Papers

We establish the use of Lagrangian diagram moves to build Lagrangian cobordisms of Legendrian knots. We then apply the moves to explore how to invert decomposable cobordisms, after adding some stabilizations. This is done by use of Reidemeister moves and handle attachments in the Lagrangian projection. As an application, we show an explicit construction of a Lagrangian concordance from a stabilized $m(9_{46})$ Legendrian knot to a stabilized unknot, i.e. a non-decomposable Lagrangian concordance of nonempty knots.

The Baringhaus–Henze–Epps–Pulley (BHEP) tests form a widely applicable class of affine invariant and consistent tests for multivariate normality. Under the null hypothesis, the test statistic converges to a weighted sum of independent chi-squared random variables. Although closed-form expressions for the first three cumulants of this limiting distribution are known for arbitrary dimension $d$ and smoothing parameter $\beta$, an explicit expression for the fourth cumulant has so far been available only in a special univariate case. We derive the fourth cumulant for arbitrary $d\geq1$ and $\beta>0$ and use the resulting first four moments to construct Johnson and Pearson approximations to the limiting null distribution. These yield essentially instantaneous analytic approximations to BHEP $p$-values, without numerical eigenvalue calculations. Using the recently obtained complete spectrum as a benchmark, we show that both four-moment approximations are highly accurate over a broad range of dimensions, smoothing parameters and upper-tail probabilities, and substantially improve on two- and three-parameter lognormal approximations. Monte Carlo results indicate good finite-sample calibration for many parameter combinations, although convergence to the limiting distribution may be slow in higher dimensions for extreme values of the smoothing parameter.

This paper investigates the singular limit problem of the one-dimensional steady compressible Euler system with heat conduction in a finite nozzle. The existence of the heat conduction solutions were established and boundary layers at one or two nozzle endpoints occur as the coefficient of heat conductivity goes to zero. The asymptotic behavior of the solutions vary depending on the state at the entrance and the value of the pressure at the exit, which classify the solutions into purely supersonic solution, purely subsonic solution and transonic solution. Moreover, we perform a formal asymptotic expansion of solutions and derive the boundary layer equations.

We study the semilinear pseudo-relativistic equation \[ \sqrt{-\Delta+m^{2}}\,Q+\omega Q=Q^{p-1}\qquad\text{in }\R^{d}, \] with $m>0$, $\omega>0$, $d\ge1$ and $2<p<2^{*}:=\frac{2d}{d-1}$ (with $2^{*}=\infty$ when $d=1$). We prove that the positive radial variational ground state is unique. The proof rests on a mass-covariant oscillation theorem for the linearized operator, established through a purely variational reduction: the Caffarelli–Silvestre–Duffin extension for $\sqrt{-\Delta+m^{2}}$ combined with a ground-state substitution reduces the linearized eigenvalue problem to a massless, potential-free weighted Steklov problem, whose second eigenfunction changes sign exactly once by strict superadditivity of the weighted Dirichlet energy under sign decomposition. The nondegeneracy of the linearization is then obtained from two orthogonality relations; the failure of scale invariance for $m>0$, which destroys the second relation used in the massless Frank–Lenzmann–Silvestre theory, is repaired by a mass-covariance identity. Global uniqueness follows by a continuation argument in the mass parameter, anchored at $m=0$ by the theorem of Frank–Lenzmann–Silvestre.

In this note, we prove that the covering radius problem for Euclidean lattices is complete for the second level of the polynomial hierarchy. The note also documents the author's first experiment with generative AI as a tool for mathematical research.

We consider the Sachdev-Ye-Kitaev model with $N$ Majorana fermions and interaction order $q$. For even $q\in \{2,4,...,N-2\}$, we give an elementary proof and show that the eigenvectors of the model are completely delocalized in all deterministic directions.

Let $G_{2n}$ be a complex symmetric random matrix whose entries above the diagonal are independent standard circular complex Gaussians, and let $H_n=\operatorname{haf}(G_{2n})$. We prove the uniform shifted anticoncentration bound $$ \Pr\!\left( \left| \frac{H_n}{\sqrt{(2n-1)!!}}-z \right| \le \varepsilon \right) \le 2\sqrt{\frac n\pi}\,\varepsilon^2 $$ for every $z\in\mathbb C$ and $\varepsilon>0$. This establishes a local anticoncentration property that supports hardness arguments for quantum advantage in Gaussian boson sampling.

Range reduction plays a crucial role in the accuracy and performance of evaluating trigonometric functions, and is often the primary bottleneck for large floating-point inputs. While fast algorithms such as Cody–Waite work efficiently over narrow intervals, the Payne–Hanek algorithm remains the standard technique for accurate reduction across large floating-point inputs. However, many existing implementations of Payne–Hanek suffer from high latency due to heavy branching, conversion overheads, and the use of multi-word integer arithmetic, which hinders SIMD vectorization. In this paper, we analyze and present a branch-free variation of the Payne–Hanek algorithm using only floating-point arithmetic. Our method operates directly over large double-precision inputs ($|x| \ge 2^{16}$) and is well suited to hardware with FMA instructions. We formulate the precision constraints in terms of a truncation error budget, construct a compact lookup table indexed by the input exponent, and prove that the scaled reduced argument has absolute error below $2^{-110}$ and relative error below $2^{-60}$ for every input, including worst cases. The same routine can serve both as the complete range reduction of a single-stage implementation and as the fast path of a correctly rounded one, achieving higher throughput than existing implementations and lower latency than those returning a double-double reduced argument. The algorithm is currently implemented in the LLVM libc project.

Closed-loop planning accounts for future observation-dependent actions but can be expensive to repeat at deployment. Bellman-gradient (BG) refinement differentiates conditional rollouts through an existing actor, corrects the current action, and stores the result in an executable policy. A backward sweep reuses deployed future feedback; full-horizon rollouts in an identified Gaussian belief model need no learned value critic. A nonlinear error recursion links quadratic continuation error, local correction, and policy storage. A controlled non-LQG example exhibits second-order policy accuracy with consistent storage, while ideal affine LQG admits exact backward recovery. In nonquadratic thrust, BG reduces actor cost by 2.47-3.20% and remains within 0.13-0.53% of the tested feedback MPC; original policies execute in 5-6 microseconds in a seed-0 native audit. With matched storage, BG attains competitive plant costs at about one ninth (arm) and one sixtieth (docking) of feedback-teacher-plus-student construction time on one GPU, excluding shared learning and node preparation. Richer common maps substantially narrow some student-BG gaps. These results expose the roles of learned feedback, local correction, and storage in executable control.

Let $d\ge2$ and let $\mathbf w=(w_1,\ldots,w_d)$ satisfy $w_1\ge\cdots\ge w_d>0$ and $\sum_i w_i=1$. Set $s_*=d-(1+w_1)^{-1}$. We prove that there exist constants $C_{d,\mathbf w}>0$ and $\varepsilon_0=\varepsilon_0(d,\mathbf w)>0$ such that for all $0<\varepsilon<\varepsilon_0$, $$ \dim_H\operatorname{DI}_{\mathbf w}(\varepsilon)\le s_*+C_{d,\mathbf w}\sqrt\varepsilon. $$ Together with the lower bound of Kim–Park, this gives the exact formula $$ \dim_H\operatorname{Sing}(\mathbf w)=s_*. $$

We study the local geometry of Euclidean submanifolds of rank at least two whose curvature normals with respect to the flat part of the normal bundle have constant length, equivalently, whose adapted third fundamental form has constant eigenvalues. The complete case was fully settled by Di Scala and the third author, who showed that such a complete submanifold has constant principal curvatures. Around a regular point, we show that $M$ is generated, via parallel manifolds, by a hypersurface $N$ whose extrinsic factors are contained in spheres and, when of dimension at least two, have rank one; conversely, such factors can be assembled to construct examples of $M$. The proof relies on the isoparametric rank theorem to rule out non-umbilical factors. As a further application, when the inner products of the curvature normals are constant, we show that there are exactly two eigendistributions, one of which is one-dimensional and autoparallel, extending, for arbitrary codimension when the normal bundle is flat, curvature-homogeneity results of Tsukada and the recent ones of Bryant, Florit, and Ziller for hypersurfaces.

Medical image segmentation remains challenging in terms of accurate boundary localization and complex-structure preservation, as target regions often exhibit weak boundaries, fine-grained structures, and irregular shapes, while high-quality images and precise annotations are usually limited. Existing generative segmentation approaches based on Flow Matching mainly learn velocity fields in the state space but lack explicit constraints on the spatial regularity and distance-field properties of the recovered representation. This limitation may lead to spatial oscillations, boundary displacement, and discontinuities in fine structures. To address these issues, we propose an image-conditioned Geometrically Regularized Flow Matching framework, termed GR-FM, which formulates segmentation as a continuous probability transport process from an initial distribution to a target implicit representation distribution. Instead of directly modeling binary masks, GR-FM adopts the SDF to describe the target structure, where each pixel is represented by its signed distance to the object boundary, resulting in a continuous geometric field. Flow Matching and ordinary differential equations are then employed to achieve deterministic and efficient distribution transport. The training objective further incorporates a biharmonic regularization term and an Eikonal constraint to enhance the spatial smoothness, structural consistency, and distance-field characteristics of the recovered representation. Moreover, we analyze the continuous transport process under geometric constraints and investigate the evolution of the zero level set. Experiments conducted on the MoNuSeg, GlaS, and DRIVE datasets demonstrate that GR-FM achieves competitive performance in both region overlap and boundary accuracy, reduces performance variation, and maintains stable segmentation results with only a small number of integration steps.

We consider an initial value problem for a time-fractional evolution equation in a Hilbert space $X$: $$ \partial_t^{\alpha} (u(t)-a) = Au(t) \qquad \mbox{for $0<t<T$}, $$ where $\partial_t^{\alpha}$ denotes a fractional differential operator of Caputo type with order $0<\alpha<1,$ $u\colon (0,T) \to X,$ $a$ describes an initial value, and $A$, with domain $\mathcal{D}(A),$ is the generator of a contraction C$_0$ semigroup in $X$. We prove a fractional Hille-Yosida theorem characterizing the unique existence of a weak solution for every $a\in X$. We also discuss the unique existence of a strong solution for $a \in \mathcal{D}(A),$ as well as the continuity of weak solutions. Our results are applicable, for example, to time-fractional transport equations and Boltzmann equations. The proofs are mainly based on constructing an approximating sequence of solutions using the Yosida approximation.

We construct and analyze a renormalized three-dimensional massive Lee model; its renormalized Hamiltonian is defined intrinsically through an abstract construction which explicitly provides the operator domain, action and a Kre\u\i n-like resolvent formula. Conservation of the total excitation number reduces the spectral analysis of the full Hamiltonian to a sector-wise study of the spectral profile of an explicit operator pencil, playing the role of an operator-valued Weyl function. We construct essential spectrum components in a sector generated by lower excitation sectors and obtain the inclusion in the expected sector-wise HVZ type formula; the full formula is proved in the first two sectors and, under explicit conditions, in every sector. Below the free or one-excitation threshold, the point spectrum in every higher sector is described by a bounded self-adjoint nonnegative Birman–Schwinger operator. In every sector with at least two excitations we prove absence of sub-threshold spectrum for arbitrary coupling when the renormalized gap is not smaller than the boson mass, and under an explicit weak-coupling condition when it is smaller. In the latter case we also derive a sufficient condition for the existence of a simple bound state in the two-excitation sector.

We build a Floer theory on the space of braids of periodic orbits of a Hamiltonian flow on a closed symplectic surface. The differential and continuation maps are defined by counting Floer isotopies of braids: tuples of Floer cylinders with pairwise disjoint graphs. With this new perspective, we prove a quantitative braid stability result that shows the persistence of braids under possibly large Hamiltonian perturbations. As an application, we show that for every $\alpha\geq 0$ there is a sequence of Hamiltonian diffeomorphisms $\phi_k$ on the two-torus $T^2$ such that $$d_H(\operatorname{Ent}_{\leq \alpha}(T^2,\omega),\phi_k)\to \infty \quad (k \to \infty),$$ where $\text{Ent}_{\leq \alpha}(T^2,\omega)\subset \text{Ham}(T^2,\omega)$ denotes the set of Hamiltonian diffeomorphisms with topological entropy at most $\alpha$ and $d_H$ the Hofer metric. We prove this result by studying only contractible periodic orbits, whereas analogous higher-genus statements were previously obtained using non-contractible orbits.

Formalizing research-level stochastic optimization in Lean requires both an algorithm model and domain theory connecting foundational libraries to convergence proofs. Revising a model to restore provability can change the mathematical claim. We introduce ProofLoom, a fully automated LLM-agent system for Proof-Obligation-Driven Theory Construction. Given a published algorithm, target theorem, and source proof, ProofLoom autonomously constructs the Lean model and supporting theory. Open proof obligations drive the development of definitions, interfaces, lemmas, and proof plans. Signature contracts record evidence and obligations for model revisions; an independent Judge rejects unsupported assumptions and weakened conclusions. Planner expands the published argument into intermediate claims, and Audit checks whether the Lean proof follows it. Across tasks, SOptLib accumulates verified mathematics and construction experience: reusable results are extracted, generalized, and verified, while modeling decisions and failed proof routes are recorded. Later tasks retrieve these results and records and contribute new developments, forming a cycle of construction, accumulation, and reuse. On fifteen textbook and research-paper tasks, ProofLoom obtains mean human ratings of 6.3/7 and 6.4/7, compared with 4.9/7 and 5.0/7 for the strongest of six baselines. Across 33 developments, it produces 490,693 lines of algorithm-local Lean code with no sorry. The formalizations also expose 28 incorrect formulas, proof gaps, and algorithm-analysis mismatches in published sources across 22 developments, each with checked evidence.

We resolve an open problem posed by Frank–Loss. Let $n\ge 3$ and let $\varphi\in L^p(\mathbb{S}^n)$, with $\frac{n}{n-1}<p<\infty$, be a nontrivial Dirac zero mode on $\mathbb{S}^n$, i.e. a nonzero spinor satisfying \begin{equation*} D\varphi = iA\cdot\varphi, \end{equation*} where $D$ is the Dirac operator and $A$ is a vector field with $\mathrm{d} A^\flat\in L^{n/2}$. We prove the sharp lower bound \begin{equation*} \|\mathrm{d} A^\flat\|_{\frac{n}{2}} \ge 2\left[\frac{n}{2}\right]^{-\frac12}\frac{n-1}{n-2}S_n = \left[\frac{n}{2}\right]^{-\frac12}\frac{n(n-1)}{2}\omega_n^{\frac{2}{n}}. \end{equation*} Equality is attainable if and only if $n$ is odd; in that case, modulo conformal and gauge transformations, $\varphi$ is a Killing spinor and $A$ is a real multiple of the Reeb field associated with $\varphi$. The case $n=3$ was proved in our recent paper using a different method. In the paper we divide the remaining cases into 3 cases: i) $n\ge 5$ is odd, ii) $n\ge 5$ is even and iii) $n=4$. All these cases need to use different methods. For $n\ge 5$, the argument crucially reduces to an improved Sobolev inequality on $\mathbb{S}^n$ under a barycenter constraint. Specifically, for $u\in W^{1,2}(\mathbb{S}^n)$ we consider \begin{equation*} \mathfrak{a}_n := \inf\Bigg\{ \frac{ \int \Big(|\nabla u|^2 + \frac{n(n-2)}{4}u^2\Big) - \frac{n(n-2)}{4}\omega_n^{\frac{2}{n}}\|u\|_{\frac{2n}{n-2}}^2 }{ \omega_n^{\frac{2}{n}}\|u\|_{\frac{2n}{n-2}}^2 - \int u^2 } \,\Bigg|\, \int x|u|^{\frac{2n}{n-2}}=0,\ \omega_n^{\frac{2}{n}}\|u\|_{\frac{2n}{n-2}}^2 - \int u^2>0 \Bigg\}, \end{equation*} and we obtain the following universal estimate \begin{equation*} \mathfrak{a}_n > \frac{n(n-2)}{4(n^2-3n+1)}, \end{equation*} which is enough for our aim. Determining the exact value of $\mathfrak{a}_n$ remains open.

In this work, we provide a complete characterisation of Pitt's inequality for metaplectic operators. The symplectic geometry underlying a metaplectic operator distinguishes effective directions, along which its action is Fourier-type, from singular directions, along which concentration is preserved. For this reason, we adopt two complementary perspectives, thereby obtaining both an isotropic Pitt's inequality, emulating the classical theorem of Pitt, and an anisotropic inequality that adapts to the geometric features of the metaplectic group. As a by-product, we obtain Pitt's inequality for Fourier transforms along subspaces of $\rd$ and new quantitative time-dependent boundedness results for metaplectic operators on homogeneous Sobolev spaces, with applications to Schrödinger evolutions generated by quadratic Hamiltonians. Additionally, we derive logarithmic and entropic uncertainty principles for the metaplectic group. We show that the classical entropic uncertainty principle fails when both the effective and singular directions are present. In this case, a generally unbounded corrective entropy term accounts for the concentration-preserving directions and restores a meaningful lower bound. For quadratic Schrödinger evolutions, this correction follows the time-dependent geometry of the nondispersive directions and adjusts the uncertainty estimate accordingly.

Exponential stabilization is investigated for a hetero-dimensional parabolic system in which a one-dimensional shoreline subsystem, subject to a constant input delay, is bidirectionally coupled with unactuated two-dimensional bathymetric dynamics. A spectral decomposition separates the finite-dimensional critical shoreline modes from the exponentially stable residual dynamics. State feedback is designed on the critical subspace and lifted to the full shoreline space. An auxiliary transport equation represents the delay, while a predictor removes the explicit input delay from the nominal predictor dynamics. Exponential stability of the full coupled system is established by combining shoreline stabilization and bathymetric coercivity using weighted input–output estimates and a small-gain condition on the hetero-dimensional interaction. Numerical simulations show that uncompensated delayed feedback can strongly amplify both shoreline and bathymetric responses, whereas predictor compensation restores decay of the critical shoreline dynamics and prevents delay-induced amplification from propagating to the unactuated bathymetric subsystem.

A major problem in geometric analysis is to understand the behaviour of the Lagrangian mean curvature flow. This has proved to be a challenging problem, in particular, not many explicit examples of flows that exist for all time and converge to a minimal Lagrangian are known. In this paper we construct new examples of Lagrangian mean curvature flows in the Kummer K3 surface that converge to special Lagrangian spheres. We reduce the construction to a scalar perturbation problem that can be solved by a fixed point argument.

For functions that are convex of higher order, classical extremalities identify Gaussian quadrature as a lower bound and Gauss–Lobatto quadrature as an upper bound for the integral. We sharpen this bracket in every odd order: if $f$ is $(2n-1)$-convex on $[-1,1]$, then the exact integral $I[f]:=\int_{-1}^{1}f(x)\,dx$ lies not merely between the $n$-point Gauss–Legendre rule $G_n[f]$ and the $(n+1)$-point Gauss–Lobatto rule $L_{n+1}[f]$, but already between $G_n[f]$ and their midpoint. This yields the certified estimator $Q_n[f]:=\frac{3}{4}G_n[f]+\frac{1}{4}L_{n+1}[f]$, with $|I[f]-Q_n[f]|\leqslant\frac{1}{4}|L_{n+1}[f]-G_n[f]|$, and the constant $\frac{1}{4}$ is sharp. We also record a complementary phenomenon in even order: for $2n$-convex functions the Gauss–Radau endpoint rules still bracket the integral, but the midpoint of the two Radau rules cannot yield a one-sided refinement. The obstruction is explained by an odd, sign-changing Radau quadrature kernel. Finally, we distinguish numerical certification from ordinary high-order approximation: we include concrete examples from spline, moment and statistical models and report experiments for truncated powers and near-pole Stieltjes kernels, where the shape certificate remains meaningful when classical smoothness-based error constants are unavailable or severely pessimistic.

The generalized packing–covering conjecture of Elimelech, Firer and Schwartz asserts that, for every linear code $\mathcal{C}$ and every admissible order $t$, the $t$-th generalized Hamming weight $d_t(\mathcal{C})$ and the $t$-th generalized covering radius $R_t(\mathcal{C})$ satisfy $d_t(\mathcal{C})\le 2R_t(\mathcal{C})+2$. We give a computer-assisted proof of the conjecture for every linear code over every finite field and every admissible order. Combining a parity-check reformulation of the conjecture, bounds on the length of putative counterexamples, and successive puncturing arguments, we settle all orders $t\ge 32$ and reduce the remaining orders to finitely many parameter tuples, which we exclude by an exact computer verification.

We establish maximal inequalities for geometric averages of operator-valued functions in noncommutative \(L^p\)-spaces associated with semifinite von Neumann algebras. For averages over a fixed smooth submanifold of finite type at the parameter origin, we prove local maximal bounds for every \(1<p\leq\infty\). For polynomial parametrizations, we obtain bounds over all positive scales without a finite-type assumption, with constants only depending on the degree and dimensions. We also prove local maximal inequalities for variable hypersurfaces in \(\mathbb R^n\), \(n\geq3\), satisfying a uniform rotational curvature condition, in the range \(p>n/(n-1)\). The finite-type and polynomial estimates rely on a weak type \((1,1)\) inequality for a regularized auxiliary family adapted to non-isotropic dilations. Its proof uses a noncommutative Calderón-Zygmund decomposition based on Cuculescu projections. Interpolation with Fourier-based \(L^2\) bounds recovers the maximal inequalities for the original averages. The variable-hypersurface result uses a separate argument based on oscillatory \(L^2\) estimates and localization. As applications, we obtain some noncommutative maximal ergodic inequalities for trace-preserving actions of \(\mathbb R^n\) (corresponding to the geometric averages considered above) and bilateral almost uniform convergence for normalized ergodic averages.

Let $F$ be a fixed graph without isolated vertices. An edge-colored graph is $F$-rainbow saturated if it contains no rainbow copy of $F$, but the addition of any missing edge in any color creates a rainbow copy of $F$. The rainbow saturation number $rsat(n,F)$ is the minimum number of edges in such a graph on $n$ vertices. We prove a dichotomy governed by isolated edges: if $F$ contains an isolated edge, then $rsat(n,F)=O(1)$ for all sufficiently large $n$, while if $F$ has no isolated edge, then $rsat(n,F)=\Theta(n)$. The linear lower bound is expressed in terms of a directed weight parameter $\eta(F)$ and establishes the linear half of the dichotomy; in several cases it also strengthens the Cameron–Puleo type coefficient. For the bounded half, we construct rainbow saturated graphs for targets of the form $H\cup K_2$. As an application of these constructions, we determine the asymptotically tight behavior for the rainbow saturation number of the generalized friendship graph $F_{t,p,q}=tK_p\vee K_q$, proving that $ rsat(n,F_{t,p,q})=(p+q-1)n+O(1)$ for fixed $t\geq 2$, $p\geq 2$ and $q\geq 1$ as $n\to\infty$.

In this paper, we study the hypersurfaces in $\mathbb{H}^m\times\mathbb{H}^n$ ($m\geq3, n\geq2$) with constant principal curvatures. Let $g$ be the number of distinct constant principal curvatures. First, we classify all such hypersurfaces with $g\leq2$. Then, we prove that a hypersurface with constant principal curvatures and constant product angle function has $g\le3$, and we obtain a complete classification of these hypersurfaces. As a corollary, we classify the isoparametric hypersurfaces in $\mathbb{H}^m\times \mathbb{H}^n$ ($m\geq3, n\geq2$) with constant principal curvatures.

This work deals with the numerical solution of parametric linear elliptic partial differential equations using the finite element method for spatial discretization and a proper orthogonal decomposition or Lagrange subspace method based ROM for treating the parameter dependency. We study a simple one-parameter model problem and prove an error estimate for these ROMs as a function of the snapshot point set. Sub-exponential convergence with respect to the number of snapshot points is established under very mild assumptions on the point set. Exponential convergence is proven for structured point sets. In contrast to existing error analysis of Lagrange subspace methods, our estimates are not based on interpolation or perturbation arguments. Rather, we define a new approximation operator from the snapshot subspace and prove error and stability estimates for it.

We investigate the relative position of von Neumann subalgebras through the Mashood–Taylor ($\mathrm{d}_{\mathrm{MT}}$) and Kadison–Kastler ($\mathrm{d}_{\mathrm{KK}}$) distances and the interior angle. For each $n\in\mathbb{N}$, we show that the hyperfinite $\mathrm{II}_1$-factor $\mathscr{R}$ contains an uncountable family of pairwise distinct regular $n\times n$ spin model subfactors. In particular, this yields a continuous family $(\mathscr{R}_{\mathsf{H}_\alpha})_{\alpha\in[0,\pi)}$ of $2\times2$ spin model subfactors which, equipped with $\mathrm{d}_{\mathrm{MT}}$, is homeomorphic to the circle group $\mathbb{T}$ and admits a natural topological group structure. We further establish \[ \mathrm{d}_{\mathrm{MT}}(\mathscr{R}_{\mathsf{H}_\alpha},\mathscr{R}_{\mathsf{H}_\beta}) =\vert\sin(\alpha-\beta)\vert. \] We prove that two such spin model subfactors form a commuting square over their intersection if and only if they are maximally distant. More generally, commuting squares of $\mathrm{II}_1$-factors, under natural index conditions, force maximal distance, yielding $\mathrm{d}_{\mathrm{KK}}=1=\mathrm{d}_{\mathrm{MT}}$. We also show that diffuse subalgebras which are orthogonal in the sense of Popa are maximally distant. In contrast, no two members of the above spin model family are Popa-orthogonal. Nevertheless, maximal distance between two such subfactors implies that their interior angle over the intersection is $\pi/2$, highlighting the distinction between angle orthogonality and Popa orthogonality. Finally, in $L(\mathbb{F}_2)=L(\langle a,b\rangle)$, for $u\in L(\langle b\rangle)$, we obtain \[ \mathrm{d}_{\mathrm{MT}}(L(\langle a\rangle),uL(\langle a\rangle)u^*) =\sqrt{1-\vert\tau(u)\vert^4}, \] and construct maximally distant masas not arising from subgroups of $\mathbb{F}_2$.

We study bounded viscosity solutions of the degenerate or singular parabolic equation \[ u_t-|Du|^{\gamma}\Delta_p^{\mathrm{N}}u=f \qquad \text{in } Q_1, \] where \[ \Delta_p^{\mathrm{N}}u := \Delta u+(p-2) \left\langle D^2u\frac{Du}{|Du|}, \frac{Du}{|Du|} \right\rangle, \qquad -1<\gamma<\infty,\quad 1<p<\infty, \] and $f\in C^0(Q_1)\cap L^\infty(Q_1)$. Given any prescribed $\alpha\in(0,1)$, we prove that there exists $\varepsilon=\varepsilon(n,\alpha)>0$ such that \[ |p-2|+|\gamma|\leq\varepsilon \quad\Longrightarrow\quad u\in C_{\mathrm{loc}}^{1+\alpha,\frac{1+\alpha}{2}}(Q_1). \] In particular, the spatial gradient Hölder exponent can be chosen arbitrarily close to $ 1 $ as $(p,\gamma)\to(2,0)$. This extends the result of Andrade and Santos (Calc. Var. Partial Differential Equations 61, Paper No. 196, 2022) for $\gamma=0$ to the joint regime $(p,\gamma)\to(2,0)$ and answers affirmatively the question raised in Remark~1.1 therein.

We study weak tiling by finite unions of pairwise non-overlapping unit cubes in $\mathbb R^d$ that are not necessarily axis-parallel. We obtain a weak-tiling analogue of Keller's classical theorem, which gives a structural restriction on the support of any weak tiling measure associated with the unit cube. This allows us to derive geometric restrictions on configurations of cubes whose union admits a weak tiling. As an application, we prove Fuglede's conjecture for unions of three non-overlapping unit squares in $\mathbb R^2$ as well as unions of two non-overlapping axis-parallel unit cubes in any dimension.

Completely positive functions are an extension of completely positive matrices. They are known to characterize maximal spherical codes and maximum-density distance-avoiding subsets of $\mathbb{R}^n$ and certain compact metric spaces. This thesis expands this framework to related classes of problems in finite measure spaces and to the sphere-packing problem. For the latter, this is sharpened to show that the optimal sphere-packing density can be approximated using Schwartz functions. Converging hierarchies of semidefinite programming bounds on the size of optimal spherical codes are known, based on approximations of completely positive functions and the Lovász theta number of a graph. This thesis extends these hierarchies to distance-avoiding sets and similar problems and to the sphere-packing problem, and proves their convergence to the maximum density. For distance-avoiding sets, additional hierarchies, such as the moment hierarchy, are introduced and shown to be stronger than the completely positive hierarchy, hence they also converge. Related hierarchies for compact packing problems are also investigate. These bounds are implemented for Witsenhausen's problem, which asks for the maximum fraction $\alpha_n$ of the $n$-dimensional unit sphere that is coverable by a set avoiding orthogonal pairs; and for the $t$-almost-equiangular-set problem: finding the maximum size $\alpha(n,t)$ of a subset of the $n$-dimensional unit sphere in which every triple contains a pair with inner product $t \in [-1,1)$. An analytic solution to this bound yields an enumeration of optimal constructions for $n = 2$ and $3$ when $t \geq 0$.