# Plenary Talks

## Friday

• Andrea Nahmod (University of Massachusetts, Amherst), “Recent progress on propagation of randomness under the NLS flow”
Abstract: In this talk we focus on the time dynamics of solutions of periodic nonlinear Schrödinger with random initial data. It is well known that in many situations, randomization improves the behavior of solutions to evolution PDEs: the key underlying difficulty is in understanding how randomness propagates under the nonlinear flow. In this context, starting with Bourgain’s seminal work on the invariance of Gibbs measures for nonlinear Schrödinger equations we survey recent methods and describe new results that offer deeper insights. In joint works with Yu Deng and Haitian Yue, we develop the theory of random tensors and prove almost-sure local well-posedness in the optimal range relative to the probabilistic scaling. We also obtain invariance of the Gibbs measure for any odd power nonlinearity in 2D, solving a major open problem since Bourgain’s seminal 1996 paper.
• Betsy Stovall (University of Wisconsin-Madison), “Fourier restriction to degenerate hypersurfaces”
Abstract:  In this talk, we will describe various open questions and recent progress on the Fourier restriction problem associated to hypersurfaces with varying or vanishing curvature.

## Saturday

• Kasso Okoudjou (Tufts University), “The HRT Conjecture for real-valued functions”
Abstract: Given a non-zero square integrable function $g$ and $\Lambda=\{(a_k, b_k)\}_{k=1}^N \subset \mathbb{R}^2$ let

 $\mathcal{G}(g, \Lambda)=\{e^{2\pi i b_k \cdot}g(\cdot - a_k)\}_{k=1}^N.$

The Heil-Ramanathan-Topiwala (HRT) Conjecture is the question of whether  $\mathcal{G}(g, \Lambda)$ is linearly independent. For the last two decades very, little progress has been made in settling the conjecture. In the first part of the talk, I will give an overview of the state of the conjecture focusing especially on the case $N\leq 4$. I will then describe some recent attempts in settling   the conjecture for some special classes of functions and special sets $\Lambda\subset \mathbb{R}^2$.

• Tatiana Toro (University of Washington) “Perturbing elliptic operators”
Abstract: To every second order elliptic divergence form operator on a nice domain one can associate a canonical measure, the elliptic measure. In the case of the Laplacian this canonical measure is the harmonic measure. In this talk we will discuss how perturbing an elliptic operator affects the properties of the elliptic measure. This is a question with a long history, which I will  put in context before describing recent work with M. Akman, S. Hofmann and J.M. Martell.

## Sunday

• Shahaf Nitzan (Georgia Institute of Technology), “What is a good definition of ‘uniform completeness’?”
Abstract: We discuss possible ways to define a notion of ‘uniform completeness’ as a dual notion for uniform minimality. We contrast these definitions with a well known density theorem of Landau, and a quantified version of this theorem due to Olevskii and Ulanovskii. We show that analogs of these results can be obtained for an appropriate notion of ‘uniform completeness’.
• Michael Brannan (Texas A&M University), “Quantum graphs and quantum Cuntz-Krieger algebras”
Abstract: In this talk I will give a light introduction to the theory of quantum graphs. Quantum graphs are generalizations of directed graphs within the framework of non-commutative geometry, and they arise naturally in a surprising variety of areas including quantum information theory, representation theory, quantum symmetries of graphs, and in the theory of non-local games. I will give an overview of some of these connections and also explain how one can generalize the well-known construction of Cuntz-Krieger C*-algebras associated to ordinary graphs to the setting of quantum graphs.  Time permitting, I will also explain how quantum symmetries of quantum graphs can be used to shed a great deal of light on the structure of quantum Cuntz-Krieger algebras.  (This is joint work with Kari Eifler, Christian Voigt, and Moritz Weber.)

# Contributed Talks

## Parallel Sessions I (Saturday 11am – 12pm EDT)

### Room 1

• El Mehdi Ainasse (Stony Brook), “Twisted Complex Brunn-Minkowski Theory”
Abstract: We will briefly introduce the Brunn-Minkowski theorem and Prékopa’s theorem, and discuss Berndtsson’s observations leading to their complex generalizations. We will then present Berndtsson’s Nakano-positivity theorem which is at the heart of his complex Brunn-Minkowski theory. After that, we will describe part of our work, generalizing Berndtsson’s theorem under weaker (complex) convexity hypotheses.
• Christina Karafyllia (Stony Brook), “Conformal Invariants and Conformal Mappings in Hardy and Bergman Spaces”
Abstract: This talk is about some classical subjects in complex analysis and geometric function theory such as properties of conformal invariants and relations between them. In particular, we will talk about some recent results on the harmonic measure and its connection with the hyperbolic distance on simply connected domains. We also present geometric conditions for a conformal mapping of the unit disk to belong to Hardy or Bergman spaces by studying the harmonic measure and the hyperbolic metric in the image region. Part of this presentation is based on joint work with D. Betsakos and N. Karamanlis.

### Room 2

• Shukun Wu (UIUC), “On the Bochner-Riesz operator in $R^3$
Abstract: We improve the Bochner-Riesz conjecture in $R^3$ to $p>3.25$.
• Benjamin Bruce (UW-Madison), “Fourier restriction to hyperboloids”
Abstract:  In this talk, I will discuss recent work on the Fourier restriction problem for hyperboloids, focusing primarily on the one-sheeted hyperboloid in three dimensions.  This surface exhibits both negative curvature and asymptotically conic behavior, making it interesting from the viewpoint of restriction theory.  Some of the results presented originate in joint work with Diogo Oliveira e Silva and Betsy Stovall.

## Parallel Sessions II (Saturday 3:30-4:30 EDT)

### Room 1

• Oleksandr Vlasiuk (FSU), “Asymptotic properties of short-range interaction functionals”
Abstract: We describe a framework for extending the asymptotic behavior of a short-range interaction from the unit cube to general compact subsets of ${\mathbb R}^d$. This framework allows us to give a unified treatment of asymptotics of hypersingular Riesz energies and optimal quantizers. We further obtain new results about the scale-invariant nearest neighbor interactions, such as the $k$-nearest neighbor truncated Riesz energy. Our generalized approach has applications to methods for generating distributions with prescribed density: strongly-repulsive Riesz energies, centroidal Voronoi tessellations, and some popular meshing algorithms. Based on a joint work with D. Hardin and E. Saff.
• Trevor Leslie (UW-Madison), “Geometric Structure of Mass Concentration Sets for Pressureless Euler Alignment Systems”
Abstract: We study the limiting dynamics of the Euler Alignment system with a smooth, heavy-tailed interaction kernel $\phi$ and unidirectional velocity $\mathbf{u} = (u, 0, \ldots, 0)$.  We demonstrate a striking correspondence between the entropy function $e_0 = \partial_1 u_0 + \phi*\rho_0$ and the limiting `concentration set’, i.e., the support of the singular part of the limiting density measure. In a typical scenario, a flock experiences aggregation toward a union of $C^1$ hypersurfaces: the image of the zero set of $e_0$ under the limiting flow map.  This correspondence also allows us to make statements about the fine properties associated to the limiting dynamics, including a sharp upper bound on the dimension of the concentration set,  depending only on the smoothness of $e_0$.

### Room 2

• Steven Senger (Missouri State University), “Point configurations given by distances and dot products”
Abstract: We explore a number of generalizations of Erdös’ original unit distance problem, which asks for upper bounds on how often a fixed distance can occur in a large finite point set in the plane. We offer novel bounds on a family of variants of this problem involving multiple points, and relationships determined by distances and dot products, in $\mathbb R^2$ and $\mathbb R^3.$
• Caleb Marshall (University of British Columbia), “Dot Product Chains”
Abstract: We study a variant of Erdös’ unit distance problem, concerning dot products between successive pairs of points chosen from a large finite point set. Specifically, given a large finite set of n points E, and a sequence of nonzero dot products (α_{1},…,α_{k}), we give upper and lower bounds on the maximum possible number of tuples of distinct points (A_{1},…,A_{k+1})∈E^{k+1} satisfying A_{j}⋅A_{j+1} =α_{j} for every 1≤j≤k. This work was completed at Missouri State University, and is joint work with Drs. Steven Senger and Shelby Kilmer.

## Parallel Sessions III (Sunday 11am – 12pm EDT)

### Room 1

• Gareth Speight (University of Cincinnati), “A $C^m$ Lusin Approximation Theorem for Horizontal Curves in the Heisenberg Group”
Abstract: The classical Lusin theorem in measure theory states that measurable functions can be approximated by continuous functions except for a set of small measure. Higher smoothness versions of this give conditions under which functions can be approximated by $C^m$ functions up to a set of small measure. We discuss a recent theorem of this type in the Heisenberg group, which is $R^3$ with a geometry in which distinguished horizontal directions and horizontal curves play a vital role.
• Geoff Bentsen (UW-Madison), “Averages over curves on the Heisenberg group”
Abstract: We investigate the $L^p$ regularity of averaging operators over certain families of curves invariant under translation in the Heisenberg group. This noncommutative setting involves degeneracies not seen in the Euclidean case. We prove $L^p$ regularity in the “worst” case of these degeneracies, a fold blowdown singularity, using almost orthogonality arguments and iterated decoupling on the cone.

### Room 2

• Bingyang Hu (Purdue), “Sparse bound of singular Radon transform”
Abstract: In this talk, we will talk about the sparse bounds of singular Radon transforms, which strengths the $L^p$ boundedness of such operators due to Christ, Nagel, Stein and Wainger in 1999.
• Changkeun Oh (UW-Madison): “A restriction estimate for surfaces with negative Gaussian curvatures”
Abstract: In this talk, we will discuss a restriction estimate for surfaces with negative Gaussian curvatures in ${\mathbb R}^3$. We will introduce notions of bad pairs and bad lines, and combine these concepts with the polynomial method developed by Guth to prove a $L^p \rightarrow L^q$ restriction estimate for surfaces with negative Gaussian curvatures for $q>3.25$.

## Parallel Sessions IV (Sunday 2pm-3pm EDT)

### Room 1

• Tomasz Z. Szarek (BCAM), “Sharp estimates of the spherical heat kernel”
Abstract: The classical spherical heat kernel is an important object in analysis, probability and physics, among other fields. It is the integral kernel of the spherical heat semigroup and thus provides solutions to the heat equation based on the Laplace-Beltrami operator on the sphere. It is also a transition probability density of the spherical Brownian motion.
In this talk we prove sharp two-sided global estimates for the heat kernel associated with a Euclidean sphere of arbitrary dimension. If time permits, we will present a generalization of this result to the compact rank-one symmetric spaces.
The talk is based on joint papers with Adam Nowak and Peter Sjögren.
• Yunbai Cao (Rutgers), “The Vlasov-Poisson-Boltzmann system in bounded domains”
Abstract: The Vlasov-Poisson-Boltzmann system (abbreviated as VPB) models the dynamics and collision processes of dilute charged particles when there is an electric field. Boundary effects plays a crucial role when modeling the system in bounded domains. In this talk, we discuss the existence and regularity results of the solutions of VPB system in bounded domains with physical boundary conditions.

### Room 2

• John MacLellan (University of Alabama), “Necessary Conditions for Two Weight Inequalities for Singular Integral Operators”
Abstract: In this talk we discuss our recent work on necessary conditions for two weight norm inequalities for singular integral operators. We prove necessary conditions on pairs of measures $(\mu,\nu)$ for a singular integral operator T to satisfy weak (p,p) inequalities provided the kernel of T satisfies a non degeneracy condition and $\mu$ satisfies a weak doubling condition. We also prove analogous results for pairs of measures $(\mu, \sigma)$ for the singular integral operator $T(f d\sigma)$. As an application of our techniques we show that in general T does not satisfy the strong endpoint estimate $T:L^1(\nu)-> L^1( \mu)$. Finally we discuss how some of these results extend to multilinear singular integral operators.
• Naga Manasa Vempati (Washington University in St. Louis), “A two weight inequality for Calderón-Zygmund operators on spaces of homogeneous type with application”
Abstract: Let $(X,d,\mu )$ be a space of homogeneous type in the sense of Coifman and Weiss, i.e. $d$ is a quasi metric on $X$ and $\mu$ is a nonzero measure satisfying the doubling condition. Suppose that $u$ and $v$ are two locally finite positive Borel measures on $(X,d,\mu )$. Subject to the pair of weights satisfying a side condition, we characterize the boundedness of a Calderón-Zygmund operator $T$ from $L^{2}(u)$ to $L^{2}(v)$ in terms of the $A_{2}$ condition and two testing conditions. For every cube $B\subset X$, we have the following testing conditions, with $\mathbf{1}_{B}$ taken as the indicator of $B$

 $\displaystyle \Vert T(u\mathbf{1}_{B})\Vert _{L^{2}(v)}\leq \mathcal{T}\Vert 1_{B}\Vert_{L^{2}(u)},$ $\displaystyle \Vert T^{\ast }(v\mathbf{1}_{B})\Vert _{L^{2}(u)}\leq \mathcal{T}\Vert 1_{B}\Vert _{L^{2}(v)}.$

The proof uses stopping intervals and corona decompositions originating in work of Nazarov, Treil and Volberg, along with the pivotal side condition.

## Parallel Sessions V (Sunday 3:30pm-4:30pm EDT)

### Room 1

• Matthew Romney (Stony Brook), “The branch set of minimal surfaces in metric spaces with a quadratic isoperimetric inequality”
Abstract: A theory of minimal surfaces in metric spaces has recently been developed by Lytchak–Wenger using the quadratic isoperimetric inequality as a basic axiom. This assumption is quite broad, being satisfied, for example, by compact Finsler manifolds, complete CAT(k) spaces, and all Banach spaces. In this talk, we discuss several new contributions to this theory related to the structure of the branch set of a minimal surface, i.e., the set of points where an energy-minimizing mapping is not a local homeomorphism. This is joint work with Paul Creutz.
• Behnam Esmayli (University of Pittsburgh), “The Coarea Inequality”
Abstract: Co-area inequality claims that the integral of the Hausdorff measures of preimages of a Lipschitz map between metric spaces is bounded by a quantity in terms of the Lipschitz constant of the function and the measure of the domain. In a joint work with Piotr Hajlasz we replace this bound by a much more refined quantity that takes into account also the local behavior of the function. While the old bound vanishes only when the domain is a null set, our bound vanishes in many important nontrivial instances. For example, for maps between Euclidean spaces, our bound vanishes on the set where the derivative is not full-rank. Our proof is new and unlike the earlier proofs does not use a difficult result of Davies’s which involves Ramsey’s theorem and ordinal numbers among other tools. If time allows, I will discuss potential applications to metric implicit function theorem.

### Room 2

• Erik Lundberg (Florida Atlantic University), “Limit cycles in random vector fields”
Abstract: We study the distribution of the limit cycles of a planar vector field whose component functions are random polynomials. We present estimates for the number of limit cycles in both perturbative and non-perturbative settings. The proofs use novel combinations of techniques from dynamical systems and random analytic functions.
• Sean Sovine (Virginia Tech), “Simplex Averaging Operators: Nontrivial and Lp-improving bounds in lower dimensions”
Abstract: We establish some new $L^p$-improving bounds for the $k$-simplex averaging operators $S^k$ that hold in dimensions $d \geq k$. As a consequence of these $L^p$-improving bounds we obtain nontrivial bounds $S^k\colon L^{p_1}\times \cdots\times L^{p_k}\rightarrow L^r$ with $r < 1$. In particular we show that the triangle averaging operator $S^2$ maps $L^{\frac{d+1}{d}}\times L^{\frac{d+1}{d}} \rightarrow L^{\frac{d+1}{2d}}$ in dimensions $d\geq 2$.