# Summer Math Research Program

This is the website for the Northeastern Summer Math Research Program (NSMRP), a paid summer program funded by the math department's Research Training Grant.

**Other undergraduate math research opportunities at Northeastern.**

## Research Experience for Undergraduates

## REU 2022

**In light of the COVID-19 pandemic, this program will be held remotely.**

## Calendar

**Mon May****9****,****9:30****AM: Kick Off Meeting****Wed****May****18****,****3-4****P****M: Colloquium****Wed May****25****, 3-4 PM: Colloquium****Fri****June****3****, 10 AM: Preliminary Presentations (20 min/group)****Wed****June 8****, 3-4 PM: Colloquium****Wed****June 1****5, 3-4 PM: Colloquium****Fri****Jun****24****,****9****AM: Final Presentations (45 min/group)****Wed****July 6: Final Paper Due**

**Final Paper:**The jointly written paper should be at least 10 pages. Should include an introduction with motivation and guiding questions, definitions, examples, a main result and proof. No maximum page limit, but writing should be clear and concise.**Final Presentation:**(45 min for each group) Should introduce audience to area of research, convince them it is interesting through examples, definitions and results. Time is too short to include everything so choose carefully what to present highlighting what you worked hardest on. Narrative and slides and/or boardwork should be clear.

## Research projects and mentors

### Equivalent matroids arising from graphs

**Participants:**Jiaqi Lu and Brandon Onyejekwe

**Mentor:**K. Finn Prideaux

**Paper:**Equivalent matroids arising from graphs

**Final presentation slides**

### Morita theorems

**Participants: **Nicholas Bidorini and Zach Greenfield

**Mentor:** Vitor Emanuel Gulisz

**Paper: **The Morita theorems

### Regularization and renormalization in quantum field theory

**Participants:** Daniel Abadjiev, Anthony Melo, and Zhengxun Liu

**Mentor: **M. Anadil Saeed Rao

**Paper:** Renormalization and regularization of φ^4 and φ^3 scalar quantum field theories

### Graph bootstrap percolation

**Participants:** Andy Babb, Stephanie Martinez, and Timothy Wang

**Mentor:** Connor Anderson

**Paper:** Approximation of spread minimization in bootstrap percolation

### Knot polynomials and Khovanov homology

**Participants: **Jordan Martino and Francisco Turdera

**Mentor: **Ryan Kannanaikal

**Post-doc coordinators: **Iva Halacheva, Matej Penciak, Josh Wen

**Faculty organizer: **Valerio Toledano Laredo

## REU 2021

**In light of the COVID-19 shutdown, this program was held entirely remotely. **

Additional funding for the Summer 2021 program was provided by the College of Science and Department of Mathematics.

## Calendar

**Mon May 10, 9 AM: Kick Off Meeting****Thu May 20, 10-11 AM: Colloquium****Thu June 3, 10 AM: Preliminary Presentation (20 min/group)****Thu June 17, 10 AM: Colloquium****Tue Jun 29, 8 AM: Final Presentations (45 min/group)****Tues July 6: Final Paper Due**

**Final Paper: **The jointly written paper should be at least 10 pages. Should include an introduction with motivation and guiding questions, definitions, examples, a main result and proof. No maximum page limit, but writing should be clear and concise.

**Final Presentation: **(45 for each group) Should introduce audience to area of research, convince them it is interesting through examples, definitions and results. Time is too short to include everything so choose carefully what to present highlighting what you worked hardest on. Narrative and slides and/or boardwork should be clear.

## Groups

### Configuration spaces of points and lines

**Participants**: Luke Boyer, Shane Calle, Nick Payne. **Mentor**: Ian Dumais

**Paper:** Configurations of Lines and Spaces

### Probabilistic simulations of IOTA cryptocurrency

**Participants**: Maria Candello, Grant Chau, Jiachen Xu. **Mentor**: Anupam Kumar

**Paper:** Probabilistic Simulations of IOTA Cryptocurrency

### Modeling of random geometric graphs

**Participants**: Maitreyee Joshi, Nicholas Thevenin, Bryan Vogt. **Mentor**: Hiu Ying Man

**Paper:** Extending Bottleneck Matching and Monotone Property Threshold Width to the Torus

### Chromatic numbers of fractional power graphs

**Participants**: Luke Drennen, Noah Lichtblau. **Mentor**: Finn Prideaux

**Paper:** Chromatic Numbers and K-Connectivity of Graph Fractional Powers

**Post-doc coordinators: **Vance Blankers, Matej Penciak

**Faculty organizer: **Valerio Toledano Laredo

## REU 2020

**In light of the COVID-19 shutdown, this program was held entirely remotely. **

## Calendar

**Mon May 4, 9AM: Kick Off Meeting****Mon May 18, 2PM: Preliminary Presentation (20 min/group)****Wed May 27, 2-3PM: Colloquium****Wed June 17, 2-3PM: Colloquium****Thr Jun 25, 2PM: Final Presentations (45 min/group)****Fri July 3: Final Paper Due**

**Final Paper: **The jointly written paper should be at least 10 pages. Should include an introduction with motivation and guiding questions, definitions, examples, a main result and proof. No maximum page limit, but writing should be clear and concise.

**Final Presentation: **(45 for each group). Should introduce audience to area of research, convince them it is interesting through examples, definitions and results. Time is too short to include everything so choose carefully what to present highlighting what you worked hardest on. Narrative and slides and/or boardwork should be clear.

## Groups

### Evolutionary Dynamics.

**Participants**: Jack Steilberg, Lauren Neudorf. **Mentor**: Jonier Antunes.

**Paper: ***Infinite Random Graphs and Evolutionary Dynamics*

### Spectral Graph Theory.

**Participants**: Ryan Keleti, Noble Mushtak. **Mentor**: Whitney Drazen.

**Paper: ***Fractional Revival and Fractional Cospectrality*

### Elliptic Curves.

**Participants:** Zoe Daunt, Xiaoying He, Xuyang Li. **Mentor: **Lei Yang.

Paper: *Elliptic Curves and Probability of $\ell$-torsion*

**Post-doc coordinators: **Rob Silversmith, Robin Walters

**Faculty organizer: **Valerio Toledano Laredo

## REU 2019

## Calendar

**May 6: First Official REU Meeting (Summer I)****May 13: Proposal Draft****May 28: REU Tea & Colloquium 3-4 PM, Lake 509****June 4: REU Tea & Group Presentations 3-4 PM, Nightingale 544****June 11: REU Tea & Colloquium 3-4 PM, Lake 509****June 21: Final Presentations, Lunch 12-1, Talks 1-4, Lake 509 (Summer I)****June 28: Final Paper Due (Summer I)****August 14: Final Presentations (Summer II)**

**Final Paper: **At least 5 pages. If two students, then paper is written together is at least 10 pages. Should include an introduction with motivation and guiding questions, definitions, examples, a main result and proof. No maximum page limit, but writing should be clear and concise.

**Final Presentation: **30 minutes (40 if group of 2). Should introduce audience to area of research, convince them it is interesting through examples, definitions and results. Time is too short to include everything so choose carefully what to present highlighting what you worked hardest on. Narrative and slides and/or boardwork should be clear.

## Groups

### Evolutionary Dynamics.

**Participants**: Hoyin Chu. **Mentor**: Jonier Antunes. Summer I.

**Paper: ***Introduction to Evolutionary Dynamics and Stochastic Calculus.*

### Knot Theory.

**Participants**: Noah Fleischmann, Kally Lyonnais. **Mentor**: Dmytro Matvieievsky. Summer I.

**Paper: ***Knots, Colors, and Polynomials***.**

### Graph Theory/Chip Firing:

**Participants:** Kristin Timothy, Michael Wang. **Mentor: **Ian Dumais. Summer I.

Paper: *Sandpiles and Chip Firing.*

### Homotopical Algebra:

**Participants:** Karthik Boyareddygari **Mentor: **Oleksiy Sorokin. Summer I.

**Paper: ***Homotopical Approach to Tensor Products.*

### Algebraic Geometry:

**Participants:** Walker Miller-Breetz. **Mentor: **Anupam Kumar. Summer II.

Paper: *Cone Points on Algebraic Curves*.

**Post-doc coordinators: **Emily Barnard, Rob Silversmith, Robin Walters

**Faculty organizer: **Valerio Toledano Laredo

## REU 2018

The inaugural year of the Northeastern REU had 9 undergraduate participants working in several different areas. The postdoc coordinator was Ivan Martino.

Student:

**Alon Duvall**

Mentor: Brian Hepler

Title:* **Abstract Regular Polytope*

Abstract: Introduction to Abstract Regular Polytopes through a combinatorial geometry approach. C-Groups, Coxeter Groups and 2^k-constructions are also studied.

Student:

**Kevin Su**

Mentor: Jonier Antunes

Title: *Graph Limits and Extremal Combinatorics*

Abstract: "One of the problems in extremal combinatorics is asking how many edges a graph on n nodes may have while avoiding some speciﬁc subgraphs. For example, Mantel’s theorem says that the most edges a graph on n nodes can have while avoiding K_3 is n^2/4. These statements can also be represented in terms of inequalities of numbers of homomorphisms or in terms of homomorphism densities. Many of these inequalities may be proved using Cauchy-Schwarz (as an inequality on sums of squares) or pictorially due to a gluing algebra on the space of graphs. We summarize some of the graph algebra background involved here and look at how graph limit theory gives a way of proving results in extremal combinatorics."

Student:

**Timothy Jackman**

Mentor: Whitney Drazen

Title: *Graph Theory*

Abstract: This is an introduction to spectral graph theory until the concepts of (quantum) state transfer.

Student:

**Felipe Castellano-Macias**

Mentor: Alex Sorokin

Title: *Leavitt Path Algebras** *

Abstract. We describe diﬀerent properties of Leavitt path algebras of a graph over a ﬁeld, providing the appropriate background. In addition, we investigate similar results about Leavitt path algebras of a graph over a commutative unital ring. We will mainly explore the connection between the diagonalizability of matrices over a given Leavitt path algebra and the structure of its underlying graph, as well as the classiﬁcation of such algebras up to isomorphism.

Name:

**Benjamin Bonenfant**

Mentor: Reuven Hodges

Student:

**Walker Miller-Breetz**

Mentor: Rahul Singh

Title: Young Tableaux

Abstract: After a short introduction on group theory and group actions, the work focuses on the symmetric group and its action on flagged spaces.

Student:

**Christina Nguyen**

Mentor: Whitney Drazen

Title: *Non-backtracking Walks on Graphs*

Abstract: The main goal of the REU was to obtain a new proof of the non-backtracking version of Ploya’s Theorem for random walks. We began with a discussion of spectral graph theory before studying the backtracking version of Ploya’s Theorem.

Student:

**Noah Lichtblau**

Mentor: Mikhail Mironov

Title: *On the Morphism of Schemes*

Student:

**Zheying Yu**

Mentor: Celine Bonandrini

Title:* **Topology*

Faculty Organizer:

**Emanuele Macri**

*Tiling Viewer*, Robert Webb's

*Stella*