The basic question there is, how large does a set in Euclidean space (or a Riemannian manifold) need to be to ensure that it contains vertices of a geometric configuration such as an equilateral triangle, an angle of a given size, or a chain of a given length. A combination of Fourier analytic and combinatorial methods play an important role. The key problem in this area is the Falconer distance conjecture which says that if the Hausdorff dimension of a compact set is half the ambient dimension, then the Lebesgue measure of the set of pairwise distances is positive.

Among the Ph.D. students who worked or are working with me in this area are Shengze Duan (current graduate student), Quy Pham (current graduate student), Donggeun Ryou (current graduate student), Brian McDonald (former graduate student and currently a postdoc at the University of Georgia), Alex McDonald (former graduate student and currently a postdoctoral researcher at Ohio State University), Nik Chatzikonstantinou (former Ph.D. student at UR and currently a postdoc in Okinawa), Bochen Liu (former Ph.D. student and currently an Associate Professor at Southern Chinese University), Steven Senger (Associate Professor at the University of Missouri), Belmiro da Silva (instructor at Rensselaer Polytechnic), Krystal Taylor (Associate Professor, Ohio State), Brianna Vick (Assistant Professor at Clayton State University).

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 are Esen Aksoy (postdoctoral research at the University of Ankara), Philipp Birklbauer (working in industry), Jeremy Chapman (former graduate student and currently a Professor at the Lyons College), David Covert (former graduate student and current an Assistant Professor at the University of Missouri-St. Louis), Doowon Koh (Associate Professor at Chungbuk University), Alex McDonald (postdoc at Ohio State), Brendan Murphy (former graduate student and currently a postdoctoral researcher at the University of Bristol), Steven Senger (Associate Professor at Missouri State University), Le Anh Vinh (Professor at the University of Hanoi).

This problem 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 (Heilbronn postdoc at the University of Bristol) 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 data science, ranging from completely applied to theoretical. I also work on applying some ideas from Learning Theory, the theoretical framework behind Data science, to analysis and combinatorics. In the applied data science realm, I am working on concrete sampling bounds for time series that would guarantee, with high probability, a forecast for a given time series can be given with high probability. I have also co-authored a series of paper recently studying the Vapnik-Chervonenkis dimension (one the key concepts of Learning Theory) of families of indicator functions of spheres centered at point of subsets of a given size of vector spaces over finite fields. This yields an interesting new angle on the finite field configuration problems mentioned above. Among the graduate students who have worked in this direction are Brian McDonald (a former graduate student and currently a postdoctoral researcher at the University of Georgia), Shashank Chorge (former graduate student and currently a postdoctoral research at IIT, India), and Pablo Bhowmick (current graduate student).

The complete list of my current and former Ph.D. students can be found here. 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, published in Inventiones.

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, published 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 give you 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, published in Moscow Journal of Combinatorics and Number Theory.

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, published in the Steklov Institute Proceedings in honor of Konyagin's 60th birthday

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

The VC-dimension and point configurations in two-dimensional vector spaces over finite fields, with Fitzpatrick, McDonald and Wyman, to appear in Discrete and Computational Geometry

Fractal dimension, approximation and data sets, with Betti, Chio, Fleischman, Iulianelli, Kirila, Martino, Mayeli, Pack, Sheng, Taliancic, Thomas, Whybra, Wyman, Yildirim, and Zhao, submitted for publication

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!