Abstract: There are several ways of defining classes of Sobolev maps from domains into an arbitrary metric space. In this talk, I will discuss some the relationships between some of these notions when the target is the sub-Riemannian Heisenberg group (or, more generally, a Lipschitz connected metric space). Moreover, I will discuss those domains for which Sobolev maps of different classes are the traces of globally defined Sobolev maps. This is joint work with Piotr Hajlasz.

Abstract: Rademacher's theorem states that every Lipschitz function is differentiable almost everywhere. A universal differentiability set (UDS) is a set which contains a point of differentiability for every real-valued Lipschitz function. As a consequence of Rademacher's theorem, every set of positive measure is a UDS. However, many spaces contain a measure zero UDS. This talk will describe what is known in Euclidean spaces, Carnot groups, and Laakso spaces.

Abstract: The classical von Neumann inequality provides a fundamental link between complex analysis and operator theory. It shows that for any contraction $T$ on a Hilbert space and any polynomial $p$, the operator norm of $p(T)$ satisfies

\[ \|p(T)\| \le \sup_{|z| \le 1} |p(z)|. \]

Whereas And\^o extended this inequality to pairs of commuting contractions, the corresponding statement for triples of commuting contractions is false.

However, it is still not known whether von Neumann's inequality for triples of commuting contractions holds up to a constant. I will talk about this question and about function theoretic upper bounds for $\|p(T)\|$.

Abstract: The study of (fractional) Sobolev spaces in the Euclidean setting has long been central to analysis, particularly in the context of the theory of partial differential equations. In recent years, there have been significant developments in extending the theory of these function spaces beyond $\mathbb{R}^n$, to the setting of metric-measure spaces. When considered in this more general setting, these function spaces exhibit a rich and complex behavior that depends heavily on geometric and measure-theoretic properties of the underlying metric-measure space. In this talk, we will discuss the extension of the classical (fractional) Sobolev spaces to metric measure spaces, investigating how the underlying geometry affects their analytical nature. In particular, we will present new results pertaining to their (compact) embedding properties, illustrating the fundamental role that the metric-measure structure plays in shaping function space behavior. This talk will be accessible to graduate students having some basic knowledge of measure theory and functional analysis.

Abstract: The Gromov-Wasserstein (GW) distances define a family of metrics, based on ideas from optimal transport, which enable comparisons between probability measures defined on distinct metric spaces. Although GW distances have proven useful for various applications in the recent machine learning literature, it has been observed that they are inherently sensitive to outlier noise and cannot accommodate partial matching. This has been addressed by various constructions building on the GW framework; in this talk, I focus specifically on a natural relaxation of the GW optimization problem, introduced by Chapel et al., which is aimed at addressing exactly these shortcomings. The goal is to present the properties of this relaxed optimization problem from the viewpoint of metric geometry. While the relaxed problem fails to induce a metric, in particular the axioms of non-degeneracy and triangle inequality are not satisfied, I will present a novel family of distances that define true metrics. These new distances, whose construction is inspired by the Prokhorov and Ky Fan distances, as well as by the recent work of Raghvendra et al. on robust versions of classical Wasserstein distance, are not only true metrics, but they also induce the same topology as the GW distances and enjoy additional robustness to perturbations.

Abstract: A set in the plane is purely unrectifiable if it intersects every rectifiable curve in a set of zero length. This is the antithesis of the notion of a rectifiable set. I will present various results concerning such sets in the plane, with an emphasis on Jordan curves. I will give examples of how one may study this notion for quadratic Julia sets. 

Abstract: The Maximum Distance Problem asks to find the shortest curve whose r-neighborhood contains a given set. Such curves are called r-maximum distance minimizers. We explore the limiting behavior of r-maximum distance minimizers as well as the asymptotics of their 1-dimensional Hausdorff measures as r tends to zero. Of note, we obtain results involving objects of fractal nature. This talk is based on is joint work with Enrique Alvarado, Louisa Catalano, and Tomás Merchán.

Abstract: The Loewner differential equation is a classical topic in complex analysis that connects growing families of compact sets in the complex plane (called hulls) to real-valued functions (driving functions). A central question in the deterministic Loewner equation is: how do the analytic properties of driving functions influence the geometric properties of Loewner hulls? Recently, interest has grown in extending this question to complex-valued driving functions.

In 2005, Lind showed that real-valued driving functions that are $1/2$-H\"older with a norm less than 4 generate hulls that are quasi-arcs, with this constant being sharp. For the complex-driven Loewner equation, Tran (2017) discovered a similar phase transition, though the sharp constant remains unknown. Lind and Utley (2022) explored this by considering driving functions of the form $c\sqrt{1-t}$ for $c\in\mathbb{C},$ showing that the phase transition occurs below 3.73.

In this talk, we will focus on driving functions that are parametrizations of squares and circles. By exploiting their symmetries, we prove that the phase transition for the complex-driven Loewner equation occurs below 2.4, providing new insight into the geometric behavior of these hulls.

Abstract: Given a metric space $X$ and a subset $E$ of $X$, when does there exist a geometrically nice curve that contains $E$ in its image?

Motivated by results of MacManus (1999) and Badger-Vellis (2019), we identify sufficient conditions in terms of the geometry of $X$ and the dimension of $E$ under which $E$ is contained in bi-Lipschitz or quasisymmetric image of $[0,1]$.

Additionally, we establish conditions to ensure that $X \setminus E$ is a uniform domain.

This is based on joint works with V. Vellis and S. Zimmerman.

Abstract: Let f denote a holomorphic self-map of the unit disk without any interior fixed points. A classical 1926 theorem of Denjoy and Wolff then asserts that the sequence of iterates 

\[f^{[n]}:=f\circ f\circ \cdots \circ f.\]

converges locally uniformly to a boundary fixed point of f, termed the Denjoy-Wolff point. 

The situation changes dramatically when one considers holomorphic fixed-point-free self-maps of the bidisk; the presence of large ``flat" boundary components will, in general, prevent the iterates from converging. The cluster set of the sequence of iterates in this setting was described in a 1954 paper of Herv\'{e}. 

In this talk, we discuss extensions of the notion of a Denjoy-Wolff point to the bidisk. While this is a topic that has already been studied by several researchers, our approach introduces work of Agler, McCarthy and Young (2012) on boundary regularity into the mix. This will allow us to obtain certain refinements of Herv\'{e}'s results. This is joint work with Michael Jury.

Abstract: Rademacher's theorem is a classical result in geometric measure theory, which states that every Lipschitz function defined on the unit interval is differentiable at almost every point of the interval. In light of this, it seems natural to ask if there is some sense of tangents for which a similar result is true for Hölder curves as well. In this talk, we present a notion of tangent to a set E in Euclidean space due to Badger and Lewis and show that an analogue of Rademacher's theorem holds for this sense of tangent. Specifically, we show that for every Lipschitz curve in Euclidean space, at almost every point the tangent space to the curve contains exactly one line through the origin. Then we construct Hölder curves of arbitrary exponent below 1 in Euclidean space with positive and finite Hausdorff measure for which at almost every point, the tangent space contains infinitely many homeomorphically distinct elements. We conclude by noting that at almost every point, these curves possess at least one tangent that features a form of self-similarity, which leads us to a conjecture relating these tangent spaces to a Hölder version of the analyst's traveling salesman problem. This talk is based on joint work with Vyron Vellis.

Abstract: Discrete complex analysis is the study of discrete holomorphic functions. These are functions defined on graphs embedded in the plane that satisfy some discrete analogue of the Cauchy-Riemann equations. While the subject is classical, it has seen a resurgence in the past 20-30 years with the work of Kenyon, Mercat, Smirnov, and many others demonstrating the power of discrete complex analysis as a tool for understanding 2D statistical physics at criticality. In this talk, we'll discuss how discrete complex analysis can be applied to solve a purely deterministic problem for a very general class of discretizations of 2D space accommodating a notion of discrete complex analysis. 

Abstract: The weak constant density condition (WCD) is a quantitative regularity property introduced by David and Semmes in their foundational work on uniformly rectifiable subsets of Euclidean space. Roughly speaking, a space satisfies the WCD if in \textit{most} balls, the space supports a measure with \textit{nearly constant} density in a neighborhood of scales and locations. In this talk, we discuss the ideas behind a proof that uniformly rectifiable \textit{metric spaces} satisfy the WCD. This theorem gives a metric space analog of a Euclidean result of David and Semmes.

Abstract: Continued fractions over the real numbers are well-studied, with applications ranging from gear ratios in clocks to properties of geodesics in hyperbolic surfaces. Less is known about their properties over complex numbers, and standard methods break down over quaternions and octonions. Adapting the work of Nakada and Hensley, we prove that natural continued fraction algorithms in multiple Euclidean spaces are ergodic (in fact, exact) with a piecewise-analytic invariant probability measure.

Abstract: Rectifiable sets extend the class of surfaces considered in classical differential geometry; while admitting a few edges and sharp corners, they are still smooth enough to support a rich theory of local analysis. However, for certain questions of global nature the notion of rectifiability is too qualitative. In a series of influential papers around the year 1990, David and Semmes developed an extensive theory of quantitative rectifiability in Euclidean spaces. A motivation for their efforts was the significance of a geometric framework for the study of certain singular integrals and their connections to removability. We will discuss recent results which lay the foundations for a theory of quantitative rectifiability in Heisenberg groups. As in the Euclidean case motivation stems from questions involving singular integrals and removability. We will see that, in certain aspects, the situation is very different than in Euclidean spaces and new phenomena appear. Based on joint works with Sean Li (UConn) and Robert Young (NYU).

Abstract: I will report on some recent progress on the problem of characterizing sets which lie in the image of a Lipschitz map from the plane into 3-dimensional Euclidean space. The new construction of Lipschitz maps is based on a simple observation about square packings. As an application, in any complete Ahlfors q-regular metric space with q>m-1, we construct an abundance of m-rectifiable doubling measures that are purely (m-1)-unrectifiable. Moreover, it is possible to prescribe the lower and upper Hausdorff and packing dimensions of the measures. Relevant concepts from geometric measure theory will be explained during the talk. This is joint work with Raanan Schul.

Abstract: Subfactors are inclusions of von Neumann algebras of trivial center (called factors), which often encode symmetries of quantum mechanical systems. Subfactors can be constructed from mathematical objects such as finite groups, Hadamard matrices, finite dimensional Hopf algebras, and other large classes of quantum groups. To each subfactor one associates an invariant called the standard invariant, which is a lattice of finite dimensional matrix algebras. We study deformations of subfactors by looking at the way the associated standard invariants can deform. Consequences are obtained for Hadamard matrices, groups and Hopf algebras.

Abstract: This talk is based on ongoing work of the speaker. We will discuss the stochastic embeddability of snowflakes of doubling metric spaces into ultrametric spaces and the induced stochastic embeddings of their hyperbolic fillings into trees. As an application, we obtain that finitely generated Gromov hyperbolic groups admit proper, uniformly Lipschitz affine actions on $L^1$.

Abstract: In 1977 and 1979, Erik Christensen published three papers which studied subalgebras of a given operator algebra that are “close” to each other. We will present Christensen’s main results, which show that under certain conditions morphisms close to identity are inner, and subalgebras close to each other are unitarily conjugate.

Abstract: We give a proof to the following theorem of Petrescu: For m prime, the Fourier matrix of size m is isolated among all normalized complex Hadamard matrices. The proof uses a directional limit and compactness argument.

Abstract: Federer's characterization states that a set is of 􏰂finite perimeter if and only if its measure theoretic boundary has 􏰂finite codimension 1 Hausdorff􏰁 measure. In this talk, we discuss the extent to which an analog of this result holds for sets of 􏰂finite s-perimeter, with 0 < s < 1, in doubling metric measure spaces. Here the nonlocal s-perimeter is defi􏰂ned via a Besov seminorm, and as shown by Dávila in R^n and Di Marino and Squassina in the metric setting, recovers the perimeter of a set as s → 1− under suitable rescaling. Time permitting, we will also consider a nonlocal minimization problem for the s-perimeter, as introduced by Caff􏰁arelli, Roquejo􏰁re, and Savin in R^n, and discuss regularity results for minimizers in the metric setting.

Abstract: In 1985, Hata showed that whenever the attractor K of a finite iterated function system is connected, it is the image of a curve, and if additionally it is contained in Euclidean space then it is the image of a 1/s-Holder continuous curve, where s is the similarity dimension of the system. We provide examples of compact and connected attractors of infinite iterated function systems (IIFS) in the plane; one which is not the image of a curve, and one which is the image of a curve but not the image of any Holder continuous curve. If time permits, we show some conditions under which the attractor of an IIFS is the image of a curve, or the image of a Holder continuous curve.

Abstract: We generalize a bi-Lipschitz extension result of David and Semmes from Euclidean spaces to complete metric measure spaces with controlled geometry (Ahlfors regularity and supporting a Poincaré inequality). In particular, we find sharp conditions on metric measure spaces X so that any bi-Lipschitz embedding of a subset of the real line into X extends to a bi-Lipschitz embedding of the whole line. Along the way, we prove that if the complement of an open subset Y of X has small Assouad dimension, then it is a uniform domain, and we prove a quantitative approximation of continua in $X$ by bi-Lipschitz curves.

Abstract: It is known how holomorphic maps change certain dimension notions. For instance, the Hausdorff dimension of compact sets does not change when a holomorphic map is applied, while the Box-counting dimension decreases (due to its Lipschitz stability). Fraser and Yu in 2018 introduced the notion of the Assouad spectrum, which is a collection of dimensions that can change in an uncontrollable way under Lipschitz maps. We prove that this notion also decreases under holomorphic maps and we provide examples showing this decrease can be strict. As a corollary, this implies upper bounds on the distortion of the Assouad spectrum (and dimension) under planar quasiregular maps, as well as the invariance of porosity of compact sets.

Abstract: A quadrilateral in the complex plane is a bounded Jordan domain (i.e. a domain whose boundary is homeomorphic to a circle) with four distinct points, called vertices, selected on its boundary and labeled in counter-clockwise order. Quasiconformal maps can be defined as homeomorphisms which only stretch quadrilaterals by a uniformly bounded amount. Motivated by the theory of quasiconformal maps, we investigate properties for families of quadrilaterals of bounded modulus. We first provided a way to approximate a quadrilateral by a linear quadrilateral (i.e. a quadrilateral with boundary consisting of finitely many line segments) without distorting its internal distances too much. This result reduces many questions to just linear quadrilaterals instead of arbitrary ones. Moreover, given a family of quadrilaterals of uniformly bounded modulus and an arbitrary quadrilateral Q in this family, we were able to show the existence of disks lying in the interior of Q, whose radius is comparable to the maximum of the internal distances of Q and the comparability constant only depends on the uniform bound of the modulus of the family. Our proof also indicates where such disks could lie inside each quadrilateral in the family. The talk is based on joint work with Aimo Hinkkanen.

Abstract: Let d be a positive integer and let f be an analytic function in a neighborhood of the origin in complex d-space. We present two independent conditions on the positive multivariable Taylor coefficients of f, which imply that f(z)=(1-g(z))^{-1} for a function g with non-negative Taylor coefficients. The results have applications in the theory of Nevanlinna-Pick kernels.

The motivations for both conditions are similar. Yet, the calculation for the second condition is using non-commutative function theory.

This is based on joint work with Jesse Sautel.

Abstract: The Riemann Mapping Theorem states that any simply connected, open, proper subset of the plane can be mapped conformally onto a disk. In 1918, Koebe proved that every domain in the plane with finitely many boundary components is conformal to a domain whose boundary components are circles or points, which we call circle domains. He conjectured that every domain is conformal to a circle domain. While there has been progress in answering this question, the general statement has yet to be proven. Modern versions of this question investigate when metric spaces can be mapped nicely onto circle domains. In particular, we will investigate when there is a quasisymmetric mapping: which requires that ratios of distances between triples of points have controlled distortion. In this talk, we’ll introduce some of the tools used to answer these questions, including modulus of curve families. The speaker will then characterize metric spaces that are quasisymmetric to countably connected circle domains. The characterization will be in terms of an analog of Heinonen and Koskela’s Loewner property for Schramm’s transboundary modulus.

Abstract: This work is joint with Fedor Nazarov and Igor Verbitsky. It is a standard fact that if a linear operator T has operator norm ∥T∥ < 1, then the Neumann series I + T + T^2 + ... converges to (I − T)^(-1). Suppose T is an integral operator on L2(Ω), where Ω is a σ-finite measure space. Suppose the kernel K of T is non-negative, measurable, symmetric, and satisfies a certain quasi-metric condition (example: the Riesz potential). The basis for our applications is a new result that the kernel of I +T +T^2 +T^3 +... is bounded below by Kexp(c1K2/K) and above by Kexp(c2K2/K), where K2 is the kernel of T^2, for constants c1,c2 >0.

We apply this conclusion to study the time-independent Schrodinger operator −△−q, for q ≥ 0, on a bounded domain in Rn. If the domain is sufficiently smooth, we obtain estimates for the Poisson kernel associated to the Schrodinger operator in terms of the Poisson kernel for the Laplacian. For more general domains, called uniform domains, we obtain corresponding estimates for the Martin kernel. Applications regarding conditions on the possibly singular potential q that guarantee the existence of positive solutions for the Dirichlet problem for the Schrodinger operator are noted.

Abstract: Two classic questions - the Erdos distinct distance problem, which asks about the least number of distinct distances determined by points in the plane, and its continuous analog, the Falconer distance problem - both focus on distance.

Here, distance can be thought of as a simple two point configuration. When studying the Falconer distance problem, a geometric averaging operator, namely the spherical averaging operator, arises naturally. Questions similar to the Erdos distinct distance problem and the Falconer distance problem can also be posed for more complicated patterns such as triangles, which can be viewed as 3-point configurations. In this talk I will go through some of the history of such point configuration questions for triangles and end with some exciting recent progress.

Abstract: The classical uniformization theorem states that any simply connected Riemann surface is conformally equivalent to the unit disk, complex plane or Riemann sphere. The non-smooth uniformization problem now asks for the strongest possible extension after replacing smooth surfaces by metric spaces. On the way to answering this question in the setting of metric surfaces of locally finite Hausdorff 2-measure, we encounter the quasisymmetric and quasiconformal uniformization results of Bonk-Kleiner and Rajala, respectively. Moreover, we will outline how the solution of Plateau‘s problem can be used to provide a more general uniformization theorem. 

Abstract: The Loewner equation gives a correspondence between curves in the upper half-plane (or disk) and continuous real functions (called the “driving function” for the equation). When the driving function is Brownian motion, the result is Schramm-Loewner Evolution (SLE). In this talk, we will discuss links between multiple Loewner evolution and Dyson Brownian motion. First, we show that multiple radial SLE is generated by Dyson Brownian motion on the circle. Second, we show that in the chordal case, scaling the Brownian term to 0 in a branching Dyson Brownian motion generates tree embeddings in the halfplane. If time allows, we will also discuss the Dyson superprocess, which is the scaling limit of this driving measure when the branching structure is conditioned to converge to the continuum random tree. (Joint work with Gregory Lawler and Govind Menon)

Abstract: In 2001 Benjamini and Schramm proved a lemma about R^2 which has proven to be useful in proving convergence results of embeddings the samples of certain probability measures into space.  The lemma is actually an equivalent condition to a doubling metric space.  The original proof in R^2 involves probability and as such introduces (probably) unnecessary constants.  We discuss this proof and plead for someone to introduce (non-probabilistic) certainty on the constants.

Abstract: In this talk we will present explicit solutions to the radial and chordal Loewner PDE and we make a study of their geometry. Specifically, we study multi-slit Loewner flows, driven by the time-dependent point masses.

Abstract: In this talk, I will give an overview of some work carried out at the intersection of Rough Path Theory and Schramm-Loewner Evolutions (SLE) Theory. Specifically, I will cover a study of the Loewner Differential Equation using Rough Path techniques (and beyond). The Loewner Differential Equation describes the evolution of a family of conformal maps. We rephrase this in terms of Singular Rough Differential Equations. In this context, it is natural to study questions on the stability, and approximations of solutions of this equation. First, I will give an introduction to these two theories, and then I will present a result on the continuity of the dynamics and related objects in a natural parameter that appears in the problem. The first approach will be based on Rough Path Theory, and a second approach will be based on a constructive method of independent interest: the square-root interpolation of the Brownian driver of the Loewner Differential Equation. This is based on joint work with Dmitry Belyaev and Terry Lyons. If time permits, I will also touch on some further applications to the simulations of the SLE traces, that is based on joint works with James Foster and Jiaming Chen.

Abstract: A Cantor set in the 3-dimensional space is called tame if there exists a homeomorphism of the whole space that takes it to the standard Cantor set; it is called wild if it is not tame. Since their discovery by Antoine a 100 years ago, wild Cantor sets have played an important role in Analysis to create counterexamples of parameterization conjectures and to construct topologically complicated maps with analytic regularity. In this talk we will discuss wild Cantor sets and recent generalizations.

Abstract: A self-similar set is a set consisting of finitely many copies of itself that have minimal overlap. For example, the unit square, the Cantor middle-thirds set, the Von Koch Snowflake, and the Sierpinski gasket are all self-similar sets. We will introduce the notion of similarity dimension, and use it to determine properties such as the Hausdorff dimension of self-similar sets, and discuss a result of Remes, that if a self-similar set is connected, then it can be parametrized as a (1/s)-Holder curve, where s is its dimension. This can be viewed as a generalization of Peano's space-filling curve onto the square to general connected self-similar sets.

Abstract: Theorems on extensions of functions are some of the first major results encountered in analysis and adjacent fields. Some examples are the Tietze extension theorem of point-set topology and the Whitney extension theorem of analysis. In the field of analysis on metric spaces, true isometries are rare, so one often relaxes things by considering biLipschitz maps, which do not stretch or shrink distances too much. In this talk we discuss a biLipschitz extension theorem given by David and Semmes. In order to understand the proof we will first discuss some common tools used in analysis on metric spaces, including homogeneity of metric spaces, Whitney decompositions, and the aforementioned Whitney extension theorem.

Abstract: For a given symbol $\varphi$ in $L^\infty$ its Toeplitz operator acting on the Hardy space $H^2(\mathbb{D})$ is defined by $T_\varphi(f) = P_{H^2}\varphi f$, i.e. multiplication by $\varphi$ followed by projection back into $H^2$. In this talk, we'll cover some of basic theory of Toeplitz operators on $H^2$. We will then shift to talking about the spectra of Toeplitz operators acting on subspaces of $H^2$ whose multiplier algebras are finite codimension subalgebras of $H^\infty$. These algebras are called \emph{constrained algebras}, and in particular we will discuss the Neil algebra and the 2-point algebra. The Neil algebra and the 2-point algebras serve as the prototypical examples of finite codimensional subalgebras. This talk is based on joint work with Chris Felder and Douglas Pfeffer.

Abstract: In this talk, we develop a few operator-theoretic analogues of the F. and M. Riesz Theorem. We first recast the classical theorem in operator theoretic terms. Next, we establish similar results in the context of representations of the Cuntz algebra and in the context of the so-called free disc system. Notable differences from the classical setting will be highlighted. This is based on joint work with R. Clouâtre, R. Martin, and M. Jury.

Abstract: In this talk, we will discuss a new approach to normal families of mappings on subdomains of R^n via a local uniform omega-continuity condition. This generalizes a viewpoint put forward by Beardon and Minda in the plane. We will have in mind applications to families of quasiregular mappings in R^n, where our condition becomes a local uniform Holder requirement. This provides a framework to recover previously known results, as well as obtain new ones. This work is joint with Dan Nicks (Nottingham).

Abstract: In this talk, we study perturbations of elliptic operators on domains with rough boundaries. In particular, we focus on the following problem: suppose that we have ``good estimates'' for the Dirichlet problem for a uniformly elliptic operator $L_0$ (with corresponding elliptic measure $\omega_{L_0}$), under what optimal conditions, are those good estimates transferred to the Dirichlet problem for uniformly elliptic operator $L$ (with corresponding elliptic measure $\omega_{L}$) which is a ``perturbation'' of $L_0$? When the domain is 1-sided NTA satisfying the capacity density condition, we show that if the discrepancy of the corresponding matrices satisfies a natural Carleson measure condition with respect to $\omega_{L_0}$ then $\omega_L\in A_\infty(\omega_{L_0})$. Moreover, we obtain that $\omega_L\in RH_q(\omega_{L_0})$ for any given $1<q<\infty$ if the Carleson measure condition is assumed to hold with a sufficiently small constant. This is a joint work with Steve Hofmann, Jose Maria Martell, and Tatiana Toro.

Abstract: In this talk, we consider de Branges-Rovnyak spaces of analytic functions of $d$ variables in which the row operator of multiplication by $z$, $M_z = (M_{z_1},\ ...,\ M_{z_d}),$ is bounded. After investigating the properties of this operator, we present a result that characterizes which abstract row operator tuples $T = (T_1,\ ...,\ T_d)$ are unitarily equivalent to $M_z$ on some de-Branges Rovnyak space $H(B)$.

Abstract: The equilateral dimension of a metric space is the maximum number of points that are equidistant from each other. For instance, the equilateral dimension of the n-dimensional Euclidean metric space is n+1. We prove that the equilateral dimension of the (first) Heisenberg group is 4. This is a joint work with Joonhyung Kim.

Abstract:  Complex Hadamard matrices generate a class of irreducible hyperfinite subfactors with integer Jones index coming from spin model commuting squares. I will prove a theorem that establishes a criterion implying that these subfactors have infinite depth. I then show that Paley type II and Petrescu's continuous family of Hadamard matrices yield infinite depth subfactors. Furthermore, infinite depth subfactors are a generic feature of continuous families of complex Hadamard matrices. In this talk I will present an outline for the proof of these results.

Abstract:  The implicit function theorem (IFT) for C^1 maps f from R^{n+m} to R^n gives that if the rank of the derivative is n on an open set then after a (local) C^1 change of coordinates G in the domain the function foG satisfies foG(x_1,x_2,...,x_n,...,x_{n+m}) = h(x_1,x_2,...,x_n) for a local C^1 diffeomorphism h. The assumption on the rank is necessary. It is a consequence of Rademacher's differentiability theorem that Lipschitz maps coincide with C^1 maps on arbitrarily large parts of their domain. Thus, it is not surprising that a version of the implicit function theorem holds for Lipschitz maps -- the open set is replaced by "a subset of positive measure" and f and h are Lipschitz and bi-Lipschitz, respectively. Azzam and Schul (2012) and Hajlasz and Zimmerman (2020) proved IFT's for maps into metric spaces under a priori very different "nondegeneracy" conditions on f. In this talk I will discuss our result that shows that both of their conditions are equivalent to the rank condition of the so-called metric derivative, defined by Kirchheim in 1994. I will also mention the implications for the geometry of the level sets of the function and the coarea formula. If time allows, I will also discuss an application to the factorization of maps through metric trees that proved a conjecture of David and Schul.

Abstract:  In this talk, I discuss a joint work with Onninen, in which we study mappings f satisfying a version of the distortion inequality of K-quasiregular and K-quasiconformal maps with an extra additive term |f|^n σ^n, where σ is an L^n-integrable weight. We resolve a question of Astala, Iwaniec and Martin, who in the planar case n=2 showed a Liouville-type uniqueness theorem for mappings satisfying such a modified distortion inequality, and then asked whether this remains the case in higher dimensions. We also obtain sharp exponents of local Hölder continuity for such maps.

Abstract:  In geometric measure theory there is interest in understanding measures by studying interactions with particular collections of sets.   Here, we will discuss a recent characterization of Radon measures on R^n which are carried by the collection of m-Lipschitz graphs. That is, we will provide necessary and sufficient conditions for a Radon measure under which there exist countably many Lipschitz graphs that capture almost all of the mass.  Our characterization will involve only countably many evaluations of the measure. This is joint work with Matthew Badger.

Abstract:  We prove that arbitrary surfaces with a length metric can be approximated in the Gromov--Hausdorff sense by polyhedral surfaces with controlled geometry. One of the main tools, of independent interest, is a recent result on decomposing metric surfaces into convex triangles. We then apply our approximation theorem to the uniformization problem for metric surfaces, which asks for the existence of local quasiconformal parametrizations by a domain in the Euclidean plane. This talk covers joint work with Paul Creutz and Dimitrios Ntalampekos.

Abstract:  I will discuss a family of maps called Quasiregular curves that are generalizations of quasiconformal maps and holomorphic curves.  I will present their basic properties and show that if they satisfy a small distortion property, then they can be approximated by Möbius transformations.  This talk is based on joint work with Susanna Heikkilä and Pekka Pankka.

Abstract: In the past decades, boundary value problems for real elliptic equations have been extensively studied. On the other hand, the literature towards complex elliptic equations and systems is relatively limited. Many powerful tools, such as the maximum principle and De Giorgi-Nash-Moser regularity theory, are not available in the complex case. A new algebraic condition (p-ellipticity) is introduced to overcome such difficulty (Dindos-Pipher, Carbonaro-Dragicevic). Recently we used this idea to study the boundary value problem of complex elliptic systems. This is a joint work with M. Dindos and J. Pipher.

Abstract: Let f be a holomorphic function on the unit disc. According to Plessner's theorem, for almost every point ζ on the unit circle, either (i) f has a finite nontangential limit at ζ, or (ii) the image f(S) of any Stolz angle S at ζ is dense in the complex plane. In this talk, we will see that condition (ii) can be replaced by a much stronger assertion. This strong form of Plessner's theorem and its harmonic analogue on halfspaces also improve classical results of Spencer, Stein and Carleson. (Joint work with Stephen Gardiner)

Abstract: While Loewner hulls for real-valued driving functions are well-understood, complex-valued driving functions are an area of new exploration. In this talk, I describe the general properties of complex-driven Loewner hulls and use them to analyze a family of Loewner hulls driven by functions c*sqrt(1-t), for complex-valued c. Specifically, I outline our proof that phase transitions in the complex case are different than in the real case. I also discuss further questions about complex-valued Loewner hulls which motivate more research. This is joint work with Dr. Lind.

Abstract: The Loewner equation provides a correspondence between growing 2-dimensional sets (such as curves) and a driving function.  This long-standing tool gained prominence after the introduction of Schramm-Loewner Evolution (or SLE), which is the family of random curves whose driving function is a multiple of Brownian motion.  In this talk, I will give an introduction to the Loewner equation and SLE, and I will highlight the work of Huy Tran about the case when the driving function is complex-valued instead of real-valued.  This talk is first of a two-part series.  In the subsequent talk, Jeffrey Utley will discuss joint work regarding complex-valued driving functions.

Abstract:  During the last ten years there has been several interesting results on diffeomorphic approximation of Sobolev homeomorphisms. Now it is known that in the plane W^{1,p} -homeomorphisms can be approximated (in W^{1,p} -norm) with smooth diffeomorphisms. These results lead to characterization of possible limits of W^{1,p}-homeomorphisms in terms of certain monotonicity conditions. When p<2 these limits are not necessarily continuous but may have a small set of discontinuities. I will discuss recent joint work with D. Campbell (U. of Hradec Kralove) and E. Radici (EPFL) where we give similar characterization of limits of homeomorphisms of bounded variation. These mappings may be discontinuous even on sets of positive 1-dimensional Hausdorff measure and  could, at least in principle, be used to model fracturing of materials.

Abstract:  In complex dynamics, Koenigs Linearization Theorem allows us to conjugate geometrically attracting or repelling fixed points of a holomorphic map to the multiplier map. We will explore a higher dimensional version of this for quasiregular maps, which are differentiable almost everywhere. To address the issue of being differentiable almost everywhere, we use the notion of generalized derivatives. As long as a quasiregular map has a single generalized derivative about the attracting (repelling) fixed point, we can use a new tool called the Zorich Transform to find the map we want to use for our conjugation of the quasiregular map.

Abstract:  The Drury-Arveson space $H^2_d$ is the space of analytic functions in the unit ball of $\mathbb C^d$ defined by the reproducing kernel $k_w(z) = \frac{1}{1-\langle z,w\rangle}$. Alternately an analytic function $f$ is in $H^2_d$ if $R^N f \in L^2((1-|z|^2)^{2N-d}dV)$ for some (and hence all) $N >(d-1)/2$, where $R$ denotes the radial derivative operator $R=\sum_{i=1}^d z_i \frac{\partial}{\partial z_i}$, and $dV$ denotes Lebesgues measure on the ball. The space has been shown to be of importance for the theory of tuples of commuting Hilbert space operators. The emerging function theory for $H^2_d$ takes advantage of the facts that the reproducing kernel has the Pick property and that the space is a weighted Besov space. In this talk I will speak about joint work with Alexandru Aleman,  Karl-Mikael Perfekt, Carl Sundberg, and James Sunkes on multipliers and cyclic vectors for $H^2_d.$

Abstract:  A metric sphere not a quasisphere but for which every weak tangent is Euclidean. For all n ≥ 2, there exists a doubling linearly locally contractible metric space X that is topologically a n-sphere such that every weak tangent is isometric to the Euclidean space R^n but X is not quasisymmetrically equivalent to the standard n-sphere. We give a sketch of the construction of such a metric space. We further show that one can make X a length space, and that any other such construction must have Hausdorff dimension n.

Abstract:  This talk discusses recent work (joint with Matthew Badger, UCONN) on generalizations of the Analysts' Traveling Salesman Theorem to uniformly smooth and uniformly convex Banach spaces (e.g., l_p spaces).  In 1990, motivated by problems in Singular Integral Operators, Peter Jones posed and solved his celebrated Analysts' Traveling Salesman Problem: namely, to characterize all subsets of rectifiable curves in the plane.  Since then, many authors have contributed, proving similar results in Euclidean spaces, Hilbert Spaces, Carnot groups, for 1-rectifiable measures, etc.  This talk will give a broad overview of some of these results and their core ideas.  In the end, we will discuss the challenges in Banach spaces and what generalizations hold there.  This talk will include lots of pictures and examples.

Abstract:  The Plateau-Douglas problem is an extension of the classical Plateau's problem. Given k disjoint Jordan curves in an ambient space, and a smooth surface M with k boundary components, it asks to find a (weakly conformal) map from M of minimal area, spanning the Jordan curves. The homotopic variant of this problem also prescribes some topological data of the map. This and related problems have been studied since the 60's by Shoen-Yau, Lemaire, White and Jost among others, who showed existence of minimizing maps in so-called 1-homotopy class under certain non-degenaracy conditions. In this talk I will discuss an existence proof which relies only on the presence of a local quadratic isoperimetric inequality,and thus allows e.g. Alexandrov spaces as ambient spaces. This approach is conceptually simple, relying on work of Wenger-Lytchak, and uses techniques from analysis on metric spaces. Joint work with Wenger.

Abstract:  We'll introduce a broad class of PDEs which arise from the Calculus of Variations. After producing specific examples of some PDEs that fall within this class and stating new results about the regularity of solutions we outline a Moser Iteration based argument to derive a harnack inequality for weak solutions. This demonstrates that for 0th order regularity, the aspect of "ellipticity" which is useful is the fixed homogeneity. This raises the question of whether or not some notion of convexity can be used to replace ellipticity and still recover a robust theory for 1st order regularity of solutions to anisotropic PDEs. 

Abstract:  A quasiconformal tree is a doubling metric tree in which the diameter of each arc is bounded above by a fixed multiple of the distance between its endpoints. Quasiconformal trees generalize the well-known notion of quasiarcs. In this talk, inspired by results of Herron-Meyer and Rohde for quasiarcs, we construct a catalog of metric trees in a purely combinatorial way, and show that every quasiconformal tree is bi-Lipschitz equivalent to one of the trees in our catalog. We then discuss how such constructions apply to a special class of metric spaces with good subdivisions. The talk is based on joint work with G. C. David.

Abstract:  We define the Fefferman-Stein sharp function and show that for a function in L^p, the L^p norm of the sharp function is equivalent to the norm of the function itself, for 1<p< infinity.  If time permits, we will discuss how that fact relates to interpolation and Calderon-Zygmund operators.

Abstract:  This talk is about some classical subjects in complex analysis and geometric function theory such as relations between conformal invariants and their connection to spaces of holomorphic functions. 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.

Abstract:  A complete classification of NxN complex Hadamard matrices remains elusive for N>5. Hadamard matrices were first studied in the 1800's, but in recent years applications for complex Hadamard matrices have emerged in the field of quantum information theory. In this talk we present a new result that gives a further restriction on the possible "directions" in which we can approach the Fourier Matrix with a sequence of complex Hadamard matrices. 

Abstract:  de Branges-Rovynak spaces are a class of reproducing kernel Hilbert spaces determined by a vector-valued, multivariable function $b$. We present a result that characterizes which de Branges-Rovynak spaces possess the useful complete Nevanlinna-Pick property. This generalizes the recent work of Cheng Chu, who answered the same question for de Branges-Rovynak spaces corresponding to scalar-valued functions of one variable.

Abstract:  We extend Carleson’s characterization of interpolating sequences to sequences of matrices with spectra in the unit disk. We will also discuss the multi-variable case by considering sequences of d-tuples of matrices, both in the commutative and the non-commutative case.

Abstract:  According to the Ahlfors-Gehring theorem, a simply connected domain D in the extended complex plane is a quasidisk if and only if there exists a sufficient condition for the injectivity of a holomorphic function in D in relation to the growth of its Schwarzian derivative. We extend this theorem to harmonic mappings by proving a univalence criterion on quasidisks. We also give sufficient conditions for the existence of homeomorphic and quasiconformal extensions to the whole plain for harmonic mappings defined on quasidisks. The Ahlfors-Gehring theorem has been generalized to finitely connected domains D by Osgood, Beardon and Gehring, who showed that a Schwarzian criterion for univalence holds in D if and only if the components of the boundary of D are either points or quasicircles. We extend this theorem to harmonic mappings.

Abstract:  A Thurston map is a branched covering map of the 2-sphere which is not a homeomorphism and for which every critical point has a finite orbit under iteration of the map. Frequently, a Thurston map admits a description in purely combinatorial-topological terms. In this context it is an interesting question whether a given map can (in a suitable sense) be realized by a rational map with the same combinatorics. This question was answered by Thurston in the 1980's in his celebrated characterization of rational maps. Thurston's Theorem roughly says that a Thurston map is realized if and only if it does not admit a Thurston obstruction, which is an invariant multicurve that satisfies a certain growth condition. However, in practice it can be very hard to verify whether a given map has no Thurston obstruction, because, in principle, one would need to check the growth condition for infinitely many curves. In this talk, we will consider a specific family of Thurston maps with four postcritical points that arises from Schwarz reflections on flapped pillows (a simple surgery of a polygonal sphere). Using a counting argument, we establish a necessary and sufficient condition for a map in this family to be realized by a rational map. In the last part of the talk, we will discuss a generalization of this result which states that, given an obstructed Thurston map with four postcritical points, one can eliminate obstructions by applying a so-called blowing up operation. These results are joint with M. Bonk and M. Hlushchanka.

Abstract:  Let $\mathbb{D}$ denote the open unit disc in the complex plane. An analytic function $B : \mathbb{D} →\ell_2$ is called a Schur function, if $\|B(z)\|_{\ell_2} < 1$ for every $z \in \mathbb{D}$. For every such Schur function we will define the associated de Branges-Rovnyak space $H(B)$, a space of analytic functions on the open unit disc.

If $\log(1 −\|B(eit)\|^2_{\ell_2})$ is integrable, then there is an essentially unique outer function a that satisfies $|a(e^{it})|^2 + \|B(e^{it})\|^2_{\ell_2} = 1$ a.e.. We will show that in the case when $B$ is rational, then $a(z) = a(0)\frac{p(1/z)}{q(1/z)}$, where $p$ and $q$ are the characteristic polynomials of two associated linear transformations that act on a canonical finite dimensional subspace of $H(B)$. If $B$ is rational and if the multiplication operator $f \to zf$ on $H(B)$ is what is called a $2n$-isometric operator, then all the zeros of $p$ lie in the unit circle and the norm of the functions in $H(B)$ can be expressed by use of what we call the $n$-th order local Dirichlet integral.

Abstract:  The classical Loewner differential equation provides a correspondence between real-valued, continuous functions and compact sets in the upper half-plane. In his 2017 paper, Huy Tran introduces the idea of solving the Loewner differential equation with complex-valued, continuous functions. In this talk, we explore the “world” of the complex-valued Loewner equation and provide the necessary background to understand Tran’s paper (which will be presented in Jack Ryan’s oral examination in May). An emphasis will be placed on examples, simulations, and comparing the complex-valued case to the real-valued case.

Abstract:  Given a Lipschitz map, it is often useful to chop the domain into pieces on which the map has simple behavior. For example, depending on the dimensions of source and target, one may ask for pieces on which the map behaves like a bi-Lipschitz embedding or like a linear projection. Things are even better if this decomposition is quantitative, i.e., with bounds independent of the particular map or spaces involved. After surveying the question of bi-Lipschitz decomposition, we will discuss the more complicated case in which dimension decreases, e.g., for maps from R^3 to R^2. This is very recent joint work with Raanan Schul, improving a previous result of Azzam-Schul.

Abstract:  During the past two weeks, Dr. Nicoara derived conditions that the first and second order "derivatives" of a sequence of commuting squares must satisfy if they exist. In the case of the commuting square obtained from the finite group Z_n, the sequence of commuting squares corresponds to a sequence of Hadamard matrices converging to the Fourier matrix F_n. We will present the main result from a paper by Dr. Nicoara and Dr. White that states the second order conditions follow automatically from the first order conditions in this case. We also state a conjecture of Bengtsson on the existence of matrices satisfying the third order conditions of the commuting square obtained from the finite group Z_n in the case where n is the product of three distinct primes.

Abstract:  Commuting squares arise as invariants and construction data in Jones’ theory of subfactors. In an attempt to understand the structure of the moduli space of commuting squares, we ask the following question: When is a commuting square the limit of a sequence of non-isomorphic commuting square? To answer this, we develop a method of taking higher order directional derivatives of the commuting square relations. We also present applications to the theory of Hadamard matrices. 

Abstract:  We will present sharp forms of Poincare's inequality and discuss how they follow from mapping properties of the Riesz potential I_1.  We then discuss Hedberg's proof of these mapping properties, using the Hardy-Littlewood maximal function.  As time allows, we will discuss weighted versions of these results, which can be proved by adapting Hedberg's technique.

Abstract:  The first talk will be an introduction to Poincare inequalities for analysts who don't have a background in PDE.  We will explain how the simplest version of the Poincare inequality gives existence of weak solutions to Poisson's equation.  We will then discuss sharp versions of the Poincare inequality and how they follow from mapping properties of the Riesz potential.  For people in the PDE community, the first talk will be familiar and probably not worth their time, but it is background for the second talk, in which we present Hedberg's proof of the Poincare inequality using the Hardy-Littlewood maximal function, and sketch how that proof can be used to prove a version of Poincare's inequality for matrix weights in the A_p class.

Abstract: In this talk, we will introduce the concept of the dimension of a Hilbert space over a von Neumann algebra in order to define the Jones index of an inclusion of finite factors $N\subset M$ (an invariant for the 'position' of N in M). We use this inclusion to perform Jones's "basic construction" to obtain a new factor containing M. Iterating this procedure yields a tower of factors satisfying remarkable properties. In Vaughn Jones's seminal paper "Index for Subfactors" these properties are used to prove the surprising result that not every real number >1 appears as a Jones Index.

Abstract: The Loewner differential equation is a tool that provides a correspondence between compact sets in the closure of the upper half-plane and continuous real-valued functions. In this talk, we will explore some of the natural questions that arise in the study of this theory and provide a brief survey of results that give partial answers to these questions. We will then illustrate a specific example of a continuous function and explore its behavior as it pertains to the relevant questions asked in the first half of the talk. 

Abstract:  The discrete Fourier transform (DFT) of any size is known to have four eigenvalues: 1,-1, i, and -i. A proof of this result will be presented utilizing projection matrices defined in terms of the DFT. We will use these projections to further prove a result concerning the multiplicities of these eigenvalues, a problem that has interesting connections to Gauss sums.

Abstract:  The traveling salesman problem, one of the most renowned problems in computer science, asks for the shortest path that passes through a given finite set of points E in the Euclidean space. What if the set E is infinite? Can we still visit all of its points in finite time? Even more generally, given an arbitrary set E in the space (possibly a fractal), when is it possible to construct a nice map (Holder, Lipschitz) from the unit interval that contains E in its image? In this talk we discuss this generalized traveling salesman problem which has been a very active field of research in geometric measure theory in the last 30 years.