number of other recent papers in this direction, but these will an idea of what is going on.

On restriction estimates for spheres in finite fields, with Koh, Lee, Pham and Shen, submitted for publication

A new bound for the Erdos distinct distances problem in a plane over a finite field, with Koh, Pham, Shen and Vinh, submitted for publication

On a quotient set of a distance set, with Koh and Parshall, submitted for publication

The Fuglede conjecture in vector spaces over finite fields, with Mayeli and Pakianathan, published in Analysis and PDE

Group action and combinatorics and vector spaces over finite fields, with Bennett, Hart, Pakianathan and Rudnev, published in Forum Mathematicum

Erdos distance problem in vector spaces over finite fields, with Rudnev, published in Transactions of the AMS

On discrete values of bilinear forms, with Roche-Newton and Rudnev, published in the Moscow Journal of Number Theory and Combinatorics

Finite point configurations in the plane, rigidity and Erdos problems, with Passant, to appear in the Steklov Institute Proceedings in honor of Konyagin's 60th birthday

A multi-parameter variant of the Erdos distance problem, with Janczak and Passant, submitted for publication

On the unit distance problem, to appear in the conference proceedings of the CIMPA 2017 conference in Buenos-Aires

It is important to note that a substantial number of the papers mentioned above and other papers that I have written can be put in multiple categories. If you become my student, you will hear exclamations about the unity of mathematics all the time!