Papers

In 2023, Richard Schwartz introduced a new dynamical system which is a marriage of two types of familiar billiards, tiling billiards and symplectic billiards. In this paper we investigate this dynamical system played on tilings of the plane which arise from non-Euclidean geometries on the torus. We review the affine analogue of the flat conformal structures on the torus through the work of Oliver Baues and William Goldman, and define an open subset of this deformation space corresponding to markings of complete affine tilings of the plane. We make this definition precise, and provide algebraic conditions on the symmetries of the tiling to define it. We then analyze the dynamics of symplectic tiling billiards played on these types of tilings and investigate the long-term dynamics of the system to prove a stability result concerning divergent trajectories. The divergence is defined in terms of geometric invariants arising from the tiling symmetry group. We argue that in some sense this divergence is a consequence of the tiles of a non-Euclidean tiling becoming 'thin' as one moves far away in the tiling. To do so we introduce a notion of thinness that is well adapted to the non-Euclidean affine tilings.

Capacity expansion modeling plays a critical role in optimizing the deployment of new generation, storage, and transmission, typically at national and regional levels. To support long-term planning, these models consider a large set of energy technologies and policies, along with decades of weather and demand data. Realistic capacity expansion models thus become high-dimensional optimization problems, with hundreds of millions of variables and constraints, which are challenging to solve. A common strategy to address this complexity is to omit non-linear, non-convex AC optimal power flow (ACOPF) constraints and instead use linearized power balance equations or transport formulations. While these simplifications improve tractability, they limit our understanding of how power flow and the physical properties of power networks impact investment decisions across generation, storage, and transmission infrastructure. This paper addresses this gap by extending the GenX capacity expansion model to incorporate fixed point theorem-based network constraints. These embed ACOPF-based considerations while maintaining the tractability of the planning model, nearly preserving the dimensionality of the transport formulation and incurring only modest runtime increases. This approach is much cheaper than embedding ACOPF directly, making it appropriate for large-scale capacity planning problems. We compare our approach to the original transport-based GenX model as well as a non-linear, non-convex version that incorporates the full ACOPF constraints, for a case study of the ISO New England grid.

Cubature rules, which approximate definite integrals as a linear combination of a set of function values, are ubiquitous and necessary for computational methods in the physical sciences. A superconvergence result for cubature rules on polytopal domains in two and three dimensions is developed, whereby a rule that is exact for all bivariate polynomials of a fixed even degree realizes an extra order of convergence under a decrease in the spacing between nodes.

We prove a sharp nonlinear analogue of the Li-Yau inequality for positive solutions of the normalized parabolic $p$-Laplacian equation \[ u_t = \Delta u + (p-2)|\nabla u|^{-2}\nabla^2u(\nabla u,\nabla u),\ \ \ 1<p<\infty, \] on a closed Riemannian manifold with nonnegative Ricci curvature, the equation being interpreted in the viscosity sense on the critical set of the solution. We also derive a sharp global Harnack inequality from the gradient estimate. The constant is best possible. Equality holds identically for an explicit self-similar solution in $\Rn$, and sharpness on closed manifolds follows by taking the limit along the flat tori $\mathbb R^n/(L\mathbb Z)^n$ as $L \to \infty$. The proof rests on an exact Bochner-type identity for a family of uniformly parabolic approximating flows. We choose the first-order term separately from the second-order regularization so that the maximum principle yields the sharp constant, uniformly in the regularization parameter and without any assumption on the critical set of the solution. Using this Bochner identity with a modified Li-Yau functional, we extend the estimate to closed manifolds with $\operatorname{Ric} \ge -\kappa$.

In Nyström-collocation discretizations of the electric field integral equation (EFIE), the surface divergence acts on surface densities that may be discontinuous across patch boundaries, which degrades accuracy and convergence. We show that this not only affects the EFIE but every formulation in which the operator occurs, either in the equation itself or in the scattered field computation, and propose a high-order-accurate continuity-enforcing scheme for smooth surfaces as a remedy for the direct and indirect EFIEs, magnetic field integral equations (MFIEs), and regularized combined field integral equations (CFIEs) alike. The scheme comprises two ingredients: i) We show how to discretize the equations via a Chebyshev-based Nyström scheme, which admits closed quadrature rules. ii) Since unknowns and test vectors are in terms of patch-local curvilinear bases, continuity is enforced by a change of basis: we construct sparse mapping matrices assembled solely from the curvilinear geometry description. In doing so, we restore the accuracy of the EFIE such that it can be combined with the MFIEs with equal weights to form CFIEs. Numerical studies for the scattering from canonical and realistic geometries show that all considered formulations individually and combined benefit from the continuity enforcement in terms of better conditioning, reduced iterations of an iterative solver, and several more digits of accuracy in the scattered fields, despite reducing the total number of unknowns.

Hechler forcing changes the ordinal theory.

Let $q$ be a primitive $n$-th root of unity, $n>1$, and let $f$ be a nonzero polynomial such that $n\nmid(j+1)$ for every $j\in\supp(f)$. Set $e=\gcd(n,\{j+1:j\in\supp(f)\})$. We show that $\Oz(B_q(f))\cong\mu_e\times\mu_e$: the defining relations are homogeneous for a $\mathbb{Z}/e\times\mathbb{Z}/e$ grading, the center sits in degree zero, and the ozone group is the character group of that grading. The determination of the ozone group only requires the central elements $u^n$, $v^n$, and $\Omega$. The regular normal elements modulo the center form the same group, generated by $u^{n/e}$ and $v^{n/e}$, so every normal element is central exactly when $e=1$. For $f=t^2$ we recover a computation of Chan, Gaddis, Won and Zhang, and for $e>1$ we obtain an infinite family of Calabi–Yau algebras with nontrivial ozone group.

We study dependent censoring within a generalized Cox first-hitting-time. framework based on compensator reductions. In the genuinely dependent case, we show that max-stability of the associated survival copula is equivalent to a common operational clock, leading to a nonparametric Marshall Olkin family. We establish sharp identification results under complete and binary observation schemes. Under binary observations, the common clock and event loading are point identified, whereas the censoring loading and the associated latent survival and dependence structures are only partially identified. We develop nonparametric estimation and inference for the identifiable components, derive an efficient restricted estimator, and propose an observable test of max-stable compatibility. Simulations and a real-data application illustrate the practical implications of the identification and efficiency results.

Given a field $\mathbb{F}$ and an integer $n \geq 2$, denote by $t_n(\mathbb{F})$ the greatest possible dimension for a vector space of $n$-by-$n$ matrices over $\mathbb{F}$ in which every element is triangularizable. It was recently proved that $t_n(\mathbb{F})=\frac{n(n+1)}{2}$ if and only if $\mathbb{F}$ is not quadratically closed, with the possible exception of finite fields with characteristic $2$ and less than $n-1$ elements. In this article, we prove that the equality $t_n(\mathbb{F})=\frac{n(n+1)}{2}$ holds for all perfect non-quadratically closed fields with characteristic $2$ – with the possible exception of fields with cardinality $2$ – and for these fields we obtain a key result for a future analysis of the spaces that have the critical dimension $t_n(\mathbb{F})$.

Let R be a commutative noetherian local ring and M and N be R-complexes with finitely generated homology. We prove that formulas for the homological dimensions of the derived Hom complex RHom(M,N), in terms of the homological dimensions of M and N, hold without the a priori assumptions on the dimensions of M or N present in classic results.

In this paper, we show that shadowing and inverse shadowing are generic properties for spaces in which there exists an $\varepsilon$-retraction onto a graph for any $\varepsilon>0$.

We study when a polynomial sequence $\{p_n(x)\}$ satisfying a linear homogeneous recurrence with polynomial coefficients admits an ordinary Riordan array as its coefficient matrix. For second-order recurrences $p_{n+2}(x)=a(x)p_{n+1}(x)+b(x)p_n(x)$, we give a complete characterization. We extend this to a necessary condition and a full factorization criterion for recurrences of arbitrary order $\ell \geq 3$. We then develop a production-matrix framework for arbitrary orthogonal polynomial sequences (OPS), not restricted to the Riordan-array-type case: for any lower-triangular invertible coefficient matrix $A$ of an OPS, the production matrix $P_{A^{-1}}=ASA^{-1}$ is always the tridiagonal Jacobi matrix encoding the three-term recurrence, and the first column of $A^{-1}$ always gives the moment sequence – even when, as for the Legendre polynomials, the recurrence coefficients are non-constant and no Riordan array representation of the coefficient matrix exists. We illustrate this with Legendre and Chebyshev polynomials as, respectively, non-Riordan-type and Riordan-type examples, and give an explicit closed-form coefficient matrix and its inverse for the generalized Gegenbauer–Humbert OPS. Finally, we extend the framework to $d$-orthogonal polynomial sequences, showing that the coefficient matrix of a $d$-orthogonal sequence is always invertible, that its associated production matrix is $(d+2)$-banded (lower Hessenberg) and encodes the corresponding $(d+2)$-term recurrence, and that the first $d$ columns of the inverse matrix recover the moment sequences of the defining vector functional – identifying $d$-orthogonality with $(d+1)$-Hessenberg generalized Riordan arrays and extending the classical $d=1$ tridiagonal correspondence.

We establish quantitative convergence to the target and uniform-in-time propagation of chaos for Langevin-regularized Stein variational gradient descent. The Stein interaction need not be small relative to the confining Langevin drift and does not generally yield a contractive particle coupling. At the mean-field level, the Stein and Langevin components dissipate the same relative entropy in the kernel-induced Stein and $2$-Wasserstein geometries, producing the squared kernel Stein discrepancy and relative Fisher information. Under a log-Sobolev inequality for the target, this yields exponential last-iterate convergence. We also derive a finite-particle entropy identity relative to the product target, giving exponential-in-time convergence of the empirical measure up to polynomial sampling errors. For propagation of chaos, we develop two complementary finite-time approaches. A synchronous coupling, combined with exponential moment estimates for the nonlinear mean-field diffusion, yields explicit single-exponential bounds in Wasserstein distance and kernel Stein discrepancy (KSD). Moving-product entropy gives joint-law relative entropy control relative to the evolving mean-field product law and, through entropy superadditivity and concentration, fixed-marginal relative entropy and total variation bounds and empirical KSD estimates. Under an additional $T_2$ inequality for the initial law, it also yields Wasserstein bounds. Combining these finite-time estimates with target convergence at a logarithmic cutoff time gives polynomial uniform-in-time propagation of chaos rates in expectation for empirical KSD and $W_2^2$, and for fixed-marginal total variation and $W_2^2$. All bounds control the last iterate in physical time. We also compare the two finite-time mechanisms and identify regimes in which each gives the sharper polynomial exponent.

We prove a similarity result for non-self-adjoint non-elliptic quadratic $\mathcal{PT}$-symmetric operators with real spectrum satisfying a dynamical averaging condition. In particular, we show that such an operator is weakly similar to a self-adjoint operator precisely when the fundamental matrix of the associated quadratic form is diagonalizable.

We develop a novel approach to matrix discrepancy based on matrix small-ball estimates. Specifically, we use a determinantal weight (obtained from the log-barrier) to scale the small-ball probability into a partition function of a tilt of the Gaussian measure. We then employ matrix-weighted Poincaré inequalities to compare this partition function to that of a pinched or diagonal part of the matrix, obtaining dimension-free constants. Our technique yields a hereditary small-ball estimate for Gaussian series that should be of independent interest. As the main application, we resolve the Matrix Spencer conjecture: for symmetric $n\times n$ matrices $A_1,\dots,A_n$ with $\|A_i\|\le1$, one can efficiently find a coloring $x\in\{\pm1\}^n$ with $\|\sum_{i=1}^n x_iA_i\|=O(\sqrt n)$.

We consider varieties whose singularities are complete intersections within toric singularities, which we call well-formed varieties. Using the construction of equivariant standard resolutions, we can extend several classical formulas in the derived category of bounded coherent sheaves to this singular setting. Finally, we also discuss the Bondal–Orlov localization for certain morphisms between terminal complex threefolds and their stacks.

We show that a certified formalization of the hitting-set-existence axiom of Atserias and Tzameret, instantiated on the parity-based Nisan-Wigderson compression class of Khaniki, is independent of the two-sorted theory $V^0_1$ of $\mathrm{AC}^0$-reasoning, unconditionally: $V^0_1$ proves neither it nor its negation. The same holds for the corresponding certified dual weak pigeonhole principle, whose refutation is witnessed by a single seed that certified-computes every string of the model simultaneously. The mechanism is a bounded-arithmetic transfer of Atserias-Tzameret's reduction from hitting sets to the dual weak pigeonhole principle: the amplification half of that reduction, the sole source of its NP-oracle, is unnecessary at the native stretch of the Nisan-Wigderson map, and the compression half becomes a $V^0_1$-provable implication once circuit evaluation is replaced by its certified $\Sigma^B_0$ unfolding. This is, to our knowledge, the first independence result for a derandomization-flavoured existence principle at the $\mathrm{AC}^0$-reasoning level, and it makes explicit the bridge between the Khaniki Nisan-Wigderson line and the Atserias-Tzameret reverse mathematics of hitting sets.

We prove that the boundary of every bounded $(M^{1,p},M^{1,q})$-extension domain in a $Q$-doubling metric measure space has measure zero whenever $0<q<p<\infty$, with the additional restriction that $p<qQ/(Q-q)$ if $q<Q$.

These lectures are aimed at junior researchers seeking to enter the field of applications of complex analysis to harmonic analysis, Fourier analysis, spectral and scattering theory for differential operators and related fields. Starting with classical completeness problems posed by Bernstein, Beurling and Malliavin, Wiener, Kolmogorov and Krein, the course progresses to the solutions of the Gap and Type Problems, inverse spectral problems for Schrödinger operators and canonical Hamiltonian systems, and pointwise convergence of the non-linear Fourier transform. Each lecture is supplemented with exercises and references.

Separable nonnegative matrix factorization (SNMF) has been widely used for low-rank representation and clustering of nonnegative data, owing to its ability to produce part-based and interpretable decompositions. In particular, SNMF is closely related to graph clustering and community detection. To enhance sparsity and identifiability of the learned factors, we propose an $\ell_1^p/\ell_2$-regularized SNMF model based on a powered ratio-of-norms regularizer. The resulting formulation is nonconvex and nonsmooth, which poses significant challenges for optimization. To address this, we develop efficient algorithms based on the difference-of-convex function algorithm (DCA) and the alternating direction method of multipliers (ADMM). The proposed methods decompose the original problem into tractable subproblems, leveraging closed-form proximal operators associated with the powered norm terms. We establish descent and limiting criticality properties for the DCA scheme and convergence under standard assumptions for the ADMM scheme. Extensive numerical experiments on synthetic datasets and hand gesture classification tasks demonstrate that the proposed approach achieves competitive or improved performance in anchor identification and classification accuracy compared with existing SNMF methods, while maintaining competitive computational efficiency.

We define a relative $\mathbb{Z}_2$-index $\mathcal{N}(\omega_1,\omega_2)$ for locally-comparable, parity-invariant pure states on a unital $C^*$-algebra. We prove that the index is well-defined, multiplicative, invariant under parity-preserving automorphisms, and locally constant in the norm topology. For the one-dimensional self-dual CAR algebra, we apply this construction to the half-chain automorphism $\sigma$ and define the many-body Majorana number $\mathcal{N}\sigma(\omega):=\mathcal{N}(\omega,\omega\circ\sigma)$ for parity-invariant $\sigma$-local pure states. In the quasi-free Hilbert–Schmidt regime, this index agrees with the usual single-body Majorana number. We then introduce symmetric local automorphism paths and prove that $\mathcal{N}_\sigma$ completely classifies the resulting automorphic-path-components. This automorphic-path equivalence retains the bulk Majorana number but may forget a relative zero-dimensional parity obstruction.

This article investigates the global structure of maximal surfaces (space-like immersions with zero mean curvature) in Lorentz-Minkowski $3$-space $\mathbb{E}^3_1$, especially focusing on the interplay between genus, the number of singular components–loci, and simple ends. We construct complete maximal maps with arbitrarily many singular components for any genus $p\geq 0$ with simple ends.

Let $d_k>0$ and let $Q_k$ be a polynomial of degree $r_k$ with nonnegative Chebyshev coefficients. We study locally uniform limits of $f_k(z)=C_kQ_k(\cos(d_kz))$, where $C_k>0$. If $f_k\to F$ locally uniformly and $F(0)>0$, the associated positive lattice measures converge weakly, their second moments converge, and their quadratic tails are uniformly integrable. If additionally all zeros of $Q_k$ lie in $[-1,1)$, $d_k\to0$, and $F$ has order below two, then for every $A>0$ with $F(A)F(-A)\ne0$, $\limsup_{k\to\infty} r_kd_k^2\le 4\sum_{\gamma_n>A}\gamma_n^{-2}$, where $\{\pm\gamma_n\}$ are the nonzero real zeros of $F$, counted with multiplicity. If, moreover, $f_k\to F$ locally uniformly with $F(0)>0$, all zeros of $Q_k$ lie in $[-1,1)$, and $F$ has no nonzero real period, has order below two, and is not of finite exponential type, then $d_k\to0$, $r_kd_k\to\infty$, and $r_kd_k^2\to0$. Equivalently, $d_k^{-1}=o(r_k)$ and $r_k=o(d_k^{-2})$. Boundary examples show the role of the hypotheses and the Gaussian boundary at order two.

We prove that a positive-definite quadratic form has nonnegative conorms if and only if it is matroidal, resolving a conjecture of Dutour Sikirić and Kummer. We show that the positive conorm support determines a regular matroid, and the positive conorms are the coefficients of a unimodular integral decomposition of the form. As a result, we show that a positive-definite form lies in the tropical Schottky locus if and only if its conorms are nonnegative and its positive conorm support matroid is cographic. This answers a question of Chua, Kummer, and Sturmfels and provides algorithms for the tropical Schottky decision and recovery problems in every genus.

We reformulate and discuss the conditions for the initial values in which a generalized Fibonacci sequence is eventually strictly monotonic. We then extend this exploration to the generalized \(k\)-bonacci sequence. We also give a lower bound on when an eventually strictly monotone generalized \(k\)-bonacci sequence becomes monotone.

In this paper, we introduce several novel approaches utilizing the circle method to obtain the asymptotic formula for the number of integral points of bounded height lying on a hypersurface in a weighted projective space. Let $F(\mathbf{x} ; \mathbf{y})$ be a given weighted form of degree $d$ in variables $\mathbf{x} \in \mathbb{R}^{s_1}$ and $\mathbf{y} \in \mathbb{R}^{s_2}$, where variables $\mathbf{x}$ and $\mathbf{y}$ have weights $w_1$ and $w_2$ with $w_1<w_2,\left(w_1, w_2\right)=1$, and $d>w_1 w_2$. Write $$ R_F(P):=\#\left\{(\mathbf{x} ; \mathbf{y}) \in \mathbb{Z}^{s_1+s_2}: F(\mathbf{x} ; \mathbf{y})=0,|\mathbf{x}| \leq P^{w_1 / d},|\mathbf{y}| \leq P^{w_2 / d}\right\} . $$ In particular, we show that whenever $$ s_1+s_2-\sigma_F>\left(1+\frac{w_2}{w_1}\right) \frac{d}{w_1} 2^{d / w_1}, $$ where $\sigma_F$ is the dimension of the affine singular locus of $F$, the quantity $R_F(P)$ has the expected asymptotic formula, that is $$ R_F(P)=c P^{s_1 w_1 / d+s_2 w_2 / d-1}+o\left(P^{s_1 w_1 / d+s_2 w_2 / d-1}\right), $$ where $c$ is the product of local densities. Furthermore, the constant $c$ is positive whenever $F(\mathbf{x} ; \mathbf{y})=0$ has a nonsingular solution over $\mathbb{R}$ and $\mathbb{Q}_p$ for every prime $p$. As a corollary, we verify the integral Hasse principle for the quasi-smooth hypersurface defined by $F(\mathbf{x} ; \mathbf{y})=0$ in a weighted projective space of sufficiently large dimensions.

Let $\Gamma$ be a property (T) group admitting an injective, non-surjective endomorphism and let $G$ be the associated ascending HNN-extension. Let $W=\left(\bigoplus_{G/\Gamma} \mathbb{Z}/2\mathbb{Z}\right) \rtimes G.$ We show that $W$ is not an MF group and that $C^*(G)$ is not a finite C*-algebra. The ideas and proofs were generated by ChatGPT 5.6 Sol, we have only refined their arguments in a hopefully more palatable form.

Let $\Gamma\subset\mathbb R^2$ be a compact convex graph and define \[ T(\Gamma) = \inf \left\{ t: \dim_{\mathrm H}(E)>t \Longrightarrow |E+\Gamma|>0 \text{ for every compact }E\subset\mathbb R^2 \right\}. \] For a graph over an interval of positive length the smallest possible value is $T(\Gamma)=1$. We ask whether this optimal conclusion can hold when pointwise Fourier decay of arclength is unavailable. The answer is yes, even for strictly convex curves. We use the Fourier transform convention $\widehat\nu(\xi)=\int e^{-2\pi i x\cdot\xi}\,d\nu(x)$. We construct a strictly convex Lipschitz graph $\Gamma$ with $T(\Gamma)=1$ such that, for every nontrivial subarc $\Gamma_0$ and every $\alpha>0$, \[ \limsup_{|\xi|\to\infty} |\xi|^\alpha \left| \widehat{H^1|_{\Gamma_0}}(\xi) \right|= \infty. \] We also give a convex example for which arclength on every nontrivial subarc fails even to be a Rajchman measure. The geometric mechanism behind these examples is a positive curved trace: if $\Gamma$ contains a positive-length subset of a $C^2$ curve whose curvature is bounded away from zero, then $|E+\Gamma|>0$ whenever $\dim_{\mathrm H}(E)>1$. For a nondegenerate graph this gives $T(\Gamma)=1$. For convex graphs it implies, in particular, that $T(\Gamma)=1$ whenever the curvature measure has a nonzero absolutely continuous part. The positive-measure proofs are in physical space and use translated-tube intersections and elementary facts about convex functions. The same overlap estimates give Mattila-type lower bounds for the average lengths of the associated curve projections of neighborhoods under the positive curved-trace hypothesis. We also prove a dimension-one endpoint result for sets with a positive-length rectifiable part and formulate the main remaining question: whether every strictly convex Lipschitz graph has the optimal threshold $T(\Gamma)=1$.

In this paper we consider some general aspects of Euclidean conformal field theories on curved spaces of dimension $D\ge 3$. We first look at the OPE of scalar primary fields on conformally flat spaces. Extending the results of \cite{Konechny:2026bqg}, we give a general construction of the descendants' contributions to the OPE in the scalar channel. We then discuss CFTs on non-conformally flat spaces in the ambient space formalism. We investigate the short-distance behaviour of three-point functions on such spaces using the general ansatz proposed by Parisini, Skenderis, and Withers \cite{Parisini:2022wkb, Parisini:2023nbd}. We find that some additional corrections need to be added to the ansatz to ensure the existence of a local covariant OPE.

Positroid varieties $\Pi_f$ provide a decomposition of the Grassmannian $Gr(k,n)$; they can be enumerated using bounded affine permutations ($f$) which have bijections with a number of interesting combinatorial objects. Furthermore, (the nonnegative part of) positroid varieties parameterize the space that is integrated over when calculating amplitudes in planar N=4 supersymmetric Yang-Mills theory. The main question we answer in this thesis is whether a positroid variety $\Pi_f$ has any geometric singularities. We show that it is sufficient to check singularity at the $T$-fixed points ($\lambda$) and we can obtain the multiplicity at these points $\lambda$ by calculating the equivariant cohomology of $\Pi_f$ restricted to the point. We give 2 ways of doing this: (1) A diagrammatic way using affine pipe dreams and (2) A computational method. For (2), we have written code that does the computation and outputs the multiplicity (files at josephflueg.github.io). We have included tables listing the multiplicities of all the points on positroid varieties up to $n=6$. We also describe an ordering on pairs $(\Pi_f,\lambda)$ given by deletion/contraction that interacts nicely with smoothness, and describe the relationship between affine pipe dreams and finite pipe dreams. In Part II of this thesis connects with physics of planar N=4 SYM. We briefly introduce quantum field theory, leading singularities, and positroids in N=4 SYM. We then explain Britto-Cachazo-Feng-Witten (BCFW) recursion and the BCFW bridge decomposition of an on-shell diagram. We describe how to build an on-shell diagram on a pipe dream using BCFW bridges. Finally, we work out some combinatorics related to inverse soft factors for on-shell diagrams and explore whether singularities in positroid varieties have relevance to amplitudes. This is a revised version of my thesis originally written in August 2024.