We study deformed Hermitian Yang–Mills (dHYM) connections on ruled surfaces explicitly, using the momentum construction. As a main application, we provide many new examples of dHYM connections coupled to a variable background Kähler metric. These are solutions of the moment map partial differential equations given by the Hamiltonian action of the extended gauge group, coupling the dHYM equation to the scalar curvature of the background. The large radius limit of these coupled equations is the Kähler–Yang–Mills system of Álvarez-Cónsul, Garcia-Fernandez, and García-Prada, and in this limit, our solutions converge smoothly to those constructed by Keller and Tønnesen-Friedman. We also discuss other aspects of our examples including conical singularities, realization as B-branes, the small radius limit, and canonical representatives of complexified Kähler classes.

We introduce the combinatorial notion of a q-factorization graph intended as a tool to study and express results related to the classification of prime simple modules for quantum affine algebras. These are directed graphs equipped with three decorations: a coloring and a weight map on vertices, and an exponent map on arrows (the exponent map can be seen as a weight map on arrows). Such graphs do not contain oriented cycles and, hence, the set of arrows induces a partial order on the set of vertices. In this first paper on the topic, beside setting the theoretical base of the concept, we establish several criteria for deciding whether or not a tensor product of two simple modules is a highest-$\ell $-weight module and use such criteria to prove, for type A, that a simple module whose q-factorization graph has a totally ordered vertex set is prime.

In this paper, we prove the algebraicity of some L-values attached to quaternionic modular forms. We follow the rather well-established path of the doubling method. Our main contribution is that we include the case where the corresponding symmetric space is of non-tube type. We make various aspects very explicit, such as the doubling embedding, coset decomposition, and the definition of algebraicity of modular forms via CM-points.

A crucial ingredient in the theory of theta liftings of Kudla and Millson is the construction of a $q$-form $\varphi_{KM}$ on an orthogonal symmetric space, using Howe's differential operators. This form can be seen as a Thom form of a real oriented vector bundle. We show that the Kudla-Millson form can be recovered from a canonical construction of Mathai and Quillen. A similar result was obtaind by Garcia for signature $(2,q)$ in case the symmetric space is hermitian and we extend it to arbitrary signature.

Inspired by a remarkable work of Félix, Halperin, and Thomas on the asymptotic estimation of the ranks of rational homotopy groups, and more recent works of Wu and the authors on local hyperbolicity, we prove two asymptotic formulae for torsion rank of homotopy groups, one using ordinary homology and one using K-theory. We use these to obtain explicit quantitative asymptotic lower bounds on the torsion rank of the homotopy groups for many interesting spaces after suspension, including Moore spaces, Eilenberg–MacLane spaces, complex projective spaces, complex Grassmannians, Milnor hypersurfaces, and unitary groups.

It is shown that the colored isomorphism class of a unital, simple, $\mathcal {Z}$-stable, separable amenable C$^*$-algebra satisfying the universal coefficient theorem is determined by its tracial simplex.

To every finite metric space X, including all connected unweighted graphs with the minimum edge-distance metric, we attach an invariant that we call its blowup-polynomial $p_X(\{ n_x : x \in X \})$. This is obtained from the blowup $X[\mathbf {n}]$ – which contains $n_x$ copies of each point x – by computing the determinant of the distance matrix of $X[\mathbf {n}]$ and removing an exponential factor. We prove that as a function of the sizes $n_x$, $p_X(\mathbf {n})$ is a polynomial, is multi-affine, and is real-stable. This naturally associates a hitherto unstudied delta-matroid to each metric space X; we produce another novel delta-matroid for each tree, which interestingly does not generalize to all graphs. We next specialize to the case of $X = G$ a connected unweighted graph – so $p_G$ is “partially symmetric” in $\{ n_v : v \in V(G) \}$ – and show three further results: (a) We show that the polynomial $p_G$ is indeed a graph invariant, in that $p_G$ and its symmetries recover the graph G and its isometries, respectively. (b) We show that the univariate specialization $u_G(x) := p_G(x,\dots ,x)$ is a transform of the characteristic polynomial of the distance matrix $D_G$; this connects the blowup-polynomial of G to the well-studied “distance spectrum” of G. (c) We obtain a novel characterization of complete multipartite graphs, as precisely those for which the “homogenization at $-1$” of $p_G(\mathbf { n})$ is real-stable (equivalently, Lorentzian, or strongly/completely log-concave), if and only if the normalization of $p_G(-\mathbf { n})$ is strongly Rayleigh.

For a closed, connected, and oriented 3-manifold with a non-singular flow, we construct its virtual knot diagram via a “flow-spine” of the manifold. Then, we introduce a coloring invariant of 3-manifolds through their virtual knot diagrams, and classify some 3-manifolds by using the invariant.

In this article, we are concerned with the tempered fractional parabolic problem

$$ \begin{align*}\frac{\partial u}{\partial t}(x, t)-\left(\Delta+\lambda\right)^{\frac{\alpha}{2}} u(x, t)=f(u(x, t)), \end{align*} $$

where $-\left (\Delta +\lambda \right )^{\frac {\alpha }{2}}$ is a tempered fractional operator with $\alpha \in (0,2)$ and $\lambda $ is a sufficiently small positive constant. We first establish maximum principle principles for problems involving tempered fractional parabolic operators. And then, we develop the direct sliding methods for the tempered fractional parabolic problem, and discuss how they can be used to establish monotonicity results of solutions to the tempered fractional parabolic problem in various domains. We believe that our theory and methods can be conveniently applied to study parabolic problems involving other nonlocal operators.

A weighted nonlinear flag is a nested set of closed submanifolds, each submanifold endowed with a volume density. We study the geometry of Fréchet manifolds of weighted nonlinear flags, in this way generalizing the weighted nonlinear Grassmannians. When the ambient manifold is symplectic, we use these nonlinear flags to describe a class of coadjoint orbits of the group of Hamiltonian diffeomorphisms, orbits that consist of weighted isotropic nonlinear flags.

In this paper, we study compactifications of the moduli of smooth del Pezzo surfaces using K-stability and the line arrangement. We construct K-moduli of log del Pezzo pairs with sum of lines as boundary divisors, and prove that for $d=2,3,4$, these K-moduli of pairs are isomorphic to the K-moduli spaces of del Pezzo surfaces. For $d=1$, we prove that they are different by exhibiting some walls.

Motivated by the near invariance of model spaces for the backward shift, we introduce a general notion of $(X,Y)$-invariant operators. The relations between this class of operators and the near invariance properties of their kernels are studied. Those lead to orthogonal decompositions for the kernels, which generalize well-known orthogonal decompositions of model spaces. Necessary and sufficient conditions for those kernels to be nearly X-invariant are established. This general approach can be applied to a wide class of operators defined as compressions of multiplication operators, in particular to Toeplitz operators and truncated Toeplitz operators, to study the invariance properties of their kernels (general Toeplitz kernels).

We investigate quantum lens spaces, $C(L_q^{2n+1}(r;\underline {m}))$, introduced by Brzeziński and Szymański as graph $C^*$-algebras. We give a new description of $C(L_q^{2n+1}(r;\underline {m}))$ as graph $C^*$-algebras amending an error in the original paper by Brzeziński and Szymański. Furthermore, for $n\leq 3$, we give a number-theoretic invariant, when all but one weight are coprime to the order of the acting group r. This builds upon the work of Eilers, Restorff, Ruiz, and Sørensen.

In this work, we obtain the pointwise almost everywhere convergence for two families of multilinear operators: (a) the doubly truncated homogeneous singular integral operators associated with $L^q$ functions on the sphere and (b) lacunary multiplier operators of limited smoothness. The a.e. convergence is deduced from the $L^2\times \cdots \times L^2\to L^{2/m}$ boundedness of the associated maximal multilinear operators.

An infinite graph is said to be highly connected if the induced subgraph on the complement of any set of vertices of smaller size is connected. We continue the study of weaker versions of Ramsey’s theorem on uncountable cardinals asserting that if we color edges of the complete graph, we can find a large highly connected monochromatic subgraph. In particular, several questions of Bergfalk, Hrušák, and Shelah (2021, Acta Mathematica Hungarica 163, 309–322) are answered by showing that assuming the consistency of suitable large cardinals, the following are relatively consistent with ZFC:

• • $\kappa \to _{hc} (\kappa )^2_\omega $ for every regular cardinal $\kappa \geq \aleph _2$,

• • $\neg \mathsf {CH}+ \aleph _2 \to _{hc} (\aleph _1)^2_\omega $.

Building on a work of Lambie-Hanson (2023, Fundamenta Mathematicae. 260(2):181–197), we also show that

• • $\aleph _2 \to _{hc} [\aleph _2]^2_{\omega ,2}$ is consistent with $\neg \mathsf {CH}$.

To prove these results, we use the existence of ideals with strong combinatorial properties after collapsing suitable large cardinals.

Let $\mathcal {F}$ be a hereditary collection of finite subsets of $\mathbb {N}$. In this paper, we introduce and characterize $\mathcal {F}$-(almost) greedy bases. Given such a family $\mathcal {F}$, a basis $(e_n)_n$ for a Banach space X is called $\mathcal {F}$-greedy if there is a constant $C\geqslant 1$ such that for each $x\in X$, $m \in \mathbb {N}$, and $G_m(x)$, we have

$$ \begin{align*} \|x - G_m(x)\|\ \leqslant\ C \inf\left\{\left\|x-\sum_{n\in A}a_ne_n\right\|\,:\, |A|\leqslant m, A\in \mathcal{F}, (a_n)\subset \mathbb{K}\right\}. \end{align*} $$

$G_m(x)$

$\mathbb {K}$

$\mathcal {F}$

$\mathcal {F}$

$\mathcal {F}$

$\mathcal {F}$

$\mathcal {F}$

$\mathcal {F}$

Furthermore, we provide several examples of bases that are nontrivially $\mathcal {F}$-greedy. For a countable ordinal $\alpha $, we consider the case $\mathcal {F}=\mathcal {S}_{\alpha }$, where $\mathcal {S}_{\alpha }$ is the Schreier family of order $\alpha $. We show that for each $\alpha $, there is a basis that is $\mathcal {S}_{\alpha }$-greedy but is not $\mathcal {S}_{\alpha +1}$-greedy. In other words, we prove that none of the following implications can be reversed: for two countable ordinals $\alpha < \beta $,

$$ \begin{align*} \mbox{quasi-greedy}\ \Longleftarrow\ \mathcal{S}_{\alpha}\mbox{-greedy}\ \Longleftarrow\ \mathcal{S}_{\beta}\mbox{-greedy}\ \Longleftarrow\ \mbox{greedy}. \end{align*} $$

Let $F_{\pi }$ be a finite Galois-algebra extension of a number field F, with group G. Suppose that $F_{\pi }/F$ is weakly ramified and that the square root $A_\pi $ of the inverse different $\mathfrak {D}_{\pi }^{-1}$ is defined. (This latter condition holds if, for example, $|G|$ is odd.) Erez has conjectured that the class $(A_\pi )$ of $A_\pi $ in the locally free class group $\operatorname {\mathrm {Cl}}(\mathbf {Z} G)$ of $\mathbf {Z} G$ is equal to the Cassou–Noguès–Fröhlich root number class $W(F_{\pi }/F)$ associated with $F_\pi /F$. This conjecture has been verified in many cases. We establish a precise formula for $(A_\pi )$ in terms of $W(F_{\pi }/F)$ in all cases where $A_\pi $ is defined and $F_\pi /F$ is tame, and are thereby able to deduce that, in general, $(A_\pi )$ is not equal to $W(F_\pi /F)$.

Let G be a compact connected Lie group, and let $\operatorname {Hom}({\mathbb {Z}}^m,G)$ be the space of pairwise commuting m-tuples in G. We study the problem of which primes $p \operatorname {Hom}({\mathbb {Z}}^m,G)_1$, the connected component of $\operatorname {Hom}({\mathbb {Z}}^m,G)$ containing the element $(1,\ldots ,1)$, has p-torsion in homology. We will prove that $\operatorname {Hom}({\mathbb {Z}}^m,G)_1$ for $m\ge 2$ has p-torsion in homology if and only if p divides the order of the Weyl group of G for $G=SU(n)$ and some exceptional groups. We will also compute the top homology of $\operatorname {Hom}({\mathbb {Z}}^m,G)_1$ and show that $\operatorname {Hom}({\mathbb {Z}}^m,G)_1$ always has 2-torsion in homology whenever G is simply-connected and simple. Our computation is based on a new homotopy decomposition of $\operatorname {Hom}({\mathbb {Z}}^m,G)_1$, which is of independent interest and enables us to connect torsion in homology to the combinatorics of the Weyl group.

An explicit transformation for the series $\sum \limits _{n=1}^{\infty }\displaystyle \frac {\log (n)}{e^{ny}-1}$, or equivalently, $\sum \limits _{n=1}^{\infty }d(n)\log (n)e^{-ny}$ for Re$(y)>0$, which takes y to $1/y$, is obtained for the first time. This series transforms into a series containing the derivative of $R(z)$, a function studied by Christopher Deninger while obtaining an analog of the famous Chowla–Selberg formula for real quadratic fields. In the course of obtaining the transformation, new important properties of $\psi _1(z)$ (the derivative of $R(z)$) are needed as is a new representation for the second derivative of the two-variable Mittag-Leffler function $E_{2, b}(z)$ evaluated at $b=1$, all of which may seem quite unexpected at first glance. Our transformation readily gives the complete asymptotic expansion of $\sum \limits _{n=1}^{\infty }\displaystyle \frac {\log (n)}{e^{ny}-1}$ as $y\to 0$ which was also not known before. An application of the latter is that it gives the asymptotic expansion of $ \displaystyle \int _{0}^{\infty }\zeta \left (\frac {1}{2}-it\right )\zeta '\left (\frac {1}{2}+it\right )e^{-\delta t}\, dt$ as $\delta \to 0$.

We study systems of two-tangle equations

$$ \begin{align*}\begin{cases} N(X+T_1)=L_1,\\ N(X+T_2)=L_2, \end{cases}\end{align*} $$

We show that every system of framed tangle equations has at most one-framed rational solution. Furthermore, we show that the Jones unknot conjecture implies that if a system of tangle equations has a rational solution, then that solution is unique among all two-tangles. This result potentially opens a door to a purely topological disproof of the Jones unknot conjecture.

We introduce the notion of the Kauffman bracket ratio $\{T\}_q\in \mathbb Q(q)$ of any two-tangle T and we conjecture that for $q=1$ it is the slope of meridionally incompressible surfaces in $D^3-T$. We prove that conjecture for algebraic T. We also prove that for rational T, the brackets $\{T\}_q$ coincide with the q-rationals of Morier-Genoud and Ovsienko.

Additionally, we relate systems of tangle equations to the cosmetic surgery conjecture and the nugatory crossing conjecture.