Mathematics

Permanent URI for this collectionhttps://digital.lib.washington.edu/handle/1773/4939

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Automorphisms and homological properties of locally gentle algebras
(2026-02-05) Ford, Sarafina; Zhang, James
In this work, we consider the infinite dimensional generalizations of string algebras, referred to as locally string algebras, giving special attention the generalizations of gentle algebras, known as locally gentle algebras. We describe the prime spectrum and Jacobson radical of a locally string algebra. We show that, up to an inner automorphism and a unique graded automorphism, an automorphism of a string algebra acts as permutations on stationary paths and decomposes into a composition of exponential automorphisms. For the locally gentle algebras, we give an explicit injective resolution and combinatorial descriptions of their homological dimensions. We classify the Artin-Schelter Gorenstein, Artin-Schelter regular, and Cohen-Macaulay locally gentle algebras, and provide analogues of Stanley's theorem for locally gentle algebras.
On Enumerating and Generalizing Higher Bruhat Orders with Connections to Machine Learning
(2026-02-05) Chau, Herman; Billey, Sara
The higher Bruhat orders B(n, k) were introduced by Manin–Schechtman to study discriminantal hyperplane arrangements and subsequently studied by Ziegler, who connected B(n, k) to oriented matroids. In this thesis, we study three topics: the enumeration of B(n, k), a generalization of B(n, k) to identity intervals in the affine symmetric group, and connectionsto machine learning techniques in mathematical research. Our work on the enumeration of B(n, k) improves upon Balko’s asymptotic lower and upper bounds on |B(n, k)| by a factor exponential in k, gives a proof of Ziegler’s formula for |B(n, n - 3)|, and relates B(n, n - 4) to totally symmetric plane partitions. Our work on the generalization of B(n, k) to identity intervals in the affine symmetric group answers Elias’s conjecture on the acyclicity of directed braid graphs on commutation classes of reduced expressions of an affine permutation in the affirmative. This generalization was inspired by the work of Elias-Hothem in generalizing B(n, k) to identity intervals in the ordinary symmetric group. Our work on exploring machine learning (ML) techniques in mathematical research involves releasing the Algebraic Combinatorics Dataset Repository to the ML community and led to two lessons learned about using ML for mathematical discovery.
Schubert Objects
(2007) Kelly, Elizabeth
Schubert polynomials arose from questions involving enumerative and algebraic geometry, representation theory, and algebraic topology. They have been studied from a variety of perspectives, each with its own combinatorial object [1, 4, 5, 17, 15, 16, 28, 50, 66]. In this dissertation, the combinatorial objects which index the monomials in a Schubert polynomial are called Schubert objects. There are many such objects and one of the main goals of this dissertation is to illumination the bijections between them. In addition to exploring the bijections between Schubert objects, we explore different methods of constructing them. The construction methods are all developed using trees of Schubert objects and taking the collection of leaves at the end of the tree. We introduce a new method to compute the decomposition of Schubert polynomials into key polynomials. We also define a new operator, called split, which provides an alternative approach of creating a tree of rc-graphs. A new Schubert object is explored, called an inversion filling. We discuss a special case of inversion fillings, the Grassmannian permutation case, which gives rise to a left divided difference operator on semistandard Young tableaux. In addition, we describe the previously known construction of skyline fillings and their connection to other Schubert objects.
Higher-Dimensional Analogues of Two Theorems of Orponen on Exceptional Sets
(2025-10-02) Bushling, Ryan Edward; Wilson, Bobby L
Let $A \subseteq \mathbb{R}^n$ be a Borel set of Hausdorff dimension $\dim_{\mathrm H} A = s$ and, for each $m$-plane $V$ in the Grassmannian $\mathbf{Gr}(n,m)$, let $\pi_V : \mathbb{R}^n \to V$ be the orthogonal projection of $\mathbb{R}^n$ onto $V$. \textit{Marstrand's projection theorem} states that $\dim_{\mathrm H} \pi_V(A) = \min \, \{s,m\}$ for almost every $V \in \mathbf{Gr}(n,m)$, and \textit{Marstrand's slicing theorem} states that the set of $\mathbf{x} \in V$ such that $\dim_{\mathrm H} \big( \pi_V^{-1}(\mathbf{x}) \cap A \big) = s-m$ has positive Lebesgue measure for almost every $V \in \mathbf{Gr}(n,m)$ provided $s > m$. Informally, for all but a few subspaces $V$, the shadow of $A$ on $V$ has the largest possible dimension, and there are many sections of $A$ orthogonal to $V$ of essentially maximal dimension. In this dissertation, we prove two results concerning the sizes of \textit{exceptional sets} of projections and slices—the null sets of parameters $V$ for which the conclusion of Marstrand's projection or slicing theorem fails, respectively. These are, in particular, higher-dimensional analogues of two theorems of Tuomas Orponen. The first result states that, if linear maps are \textit{almost dimension conserving} for $A$, then the exceptional sets of orthogonal projections of $A$ have small packing dimension. This applies, for example, to (weakly) Furstenberg homogeneous sets and to certain self-similar and graph-directed sets. The second concerns slices by fibers of the \textit{generalized projections} of Peres and Schlag. Roughly, the conclusion of Marstrand's slicing theorem holds for very general families of nonlinear projections, and the exceptional sets of slices have small Hausdorff dimension. In fact, not only is the set of exceptional slices of $A$ small, but the union of the exceptional sets of all positive-$\mathcal{H}^s$-measure Borel \textit{subsets} of $A$ is small.
Parameterizations and rectifiability via geometric functions and singular integral operators
(2025-08-01) Casey, Emily; Toro, Tatiana; Wilson, Bobby
Geometric measure theory provides the framework to examine the geometry of sets and measures that are not ``smooth'' enough to be studied using the classical methods of differential geometry. These rough sets arise naturally in many settings, for instance as the minimizers of certain geometric variational problems, e.g \cite{R60}. A relatively recent technique used to analyze the fine properties of sets of this type is through the study of geometric square functions, which capture simple geometric information about the set at each scale. This technique has gained popularity starting with the work of Jones \cite{J90} and David-Semmes \cite{DS91}. In \cite{Bi87} a discontinuous geometric function appeared, known as the Carleson $\varepsilon$-function. In \cite{JTV21} and \cite{FTV23} the authors prove that qualitative control on the Carleson $\varepsilon$-function, and higher-dimensional analogues, characterize tangent points of certain domains. In this thesis, we study higher regularity versions of Carleson's conjecture in the plane and in higher dimensions. As for the study of the ``smoothness'' of a given measure on $\mathbb{R}^n$ a commonly used tool in geometric measure theory are \textit{singular integral operators}. The fine properties of measures involve two things: the ``smoothness'' of the set on which the measure lives and the behavior of the measure on that set. In \cite{DS91}, the authors prove that uniformly rectifiable measures are characterized by the $L^2$-boundedness of all Calder\'{o}n-Zygmund operators. In \cite{M95} and \cite{MP95}, the authors prove that the almost everywhere existence of principal values of the Riesz transform characterizes rectifiable measures. In this thesis we extend the work of \cite{M95, MP95} to a broader class of singular integral operators, in particular, a class of anisotropic Riesz kernels.
Growth of Reciprocal pseudo-Anosovs on Lattice Surfaces
(2025-08-01) Helms, Paige; Athreya, Jayadev
Motivated by number theory, Reciprocal geodesics were first introduced by Sarnak [23], who studied theirasymptotic growth on the modular curve. Erlandsson-Souto [7] gave a geometric interpretation and gener- alization of reciprocal geodesics and a dynamical proof of asymptotic counting results in the more general setting of hyperbolic orbifolds H2/Γ where Γ is a lattice. We introduce the notion of reciprocal pseudo-Anosov maps of translation surfaces and establish a correspondence between such maps and reciprocal geodesics. We then show how to apply the Erlandsson-Souto results to compute the asymptotic growth for particular families of highly symmetric surfaces known as lattice surfaces or Veech surfaces [26], and to in fact compute the constants for the asymptotic growth of pseudo-Anosov maps on certain families of lattices surfaces, called Bouw-MÂ¨oller [5] and primitive square-tiled surfaces [24]
SchrÃ¶dinger Operators with Lattice Invariant Potentials
(2025-08-01) Lyman, Curtiss Frank; Drouot, Alexis
We develop a systematic framework to study the dispersion surfaces of SchrÃ¶dinger operators H = −∆+V, where the potential V is both periodic with respect to a lattice Λ and respects its symmetries. Our analysis relies on an abstract result, previously proven by Franz Rellich [Rel40] and which we prove using an alternative approach inspired by methods developed by Tosio Kato [Kat95]: if a self-adjoint operator depends analytically on a parameter, then so do its eigenvalues and eigenprojectors in a neighborhood of the real line. Using this and techniques from Floquet-Bloch theory and representation theory, we prove a series of results that can be used to analyze the operator H where the lattice Λ is arbitrary. As an application of this framework, we describe the generic structure of some singularities in the band spectrum of SchrÃ¶dinger operators invariant under various two- and three-dimensional lattices. Specifically, we study the square, hexagonal, rectangular, simple cubic, body-centered cubic, face-centered cubic, and stacked hexagonal lattices, in the process reproducing results due to [Kel+18] and [FW12], and also proving a conjecture of [GZZ22].
Representations of Configurations and their Moduli
(2025-08-01) Salinas, Juan Jose; Lieblich, Max D
This thesis develops a scheme-theoretic framework for studying configurations—collections of points and blocks with incidence relations—and their representations in algebraic geometry. Generalizing classical notions, we define configurations as triples of schemes over a base and show that the moduli functor of representations into a geometric configuration is representable by a scheme. We construct fine moduli spaces for nondegenerate representations and realizations, and study their behavior under deformation. Applications include a modern perspective on MnÃ«v–Sturmfels universality and connections to classical projective theorems, highlighting new interactions between configuration theory, moduli spaces, and algebraic geometry.
Moduli of Very Ample Line Bundles
(2025-08-01) Nugent, Brian; KovÃ¡cs, SÃ¡ndor
Let X be a projective variety over a field. In this paper, we will construct a modulispace of very ample line bundles on X. In doing so, we develop a generalization of Fitting ideals to complexes of sheaves on X. We give other applications of these Fitting ideals such as constructing Brill-Noether spaces for higher dimensional varieties and giving a scheme structure to the locus where the projective dimension of a module jumps up.
The Convex Algebraic Geometry of Higher-Rank numerical Ranges
(2025-08-01) Nino-Cortes, Jonathan Andres; Vinzant, Cynthia
The higher-rank numerical range is a convex compact set generalizing the classical numerical range of a square complex matrix, first appearing in the study of quantum error correction. In this thesis, we will discuss some of the real algebraic and convex geometry of these sets, including a generalization of Kippenhahn’s theorem, and describe an algorithm to explicitly calculate the higher-rank numerical range of a given matrix. We will also discuss the inverse field of values problem, an inverse problem on the numerical range. We focus on the geometric properties of the set of solutions. Finally, we consider an analogous problem for higher-rank numerical ranges and show how to solve it using the ideas behind the proof of convexity for these sets.
Time-dependent Time Fractional Equations and Probabilistic Representation
(2025-05-12) Choi, Joon Yong; Chen, Zhen-Qing
We study the nonlocal initial-value problem of the form \begin{align*} \sL u(x, t) &= h(x, t) \quad \text{for}\ (x,t)\in \R^d\times (0,\infty), \\ u(x,t) &= f(x) \quad \text{on}\ \R^d\times (-\infty, 0] \end{align*} where $\sL$ is an integro-differential operator given by \begin{eqnarray*} && \hskip -0.2truein \mathcal L \varphi(x,t) \nonumber \\ & =& \frac{1}{2}\sum_{i,j=1}^{d} a_{ij}(x,t)\:\partial_{x_ix_j}^{2} \varphi(x,t) + \sum_{i=1}^{d} b_i(x,t)\:\partial_{x_i} \varphi (x,t) % + \gamma (x, t) \partial_t u(x, t) \\ & &+ \int_{\R^d\times \R\setminus\{(0,0)\}} \bigg[ \varphi (x+y,t-s)-\varphi (x,t) -\nabla_x \varphi(x,t) \cdot y \: \mathbf 1_{\{|y|\le1\}} \bigg] J(x,t;dy,ds). \nonumber \end{eqnarray*} For the case where the jump measure takes form $J(x,t; dy, ds) = j(x,t)\delta_0(dy)\nu(ds)$ for some L\'evy measure $\nu$ on $\R$, if $f\in C_{b}^{2,\alpha}(\R^d)$ and $h\in \text{Lip}(\R^d\times \R)$ satisfies $h(\cdot, t) \in C^{2,\alpha}(\R^d)$ for each $t\in \R$ for some $\alpha \in (0,1)$, then the above parabolic equation has a unique classical solution. See Theorem \ref{TimeFracEqn} for precise statement. When the joint process $(X,Z)$ generated by $\sL$ is a L\'{e}vy process, i.e. $a_{ij}(x,t)=a_{ij}$, $b_i(x,t) = b_i$ are constants and $J(x,t; dy, ds) = J(dy, ds)$ is a L\'{e}vy measure on $\R^{d} \times \R$, and if $f\in C_{b}^{2}(\R^d)$ and $h(x,t) = \int_{\R^d \times [t,\infty)} (f(x+y)-f(x))J(dy, ds)$, then the above parabolic equation also has a unique classical solution. In this case, the solution $u$ is a bounded and continuous function on $\R^d\times \R$ and $u(\cdot, t)\in C_{b}^{2}(\R^d)$ for each $t\in \R$. See Theorem \ref{MainTheorem4LevyCase} for precise statement. Our method is probabilistic and direct. Probabilistic representation of solutions to the time-fractional equations is given.
Applications of Discrepancy Theory to Machine Learning
(2025-05-12) Heck, Rainie; Rothvoss, Thomas
In the combinatorial discrepancy theory problem, one is given a base set [n] and a collectionof subsets S_1, ... , S_m ⊆ [n] and asked to color the elements of [n] so that each set S_i is as balanced as possible. This simple set-system based question has spawned a multitude of generalizations and found many recent applications in various areas of machine learning. In this dissertation, we introduce the discrepancy problem and its geometric generalization, the vector balancing problem, and then prove two sets of results about applications of the discrepancy problem to machine learning. To conclude, we prove a more abstract result about the vector balancing constant for zonotopes. The first application to machine learning– coresets for kernel density estimators–gives both improved bounds over existing results for a variety of applications of interest, as well as a new chaining-based technique that allows for a more data-driven approach to the problem. The second application–to quantization of neural networks–is a new application of discrepancy theory that provides improvements over existing algorithmic approaches to the problem. Finally, our results for vector balancing for zonotopes address and nearly resolves an open conjecture, leaving only a log log log d gap.
Reciprocity and local to global principles on $p$-adic function fields
(2025-01-23) Rivera, Carlos; Viray, Bianca
In 2021, Olivier Wittenberg introduced a new obstruction to the local to global principle for varieties over $p$-adic function fields, the reciprocity obstruction, analogue to the Brauer-Manin obstruction for number fields, but unlike previous versions of it, including all divisorial valuation of the $p$-adic function field. We show a refinement theorem for the obstruction over blow ups of integral models of the $p$-adic function field, compare the obstruction with previous ones, construct some examples, and show the obstruction is the only one for connected $0$-dimensional varieties. We also define a reciprocity obstruction to the field patching method of Harbarter, Hartmann and Krashen, and use the refinement theorem to show it agrees with Wittenberg's version when both apply. This reduces the computation of the reciprocity pairings over all closed points of all integral models to a finite one. Finally, we analyse the existence of patch points in the field patching method from an analytic point of view, providing a conjectural reason for their existence.
Harnack inequality for nonlocal operators
(2025-01-23) Meng, Xiangqian; Chen, Zhenqing
Harnack inequalities are a fundamental property in both probability theory and analysis.The scale-variant Harnack inequalities play an important role in studying various properties, such as the regularity of harmonic functions in probability and analysis. This thesis focuses on scale-invariant Harnack inequalities for a class of nonlocal operators and a class of weakly coupled nonlocal operators. In Chapter 1, we show the scale-invariant elliptic Harnack inequality holds for a class of nonlocal operators, which are second order elliptic diﬀerential operators perturbed by non- local operators. We assume the existence of a conservative Hunt process corresponding to each operators $L$ in that class, and establish the scale-invariant Harnack inequalities for nonnegative functions that are $L$-harmonic. This is achieved by using Krylov estimate approach, where the comparison constant depends solely on the parameters of the class of the operators. We also establish the HÃ¶lder regularity for bounded $L$-caloric functions. In addition, we demonstrate that the scale-invariant parabolic Harnack inequality holds for nonnegative $L$-caloric functions and establish HÃ¶lder regularity for bounded $L$-caloric functions. In Chapter 2, utilizing the result from Chapter 1, we consider a system $G$ of nonlocaloperators $\{L_i,i = 1,..m\}$, as discussed in Chapter 1, connected by an index switching process $\{Λi,i = 1,...,m\}$ determined by its switching rate matrix $Q$. Such a system of operator whose coupling terms do not involve the derivatives of the unknown functions is called a weakly coupled nonlocal system. Weakly coupled systems are widely investigated in the field of physics, finance and engineering, etc. Through a piecing-together procedure, there exists a Hunt process corresponding to the weakly coupled operator $G$ wthin a certain class. Using the two-sided scale-invariant Green function estimates, we prove the scale-invariant Harnack inequalities for the weakly coupled nonlocal operators $G$. Under the irreducibility assumption of the switching matrix, we further derive a full rank scale-invariant Harnack inequality for this class of weakly coupled operators. The Appendix A serves as a complimentary part to Chapter 1. One of the essentialintermediate components in Krylov’s estimate approach presented in Chapter 1 is a lower bound of the hitting probability estimate. This result, along with other related important theorems, all ultimately relies on the equivalence between the Martingale problems and SDE for this class of nonlocal operators. However, there is limited literature available on this topic in English. Therefore, for both learning purposes and the reader’s convenience, we summarize and provide the detail explanations of some existing results originally in French. In addition, the proof of the support theorem for diﬀusion processes and the Krylov’s estimate for diﬀusion operators have also been rewritten and clarified in details.
Towards Cohomology of Real Closed Spaces
(2024-10-16) Clarke-James, Tafari; KovÃ¡cs, SÃ¡ndor
It was shown by Claus Scheiderer prior to 1994 that real closed spaces have \'{e}tale cohomology. Following Scheiderer, study of real closed spaces fell out of fashion and o-minimal geometry became the focus for those at the intersection of model theory and geometry. I decided to breathe new life into the theory of real closed rings and spaces, as studied by Schwartz in 1989. In Section 1, I build the fundamentals of the theory using as little machinery as possible, and presented them as clearly as I could. Hidden gems include a full proof that real closed rings are closed under limits and colimits. In Section 2, I give an introduction to the category of real closed spaces in the first half. In the second half, I construct an equivalence of topoi between Scheiderer's sheaves on the real \'{e}tale site, and sheaves on a real \'{e}tale site $\rce/X$ of my creation. Since $\text{Sh}(\rce/X)$ can be defined without the use of $G$-topoi, the equivalence of topoi renders Scheiderer's theory computable. I end with a discussion of how one might use motivic cohomology to better understand recent results of Annette Huber in \cite{no_deRham_huber}.
Statistics of the Minimal Denominator Function and Short Holonomy Vectors of Translation Surfaces
(2024-10-16) Artiles Calix, Albert Alejandro; Athreya, Jayadev S
The main results of this dissertation are the computation of the limiting distribution of the minimal denominator function, the computation of Chen-Haynes distributions for a broad class of equivariant processes, and the computation of the probability density distribution of short holonomy vectors of Veech surfaces.
Discrete Approximations to Time-changed Brownian Motions
(2024-10-16) Yu, Yang; Chen, Zhen-Qing
We give a general discrete approximation scheme of time-changed Brownian motions on $\mathbb{R}^d$. Our approximation scheme works for any smooth measure with full quasi-support on $\mathbb{R}^d$ with suitable initial distributions. Under some mild conditions on the smooth measure, the discrete approximation scheme works for every starting point.
Stochastic Analysis on Graphons
(2024-10-16) Tripathi, Raghavendra; Pal, Soumik Prof.
We say that a function on the space of symmetric matrices is \emph{invariant} if it is invariant under the simultaneous permutation of rows and columns of the input matrix by the same permutation. Homomorphism densities of finite simple graphs offer a rich class of examples of invariant functions that prominently feature in several areas of mathematics. In this thesis, we study two natural classes of interrelated problems. The first problem concerns the optimization of invariant functions in large dimensions. The second problem concerns the analysis of the dynamics on graphs/matrices where the evolution of coordinates depends on the full graph/matrix via an invariant function as the dimension goes to infinity. An important theme of the present thesis is that due to the symmetry of invariant functions, their optimization and dynamics can be reduced to optimization and dynamics on the space of graphons as the dimension of the underlying space goes to infinity. The rich geometry and the analytical properties of the space of graphons make the problems on the space of graphons more tractable. We develop a notion of gradient flow on the space of graphons following the general theory of gradient flows in metric spaces. We show that under mild differentiability assumption, any invariant function on the space of graphons admits a gradient flow which is an absolutely continuous curve with respect to the invariant $L^2$ metric. Furthermore, under appropriate convexity and differentiability assumptions, we show that the Euclidean gradient flows of invariant functions converge to the gradient flow of a suitable function on the space of graphons. We then consider a class of symmetric $n\times n$ matrix-valued diffusions where the drift is given by an invariant function. Such diffusions arise, for example, as the scaling limits of stochastic gradient descent of an invariant function. We establish a propagation of chaos phenomenon for such matrix-valued processes. That is, we show that any finite collection of coordinates of such processes becomes conditionally independent as $n\to \infty$ and that a uniformly random coordinate of such processes satisfies a novel graphon McKean-Vlasov SDE, in $n\to \infty$ limit. As a consequence of this, we obtain that these matrix-valued processes converge to a deterministic curve on the space of graphons. We also construct a Metropolis chain, with a novel relaxation step, whose state space is the stochastic block model with $r$ communities and $n$ individuals in each community. We show that fixed $r$, under appropriate scaling of parameters, the $r\times r$ matrix of connection probabilities between communities converges to a diffusion of the previous type. In particular, as $r\to \infty$, the connection probability between communities becomes conditionally independent. This allows us to prove that the trajectory of this Metropolis chain is concentrated near a deterministic curve of graphons. This allows us to approximate the gradient flow of function on graphons by suitable Markov chains on stochastic block models. Towards the end of the thesis, we also consider the scaling limit of the iterated product of matrices that are small perturbations of the identity matrix as the dimension of these matrices goes to infinity. In the fixed dimension, the scaling limit of the iterated product of such matrices is described by a non-commutative exponential of a matrix-valued semimartingale. Suppose that the bounded variation part of these semimartingales converges to some graphon as the dimension of these matrices goes to infinity. Then, we show that non-commutative exponentials converge to an infinite exchangeable array whose coordinates are Gaussians and whose mean and covariances can be described explicitly in terms of the limiting graphon.
Generalizations of the Exterior and Symmetric Power Functors on Categories of Modules using Coinvariants of Tensor Power Functors
(2024-10-16) Koch LaRue, Casey; KovÃ¡cs, SÃ¡ndor
This work, broadly speaking, is a study of coinvariants of abstract group actions on functors. We discuss background on coinvariants of group actions on objects in categories in general, as we benefit from taking the perspective of a functor $F:C\to D$ as an object of the category of functors from $C$ to $D$. That said, the focus of this work concerns the $n$th tensor power functor $T^n$ on the category of modules over a commutative unital ring $k$. We show that the $k$-algebra of endomorphisms of $T^n$ is isomorphic to the group algebra $k S_n$. This allows us to identify specific groups of automorphisms of $T^n$ and investigate the resulting functors of coinvariants. For example, the $n$th symmetric power functor $Sym^n$ and the $n$th exterior power functor $\wedge^n$ are coinvariants of $T^n$ with respect to actions of the symmetric group $S_n$. Hence, the study of coinvariants of $T^n$ with respect to group actions is a way of generalizing these familiar functors. We give examples of groups $G_1$ and $G_2$ of automorphisms of $T^n$ over $\mathbb{Z}$ such that $|G_1|$ is countably infinite whilst $|G_2|$ is finite, and they give rise to canonically isomorphic coinvariants of the functor $T^n$. We also explore the topic of sequences of groups $G_n$ with actions on components of a graded algebra $S$. We provide sufficient conditions for when the resulting direct sum of modules of coinvariants is a quotient algebra of $S$. For example, this condition implies that for any $R$-module $M$, the actions of the sequence of alternating groups $A_n$ on the components of the tensor algebra $T(M)$ induces an algebra $C$ of coinvariants. While $C$ is distinct from the algebras $Sym(M)$ and $\wedge(M)$, we observe that for finitely generated $R$-modules $M$ and sufficiently large $n$, there are isomorphisms $C_n\cong Sym^n(M)$.
The Anisotropic Gaussian Isoperimetric Inequality and Ehrhard Symmetrization
(2024-09-09) Yeh, Kuan-Ting; Toro, Tatiana
In this thesis, we establish the isoperimetric inequality for the anisotropic Gaussian measure and characterize the cases of equality. Additionally, we present an example demonstrating that Ehrhard symmetrization fails to decrease for the anisotropic Gaussian perimeter and introduce a new inequality that includes an error term. This new inequality, in particular, provides a clue to a uniqueness result for the Ehrhard measure within the class of anisotropic Gaussian measures. Our final result, a collaboration with Sean McCurdy, expands the class of measures to which the previous uniqueness result applies.

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