Vertical Integration Workshop, April 16-19





We had so much fun when ran this workshop in October, we decided to do it again! This time, we are going to have faculty, postdocs, graduate students and undergraduates listening to each other's talks and interacting throughout. The talks are labeled (F) for faculty, (P) for postdoc, (G) for gradstudent, and (U) for undergrad. The abstracts can be found at the bottom of this page.


Thursday, April 16 (Hylan 1106A)

3.30 p.m. Sarah Tammen (P), University of Wisconsin (this talk will double as as the Combinatorics Seminar lecture)
Title: Incidence estimates for slabs 

4.30 p.m. Question and answer session with Sarah Tammen about her talk and her work.


Friday, April 17 (Dewey 2110E)

9.00 a.m. Camil Muscalu (F), Cornell University (this talk will double as a Analysis Seminar lecture)
Title: On a Brascamp-Lieb type theory for singular integrals

10.00 a.m. Question and answer session with Camil Muscalu about his talk and his work.

10.30 a.m. Quy Pham (G), University of Rochester
Title: Fourier Integral Operators, Point Configurations, and Elekes-Ronyai Type Problems

11.00 a.m. Lunch break

12.45 p.m. Eyvindur Palsson (F), Virginia Tech University (this talk will double as an Analysis Seminar lecture)
Title: Falconer type results for distance graphs

1.45 p.m. Question and answer session with Eyvi Palsson about his talk and his work.

2.15 p.m. Will Burstein (G), University of Rochester
Title: The Fourier ratio: a notion of complexity

2.45 p.m. Hari Nathan (G),  University of Rochester
Title: On a Fractal Variant of Dimensionality Reduction

3.15 p.m. Chamsol Park (P), University of Rochester
Title: Estimates for the Laplace eigenfunctions on manifolds

4.15 p.m. Question and answer session with Chamsol Park about his talk and his work.

4.45 p.m. Akshay Sant (G), University of Rochester
Title: The Mobius function and learning theory



Saturday, April 18 (Hylan 1106A)

8.00 a.m. Neeraja Kulkarni (P), University of Rochester
Title: Full measure universality versus measure universality in the Erdos similarity problem

9.00 a.m. Question and answer session with Neeraja Kulkarni about her talk and her work.

9.30 a.m. Shengze Duan (G), University of Rochester
Title: Spectral synthesis and fractals

10.00 a.m. Zhangze Li (G), Ohio State University
Title: Rectifiability and Projection

10.30 a.m. Nate Shaffer (U), University of Rochester
Title: Sparsity, the Fourier Ratio, and Renyi Entropy

11.00 a.m. Steven Senger (F), Missouri State University
Title: Discretized entropy bounds for sum-product phenomena for infinite subsets of real numbers.

12.00 p.m. Question and answer session with Steven Senger about his talk and his work.

12.30 p.m.- 1.30 p.m. Lunch

1.30 p.m.  Krystal Taylor (F), Ohio State University
Title: Nonlinear Projections and Quantitative Rectifiability

2.30 p.m. Question and answer session with Krystal Taylor about her talk and her work.

3.00 p.m. Zhihe Li (G), University of Rochester
Title: Fourier Ratio in the Euclidean setting

3.30 p.m. Ella Yu (G), University of Rochester
Title: Generalized Salem sets and Bourgain's inequality

4.00 p.m. Roan James (G), University of Rochester
Title: Simple Random Walks and Discrete Stochastic Differential Equations

4.30 p.m. Masha Gordina (F), Ohio State University (zoom talk)
Title: Limit laws  on metric measure spaces

5.30 p.m. Question and answer session with Masha Gordina about her talk and her work.

6.00 p.m. Pizza Party!!!


Sunday, April 19 (Hylan 1106A)

8.00 a.m. Vishal Gupta (P), University of Rochester
Title: A lower bound on the smallest eigenvalue of a graph and its application to the associahedron graph

9.00 a.m. Question and answer session with Vishal Gupta about his talk and his work.

9.30 a.m. Nathaniel Kingsbury-Neuschotz (G), University of Rochester
Title: Orbits on some Markoff-Type Surfaces

10.00 a.m. Shantanu Deodhar (G), University of Rochester
Title: Spectral synthesis with the complexity parameter

10.30 a.m. Emmett Wyman (F), SUNY Binghamton
Title: A tutorial for lifting classical harmonic analysis results to manifolds: The Agranovsky-Narayanan Theorem

11.30 Question and answer session with Emmett Wyman about his talk and his work.


Abstracts:

Sarah Tammen: I will discuss incidence estimates for slabs which are formed by intersecting small neighborhoods of well-spaced hyperplanes in R^d with the unit cube [0,1]^d. My work is an analogue of a theorem of Guth, Solomon, and Wang, who proved a version of the Szemerédi- Trotter theorem for thin tubes that satisfy a certain strong spacing condition. My proof uses induction on scales and the high-low method of Vinh, along with geometric insights.

Akshay Sant: We are going to discuss some connections between the Mobius function and learning theory. In particular, we are going to show using the Fourier ratio, some recent exponential sums estimates, and Vapnik Chervonenkis theory that a natural class of functions that includes the Mobius function restricted to [1,N] cannot be learned in fewer than CN steps.

Camil Muscalu: The plan of the lecture is to describe some examples of natural estimates for singular integrals, that are analogous to the classical Brascamp-Lieb inequality. Joint work with Cristina Benea.

Will Burstein: We shall discuss the role of the Fourier ratio as a complexity parameter in a variety of problems in analysis and signal recovery.

Eyvi Palsson: The Falconer distance problem, a major open problem on the interface of geometric measure theory and harmonic analysis, has seen much progress in the last decade. There are many variants of it, including pinned and non-empty interior ones, as well as multi-point configuration analogues. In this talk I will give a brief introduction and report on some recent results on general distance graph variants of this classic question.

Quy Pham: In this talk, I will discuss results showing that various $k$-point configuration sets of thin sets have positive Lebesgue measure, obtained by exploiting optimal $L^2$-based Sobolev estimates for the associated family of Fourier integral operators. Extending the framework developed by Greenleaf, Iosevich, and Taylor, we obtain Falconer-type results for many configuration sets for which the method would be vacuous if one were to demand nonempty interior.

I will also discuss results on ``dimension expansion'' versions of Elekes-Ronyai theorems for bivariate and trivariate real analytic functions.

Hari Nathan: In this presentation, we propose the idea of using iterated function systems (IFS's) for dimensionality reduction. IFS's are well-known tools for producing fractals whose Minkowski can be easily upper bounded. We show that one can find an IFS that generates a fractal of at most a given dimension that is ``close to'' a data set and suggest that moving all the points on that data set to the nearest point on the fractal can be uses for dimensionality reduction.

Chamsol Park: In this talk, we will briefly discuss how the L^q estimates on the eigenfunctions of the Laplace-Beltrami operators (with or without potentials) have been studied so far. These will include the L^q estimates over the full manifolds and restricted along the submanifolds.

Neeraja Kulkarni: A set E \subset \mathbb{R} is called measure universal if every set of positive Lebesgue measure contains an affine cope of E.  The Erdos similarity conjecture says that no infinite sets are measure universal.

A set E \subset \mathbb{R} is called full measure universal if every set of full Lebesgue measure contains an affine cope of E. Clearly, every measure universal set is also full measure universal. In this talk, I will explore the following questions: (1) Does there exist any set which is full measure universal, but not measure universal? (2) If such sets exist, how can we characterize their structure? The motivation for studying these questions is to try to find "almost counterexamples" for the Erdos similarity problem and to study their properties.

Shengze Duan: If the Fourier transform of an (L^p)-function is supported on a thin subset of (\mathbb{R}^d), when must the function vanish? Agranovsky--Narayanan answered this question when the support is contained in a submanifold, and Raani later extended this perspective to sets of fractal dimension. More recently, Guo--Iosevich--Zhang--Zorin-Kranich improved the relation between the dimension of the support and the (L^p)-exponent for the moment curve. In this work, we study fractal subsets of the moment curve.

Zhangze Li: This talk discusses the connection between rectifiability and projections in geometric measure theory. After reviewing the Besicovitch projection theorem and the classical two-projection theorem, I will present a generalized version for broader families of projection-type maps, giving a criterion for pure 1-unrectifiability. As an application, I will also discuss consequences for pinned distance sets.

Nate Shaffer: We introduce a one parameter generalization of the Fourier ratio related to Renyi entropy, demonstrating that it is an effective measure of sparsity. We prove upper and lower bounds on established measures of approximate sparsity in terms of Renyi entropy, show that Renyi entropy obeys an uncertainty principle, as well as give a qualitative description of functions with maximal entropy.

Steve Senger: The classical sums and products problem essentially posits that given a finite set of real numbers, pairs of its members must either have many distinct sums or many distinct products. We discuss related phenomena for wide classes of infinite subsets of the reals. Our viewpoint is through entropy of certain random variables. Our methods employ tools from geometric measure theory, probability, additive combinatorics, and incidence geometry.

Masha Gordina: We will survey recent results on limit laws for stochastic processes on metric measure spaces. The main object is a stochastic process corresponding to a Dirichlet form on such a space.  Limit laws include small deviations,  large deviations principle, heat content asymptotics, Chung's law, as well as finding an Onsager-Machlup functional. Many of these results are closely related to the boundary problems for the corresponding infinitesimal generator in a metric ball.  This setting includes a number of examples: Riemannian manifolds, sub-Riemannian manifolds including Carnot groups, singular spaces such as fractals, diffusions and fractional sub-Laplacians.

Zhihe Li: In this talk, I will introduce a continuous analogue of the Fourier ratio for compactly supported Borel measures, defined as the ratio of the $L^1$ and $L^2$ norms of a regularized Fourier transform at scale $R$. This quantity interpolates between $L^1$ and $L^2$ Fourier information and connects uncertainty principles, Fourier restriction, and approximation by trigonometric polynomials. I will present sharp bounds in terms of geometric properties of supports, discuss several interesting examples, and derive a fractal uncertainty principle.

Ella Yu: In classical Fourier analysis, Salem sets are characterized by uniform bounds on the Fourier transform. In recent work, Jonathan Fraser introduced a framework that replaces such $L^\infty$ bounds with $L^p$ average estimates of the Fourier transform, leading to a natural generalization of Salem sets in the finite field setting. In this talk, I will present this perspective, discuss several examples, and explain its connections to sumset problems and Fourier restriction theory. I will also highlight how this framework is related to Jean Bourgain’s $\Lambda(p)$ inequality.

Roan James: We study simple random walk models in random environments and their connection to stochastic partial differential equations. We discuss how suitably scaled partition functions can be interpreted as discrete analogues of solutions to the Parabolic Anderson Model (PAM). This perspective is a natural framework for constructing SPDE solutions via discrete models. This is joint work with Arjun Krishnan.

Krystal Taylor: From the delicate geometry found in a snowflake to the intricate patterns of a coastal shoreline, nature holds infinite patterns and scales.  The world is not easily described using mere lines and cones, and classic Euclidean geometry falls short.  The notion of fractals gives us a language and a set of tools to understand more complex phenomena. 

Vishal Gupta: In this talk, I will discuss a lower bound for the smallest eigenvalue of a regular graph containing many copies of a smaller fixed subgraph. This generalizes a result of Aharoni, Alon, and Berger in which the subgraph is a triangle. We will then apply the result to obtain a lower bound on the smallest eigenvalue of the associahedron graph and prove that this bound gives the correct order of magnitude of this eigenvalue. If time allows, I will also discuss what is known regarding the second-largest eigenvalue of this graph.

Nathaniel Kingsbury-Neuschotz: The classical Markoff Equation is the Diophantine equation

X^2 + Y^2 + Z^2 = 3XYZ.

This equation was discovered by Markoff in connection with Diophantine approximation, and has since repeatedly re-emerged in relation to geometry and group theory. In 2016, Bourgain, Gamburd, and Sarnak proved that a certain group of symmetries generated by so-called Vieta Involutions acts transitional on the nonzero mod p solutions, at least for density one of primes p, and thereby showed that almost all Markoff numbers are composite.

In this talk I will discuss some analogous results of mine in the more general family of surfaces

 
X^2 + Y^2 + Z^2 = XYZ + AX + BY + CZ + D

for fixed integer parameters (A, B, C, D). Time permitting, I will mention connections to geometry, via an interpretation of Vieta involutions as the action of the pure mapping class group on the character variety of the four-times punctured sphere, and sketch the techniques in the proof of my result.

Shantanu Deodhar: Agranovsky and Narayanan proved that if a function $f \in L^p(\mathbb{R}^n)$ for $p \leq \frac{2n}{d}$ and its Fourier transform is supported on a $d$-dimensional sub-manifold, then $f \equiv 0$. We show that the exponent $p$ is governed by a quantitative spectral complexity parameter, the Fourier Ratio, in addition to the geometric size of the Fourier support. In the Euclidean setting, with additional information about Fourier ratio decay $\kappa$, the classical synthesis threshold improves from $\frac{2n}{d}$ to $\frac{2(n-2\kappa)}{d-2\kappa}.$ In particular, we show how the Fourier ratio naturally captures the curvature of the manifold, leading to a sharper integrability threshold. As an application, we compute $\kappa$ for a class of co‑dimension 2 manifolds in terms of their curvature, yielding an explicit improved bound. This demonstrates that the Fourier ratio captures curvature information and sharpens the synthesis exponent.

Emmett Wyman: A classical result of Agranovsky and Narayanan states there is no nontrivial measure along a $k$-dimensional manifold of $\mathbb R^d$ with Fourier transform in $L^p$, for $1 \leq p \leq \frac {2d} k$. One should think of this as an uncertainty principle preventing the simultaneous concentration of a function in both space and frequency. In recent work with Iosevich and Mayeli, we extend this result to the manifold setting.

In this talk, I will present a proof of a baby version of this theorem in Euclidean space and, after reviewing just a bit of Riemannian geometry, transfer the statement and proof to the manifold setting.