The Meta Home Paige

Geometric Measure Theory and Fourier Analysis

7. Spread Furstenberg Sets, with Manik Dhar

   We study a variant of the Furstenberg set problem for high dimensional flats in Euclidean space. The "standard" Furstenberg set problem asks the following: Given $X\subset {\mathbb{R}}^n$ Borel, suppose there exists a nonempty $t$-dimensional set of affine $k$-planes $\mathcal{P}$ such that for each $P\in \mathcal{P}$, $\dim (X\cap P)\ge s$. How large must $\dim X$ be in terms of $s,t,n,$ and $k$? Such sets are referred to as $(s,t;k)$-Furstenberg sets. The variant we study involves $(s,t;k)$-spread Furstenberg sets, defined as follows: Given $X\subset {\mathbb{R}}^n$ Borel, suppose there exists a nonempty $t$-dimensional set of $k$-dimensional subspaces $\mathcal{P}$ such that for each $P\in \mathcal{P}$ there exists a translation vector $a_P\in {\mathbb{R}}^n$ such that $\dim (X\cap (P+a_P))\ge s$. How large must $\dim X$ be in terms of $s,t,n,$ and $k$? The key difference here is that the set of $k$-dimensional flats are "spread out" in different directions (by assuming they come from translatins of subspaces). We find lower bounds for $(s,t;k)$-spread Furstenberg sets over Euclidean space, following the methodology of Dhar-Dvir-Lund over finite fields. This also raises interesting questions regarding the relationship between the original problem and this given variant discussed within the paper.

• [arXiv:2412.18193] [Slides: Furstenberg Survey]
8. Pinned Dot Product Set Estimates, with Caleb Marshall and Steven Senger

   We study a variant of the Falconer distance problem for dot products. In particular, given $A\subset {\mathbb{R}}^n$ Borel, and $a,x\in {\mathbb{R}}^n$, we study sets of the form ${\mathrm{\Pi }}_x^a(A):=\{(a-x)\cdot y:y\in A\}$ where $\cdot$ here is the standard dot product over ${\mathbb{R}}^n$. We are able to show that for $\dim A$ sufficiently large, ${\mathrm{\Pi }}_x^a(A)$ must be large for many $a\in A$, where here "largeness" is in terms of positive measure, large Hausdorff dimension, or non-empty interior. Interestingly, our approach for all three senses of the word "large" is identical, making use of exceptional set estimates (for orthogonal projections) and radial projection estimates (both classical and recent).

• [arXiv:2412.17985] [Slides: Dot Products]
9. A continuum Erdős-Beck theorem, with Caleb Marshall

   Let $X\subset {\mathbb{R}}^n$ be a Borel set, and consider the set of lines, $\mathcal{L}(X)\subset \mathcal{A}(n,1)$, containing at least 2 points in $X$. In recent work of Orponen-Shmerkin-Wang and Ren, radial projection estimates and (dual) Furstenberg set bounds have been used to lowerbound $\dim \mathcal{L}(X)$ in terms of $\dim X$. Such work is seen as a continuum version of Beck's theorem from point-line incidence geometry. Utilizing results of B.-Fu-Ren, together with Caleb Marshall, we find a continuum Erdős-Beck theorem. We motivate such work via their discrete counterparts, and discuss a new conjecture regarding the dimension of line sets.

• [arXiv:2406.10058] [Slides: Erdős-Beck] [YouTube Video]
10. Radial projections in ${\mathbb{R}}^n$ revisited, with Yuqiu Fu and Kevin Ren [arXiv:2406.09707]

    We generalize recent work of Orponen-Shmerkin-Wang using two different methods. In particular, we show the following bilinear radial projection estimates: Given $X,Y\subset {\mathbb{R}}^n$ Borel sets and $X\ne 0$. If $\dim Y\in (k,k+1]$ for some $k\in \{1,\dots ,n-1\}$, then $$\underset{x\in X}{sup}\dim {\pi }_x(Y\setminus \{x\})\ge min\{\dim X+\dim Y-k,k\}.$$ The first method is shorter, and proves the equivalence between this result and a result of B.-Gan/OSW. Our second method, though longer, follows the original methodology of Orponen-Shmerkin-Wang, and requires a higher dimensional incidence estimate and a dual Furstenberg set estimate for lines. These new estimates may be of independent interest.

11. A study guide to "Kaufman and Falconer estimates for radial projections", with Ryan Bushling, Caleb Marshall, and Alex Ortiz [arXiv:2402.11847]

    This study guide on recent work of Orponen-Shmerkin-Wang was written during the UPenn Study Guide Writing Workshop 2023 under the mentorship of Josh Zahl. We expound upon the major themes and clarify technical details of the paper "Kaufman and Falconer estimates for radial projections and a continuum version of Beck's theorem" of Orponen-Shmerkin-Wang.

12. Exceptional set estimates in finite fields, with Shengwen Gan [arXiv:2302.13193]

    We study exceptional set estimates for orthogonal projections over finite fields, ${\mathbb{F}}_q^n$. In particular, given $A\subset {\mathbb{F}}_q^n$ with $|A|=q^a$, and $0<s<min(k,a)$, we study sets of the form $$E(A):=\{V\in G(k,{\mathbb{F}}_q^n):\mathrm{\#}{\pi }_V(A)<q^s\}.$$ Such sets are known as exceptional set estimates. We also show that the bound $|E(A)|\lesssim p^{2s-a}$, conjectured to be true for all $A\subset {\mathbb{F}}_p^n$ with $|A|=q^a$, is sharp (if $p=q$ is prime).

13. Exceptional set estimates for radial projections in ${\mathbb{R}}^n$, with Shengwen Gan. Annales Fennici Mathematici, 49(2), 631-661, 2024.

    • [Annales Fennici Mathematici] [arXiv:2208.03597] [Slides: Orthogonal Projections] [Slides: Radial Projections]

    This research was conducted during SPUR (the Summer Program for Undergraduate Research in mathematics) 2022 at MIT, and was published in Annales Fennici Mathematici in 2024.
    
    We study exceptional set estimates for radial and orthogonal projections over Euclidean space, ${\mathbb{R}}^n$. Given a Borel set $A\subset {\mathbb{R}}^n$, with $\dim A=a$, how often is the shadow of $A$ onto a $m$-dimensional subspace, $V\in G(n,m)$ large? Marstrand's projection theorem, which states that for almost every $V\in G(n,m)$, $\dim {\pi }_V(A)=min\{a,m\}$. Exceptional set estimates try to quantify this statement further. In particular, given $0<s\le min\{a,m\}$, how can we bound $$\dim \{V\in G(n,m):\dim {\pi }_V(A)<s\}?$$ Two such bounds were proven by Kaufman and Falconer. The first half of this SPUR project involved reproving the results of Kaufman and Falconer via delta-discretization, a high-low argument from Fourier analysis, and a counting argument.
    
    One can study similar types of exceptional set estimates for radial projections. Utilizing a similar high-low argument from the first half of the project, we proved two open conjectures by Lund-Pham-Thu and Liu. The statements of said radial projection conjectures were first proven in ${\mathbb{R}}^2$ by Orponen-Shmerkin, and we generalized these statements to ${\mathbb{R}}^n$ following a similar framework. Notably, using different methods, Orponen-Shmerkin-Wang proved the radial projection statements proven in this paper shortly after this paper was released.
    
    See slides linked above to see more sources/statements regarding exceptional set estimates for orthogonal and radial projections.

Combinatorics

3. On a radial projection conjecture in ${\mathbb{F}}_q^d$, with Ben Lund and Thang Pham [arXiv:2311.05127]

   The work of B.-Gan (see above) was motivated by radial projection conjectures due to Lund-Pham-Thu and Liu. My initial work with Shengwen Gan proved the conjectures over Euclidean space, while Lund-Pham-Thu's conjecture was initially stated over finite fields. Hence, in this work with Ben Lund and Thang Pham, we prove the finite field radial projection conjectures.

4. Generalized point confiruations in ${\mathbb{F}}_q^d$, with X. Fang, B. Heritage, A. Iosevich, T. Jiang, H. Parshall, M. Sun [Finite Fields and Their Applications, 2024] [arXiv:2308.10853]

   This research was conducted during Williams College's SMALL REU 2023 (Research Experience for Undergraduates) under the mentorship of Alex Iosevich and Eyvi Palsson.
   
   One can study point configuration problems over finite fields. See the explanation in the next bullet point for further detail. In the previous paper on embedding certain configurations (see below), we studied embeddings with a "dot product" distance. In this work, we generalize the statements to distances defined via a general non-degenerate bilinear form or quadratic form.

5. Improved bounds for embedding certain configurations in subsets of vector spaces over finite fields, with X. Fang, B. Heritage, A. Iosevich, M. Sun [arXiv:2308.09215]

   This research was conducted during Williams College's SMALL REU 2023 (Research Experience for Undergraduates) under the mentorship of Alex Iosevich and Eyvi Palsson.
   
   One can study point configuration problems over finite fields in the following sense. Let $G$ be a connected graph, and assign weights to each edge in ${\mathbb{F}}_q$. We call such graphs distance graphs. We study the following problem: Given $E\subset {\mathbb{F}}_q^d$, how large does $|E|$ need to be until it contains at least one isometric copy of the distance graph $G$? We study this problem for specific types of distance graphs, with a "distance" defined over ${\mathbb{F}}_q^n$ being (roughly) given by the standard dot product mod $q$. This builds upon work of Iosevich-Parshall.

Mathematics Education and Pedagogy

5. "18.S063: Matrix Calculus" Lecture Notes, taught by Profs. Alan Edelman and Steven G. Johnson [OCW Course] [arXiv:2501.13787]

   I typed lecture notes for the Matrix Calculus class (2023) taught by Alan Edelman and Steven G. Johnson. This course discusses how the ideas from multivariable calculus generalizes to matrices.

6. "18.S190: Introduction to Metric Spaces", taught IAP 2022 and 2023 [OCW Course] [Blogpost by Peter Chipman]

   I created and taught a bridge class between two different real analysis courses at MIT. The first, 18.100A/18.100P, focusses on real analysis over Euclidean sapces ${\mathbb{R}}^n$, and the latter, 18.100B/18.100Q, focusses on real analysis over metric spaces.
   
   I obtained the MIT Mathematics Department Teaching and Learning Award for this work, and the course's material is hosted on OpenCourseWare.

7. "Communication is the Whole Game", Chalk Radio interview with Prof. Haynes Miller [Podcast]

   I guest hosted for OCW's podcast Chalk Radio. In this podcast, I interview Haynes Miller about communication and mathematics. I created and taught a bridge class between two different real analysis courses at MIT.

8. "18.100A: Real Analysis" Lecture Notes, taught by Prof. Casey Rodriguez [OCW Course]

   Typed lecture notes for 18.100A on OCW, taught by Casey Rodriguez. The course covered real analysis over Euclidean spaces.

9. "When Students Create OER...", with Ashay Athalye, Sarah Hansen, and Curt Newton [YouTube Video]

   A discussion on student contributions to open educational resources (OERs), in particular with respect to OpenCourseWare (OCW).

• 3/3--3/7/25: SLMath: "Hot Topics: Interactions between Harmonic Analysis, Homogeneous Dynamics, and Number Theory"
• 5/16--5/18/25: AWM Research Symposium: "Nonlinear Constraints: A Catalyst for Creativity in Analysis and its Applications"
• Co-organizing this special session with Marjorie Drake and Vinh Nguyen. Please reach out if you'd like to participate!

• 1/11/25: JMM: SLMath (MSRI) Special Session on "At the Intersection of Harmonic Analysis and Fractal Geometry"
• "Pinned Dot Product Sets," joint work with Caleb Marshall and Steven Senger
• 12/1/24: CMS: Scientific Session on "Incidence Problems in Analysis"
• "Dual Furstenberg Sets," joint work with Yuqiu Fu and Kevin Ren
• 11/30/24: CMS: Scientific Session on "Harmonic Analysis and Geometric Measure Theory"
• "A Continuum Erdős-Beck Theorem," joint work with Caleb Marshall, and a brief discussion of ongoing work with Alex Ortiz and Dmitrii Zakharov
• 10/23/24: UBC: Harmonic Analysis and Fractal Geometry Seminar
• "Continuum Beck-Type Problems," joint work with Caleb Marshall, and a brief discussion of ongoing work with Alex Ortiz and Dmitrii Zakharov
• 8/28/24: Harmonic Analysis People's Presentations on YouTube's (HAPPY's) "Hello, World!" Series
• "A continuum Erdős-Beck theorem," joint work with Caleb Marshall [Slides: Erdős-Beck] [YouTube Video]
• 1/4/24: JMM: AWM Special Session on "Recent Developments in Harmonic Analysis"
• "Recent Developments in Radial Projections" [Slides: Radial Projections]
• 1/3/24: JMM: AMS Special Session on "Harmonic Analysis, Geometric Measure Theory, and Fractals"
• "Exceptional Set Estimates for Orthogonal Projections" [Slides: Orthogonal Projections]
• 11/28/23 and 12/5/23: Graduate Lecture Series in Analysis and PDEs at Brown
• 11/28: "From Discrete Geometry to Geometric Measure Theory"
• 12/5: "Exceptional Set Estimates"
• 11/21/23: Online Undergraduate Research Seminar at UNC with Alex Ortiz
• "Incidences and Projections: Discrete and Continuous"
• 5/11/23: MIT International Women in Math Day
• "Exceptional Set Estimates for Orthogonal and Radial Projections"
• 11/1/22: MIT Undergraduate Mathematics Association
• "Topics in Orthogonal Projections in Euclidean Space"