Vertical Integration of Research

tree of knowledge


This page is dedicated to an ongoing project at the University of Rochester to integrate research activities involving undergraduate and graduate students, postdoctoral researchers and faculty in a unified laboratory setting. The fundamental idea behind this project is to create a research environment where graduate student, undergraduate students and postdoctoral researchers are integrated into a common structure where ideas are shared, research projects are created, and a constant effort is made to integrate the research direction in a seamless symbiosis.

The program is also going to have a high school component. We are going to train advanced local high school students to think about problems in elementary mathematics from a broader point of view. We are going to create research projects for them that are going to slowly increase in technical difficulty and sophistication, ultimately leading to work on open problems.



Faculty members of the lab: Alex Iosevich (UR), Steven Kleene (UR), Azita Mayeli (CUNY)

Postdoctoral members of the lab: Matthew Dannenberg and Nerraja Kulkarni

Graduate student members of the lab: Pablo Bhowmik, Will Burstein, Shantanu Deodhar, Shengze Duan, Nathaneal Grand, Anirudh Gurjale, Tianxiao Hu, Nathaniel Kingsbury (CUNY), Zhihe Li, Sreedev Mamikoth, Hari Nathan, Svetlana Pack (Penn State), Quy Pham, Nathan Skerrett, Ella Yu.

Undergraduate members of the lab: Karam Aldaleh, Nicholas Arnold, Abdelwahab ElNaggar, Noah Ernst, William Hagerstrom, Gabe Hart, Giovanni Garza, Karina Gurevich, Josh Iosevich (RIT), Alhussein Khalil, Anuraag Kumar, Kelvin Nguyen, Aidan Rohrbach, Nate Shaffer, Lily Stolberg, Zachary Tan, June Terzioglu,Terrance Wong

High school members of the lab: (to be determined)

Meeting times Fall 2024: Fridays starting late morning

Vertical Integration Weekend Workshop: December, 2024 - exact time and place to be announced



Current projects:

i) Signal recovery

Let f:ℤNd→ℂf: {\mathbb Z}_N^d \to {\mathbb C} be a signal, and let f^(m)=N−d2∑x∊ℤNdχ(−x⋅m)f(x)\widehat{f}(m)=N^{-\frac{d}{2}} \sum_{x \in {\mathbb Z}_N^d} \chi(-x \cdot m) f(x)denote its Fourier transform, where χ(t)=e2πitN\chi(t)=e^{\frac{2 \pi i t}{N}}
is the standard character on ℤN.{\mathbb Z}_N.The question we ask is, if {f^(m)}m∊S{\{\widehat{f}(m)\}}_{m \in S}
unobserved due to to noise or other interference where S is a subset of  ℤNd, then under what reasonable assumptions can we recover the original signal f exactly and uniquely?
{\mathbb Z}_N^d.

ii) Erdos distance problem on manifolds

Let E be a finite subset of a compact two-dimensional Riemannian manifold M without a boundary. Let Δ(E)={ρ(x,y):x,y∊E}\Delta(E)=\{\rho(x,y): x,y \in E\}, where ρ\rho is the Riemannian metric on M. The question we ask is, what is the smallest possible size of Δ(E) \Delta(E)                           if|E|=n? if |E|=n?                    

In Euclidean space it is known (Katz and Tao 2011) that |Δ(E)|≥nlog(n).|\Delta(E)| \ge \frac{n}{\log(n)}. On manifolds, the bound |Δ(E)|≥cn|\Delta(E)| \ge c \sqrt{n}     
was established by Nathan Skerrett in his honors thesis. The bound |Δ(E)|≥cn23|\Delta(E)| \ge cn^{\frac{2}{3}} 
was established during StemForAll2024 by Alex Iosevich, Steven Kleene, Nate Shaffer, Nathan Skerrett, Lilian Stolberg, and June Terziooglu.

We are now working on improving this bound further.