We study the so-called averaging functors from the geometric Langlands program in the setting of Fargues’ program. This makes explicit certain cases of the spectral action which was recently introduced by Fargues-Scholze in the local Langlands program for $\mathrm {GL}_n$. Using these averaging functors, we verify (without using local Langlands) that the Fargues-Scholze parameters associated to supercuspidal modular representations of $\mathrm {GL}_2$ are irreducible. We also attach to any irreducible $\ell $-adic Weil representation of degree n an Hecke eigensheaf on $\mathrm {Bun}_n$ and show, using the local Langlands correspondence and recent results of Hansen and Hansen-Kaletha-Weinstein, that it satisfies most of the requirements of Fargues’ conjecture for $\mathrm {GL}_n$.

This paper develops a geometric and analytical framework for studying the existence and stability of pinned pulse solutions in a class of non-autonomous reaction–diffusion equations. The analysis relies on geometric singular perturbation theory, matched asymptotic method and nonlocal eigenvalue problem method. First, we derive the general criteria on the existence and spectral (in)stability of pinned pulses in slowly varying heterogeneous media. Then, as a specific example, we apply our theory to a heterogeneous Gierer–Meinhardt (GM) equation, where the nonlinearity varies slowly in space. We identify the conditions on parameters under which the pulse solutions are spectrally stable or unstable. It is found that when the heterogeneity vanishes, the results for the heterogeneous GM system reduce directly to the known results on the homogeneous GM system. This demonstrates the validity of our approach and highlights how the spatial heterogeneity gives rise to richer pulse dynamics compared to the homogeneous case.

To investigate multiple effects of the interaction between V. cholerae and phage on cholera transmission, we propose a degenerate reaction-diffusion model with different dispersal rates, which incorporates a short-lived hyperinfectious (HI vibrios) state of V. cholerae and lower-infectious (LI vibrios) state of V. cholerae. Our main purpose is to investigate the existence and stability analysis of multi-class boundary steady states, which is much more complicated and challenging than the case when the boundary steady state is unique. In a spatially heterogeneous case, the basic reproduction number $\mathscr{R}_{0}$ is defined as the spectral radius of the sum of two linear operators associated with HI vibrios infection and LI vibrios infection. If $\mathscr{R}_{0}\leq 1$, the disease-free steady state is globally asymptotically stable. If $\mathscr{R}_{0}\gt 1$, the uniform persistence of phage-free model, as well as the existence of the phage-free steady state, are established. In a spatially homogeneous case, when $\ \;\widetilde{\!\!\!\mathscr{R}}_{0}\gt 1$, the global asymptotic stability of phage-free steady state and the uniform persistence of the phage-present model are discussed under some additional conditions. The mathematical approach here has wide applications in degenerate Partial Differential Equations.

Given $n$ convex bodies in the Euclidean space $\mathbb{R}^d$, we can find their volume polynomial which is a homogeneous polynomial of degree $d$ in $n$ variables. We consider the set of homogeneous polynomials of degree $d$ in $n$ variables that can be represented as the volume polynomial of any such given convex bodies. This set is a subset of the set of Lorentzian polynomials. Using our knowledge of operations that preserve the Lorentzian property, we give a complete classification of the cases for $(n,d)$ when the two sets are equal.

The hard-core model has as its configurations the independent sets of some graph instance $G$. The probability distribution on independent sets is controlled by a ‘fugacity’ $\lambda \gt 0$, with higher $\lambda$ leading to denser configurations. We investigate the mixing time of Glauber (single-site) dynamics for the hard-core model on restricted classes of bounded-degree graphs in which a particular graph $H$ is excluded as an induced subgraph. If $H$ is a subdivided claw then, for all $\lambda$, the mixing time is $O(n\log n)$, where $n$ is the order of $G$. This extends a result of Chen and Gu for claw-free graphs. When $H$ is a path, the set of possible instances is finite. For all other $H$, the mixing time is exponential in $n$ for sufficiently large $\lambda$, depending on $H$ and the maximum degree of $G$.

In this paper, we prove the following result advocating the importance of monomial quadratic relations between holomorphic CM periods. For any simple CM abelian variety A, we can construct a CM abelian variety B such that all non-trivial Hodge relations between the holomorphic periods of the product $A\times B$ are generated by monomial quadratic ones which are also explicit. Moreover, B splits over the Galois closure of the CM field associated with A.

In this paper, we establish variational principles for the metric mean dimension of random dynamical systems with infinite topological entropy. This is based on four types of measure-theoretic ϵ-entropies: Kolmogorov-Sinai ϵ-entropy, Shapira’s ϵ-entropy, Katok’s ϵ-entropy and Brin–Katok local ϵ-entropy. The variational principle, as a fundamental theorem, links topological dynamics and ergodic theory.

Let $G = K \rtimes \langle t \rangle $ be a finitely generated group where K is abelian and $\langle t\rangle$ is the infinite cyclic group. Let R be a finite symmetric subset of K such that $S = \{ (r,1),(0,t^{\pm 1}) \mid r \in R \}$ is a generating set of G. We prove that the spherical conjugacy ratio, and hence the conjugacy ratio, of G with respect to S is 0 unless G is virtually abelian, confirming a conjecture of Ciobanu, Cox and Martino in this case. We also show that the Baumslag–Solitar group $\mathrm{BS}(1,k)$, $k\geq 2$, has a one-sided Følner sequence F such that the conjugacy ratio with respect to F is non-zero, even though $\mathrm{BS}(1,k)$ is not virtually abelian. This is in contrast to two-sided Følner sequences, where Tointon showed that the conjugacy ratio with respect to a two-sided Følner sequence is positive if and only if the group is virtually abelian.

Let $\Omega\subset\mathbb{R}^N$, $N\geq 1$, be an open bounded connected set. We consider the indefinite weighted eigenvalue problem $-\Delta u =\lambda m u$ in Ω with $\lambda \in \mathbb{R}$, $m\in L^\infty(\Omega)$ and with homogeneous Neumann boundary conditions. We study weak* continuity, convexity and Gâteaux differentiability of the map $m\mapsto1/\lambda_1(m)$, where $\lambda_1(m)$ is the principal eigenvalue. Then, denoting by $\mathcal{G}(m_0)$ the class of rearrangements of a fixed weight m0, under the assumptions that m0 is positive on a set of positive Lebesgue measure and $\int_\Omega m_0\,dx \lt 0$, we prove the existence and a characterization of minimizers of $\lambda_1(m)$ and the non-existence of maximizers. Finally, we show that, if Ω is a cylinder, then every minimizer is monotone with respect to the direction of the generatrix. In the context of the population dynamics, this kind of problems arise from the question of determining the optimal spatial location of favourable and unfavourable habitats for a population to survive.

We define two types of the α-Farey maps Fα and $F_{\alpha, \flat}$ for $0 \lt \alpha \lt \tfrac{1}{2}$, which were previously defined only for $\tfrac{1}{2} \le \alpha \le 1$ by Natsui (2004). Then, for each $0 \lt \alpha \lt \tfrac{1}{2}$, we construct the natural extension maps on the plane and show that the natural extension of $F_{\alpha, \flat}$ is metrically isomorphic to the natural extension of the original Farey map. As an application, we show that the set of normal numbers associated with α-continued fractions does not vary by the choice of α, $0 \lt \alpha \lt 1$. This extends the result by Kraaikamp and Nakada (2000).

For each set X, an X-split is a partition of X into two parts. For each X-split S and each subset $Y\subseteq X$, the restriction of S on Y is the Y-split whose parts are obtained by intersecting the parts of S with Y. For a graph G with vertex set V, the G-coboundary size of a V-split S is the number of edges in G having non-empty intersections with both parts of S. Let T be a tree without degree-two vertices, and let V and L denote its vertex set and leaf set, respectively. For each positive integer k, a k-split on T is an L-split that is the restriction of a V-split with T-coboundary size k, while a score-k split on T is a k-split on T that is not any k′-split for any integer $k' \lt k$. Buneman’s split equivalence theorem states that the tree T is entirely encoded by its system of score-1 splits. We identify the unique exceptional case in which the tree T is not determined by its score-2 split system. To explore how our work can be extended to more general tree isomorphism problems, we propose several conjectures and open problems related to set systems and generalized Buneman graphs.

We prove the formula

\begin{equation*} \text{cat}_G(X\vee Y)=\max \{\text{cat}_G(X),\text{cat}_G(Y)\} \end{equation*}

$X\vee Y$

$\bigvee _{i=1}^m X_i$

$G$

$X_i$

$G$

$i=1,\ldots ,m$

$X/A$

$G$

$X$

$A$

$A\hookrightarrow X$

$G$

$\overline {x_0}\,:\,A\to X$

$x_0\in X^G$

\begin{align*} \text{TC}_G(X\vee Y)&=\max \{\text{TC}_G(X),\text{TC}_G(Y),\text{cat}_G(X\times Y)\},\\ \text{TC}^G(X\vee Y)&=\max \{\text{TC}^G(X),\text{TC}^G(Y),_{X\vee Y}\text{cat}_{G\times G}(X\times Y)\}, \end{align*}

$G$

$G$

$X$

$Y$

We present an explicit subset $A\subseteq \mathbb{N} = \{0,1,\ldots \}$ such that $A + A = \mathbb{N}$ and for all $\varepsilon \gt 0$,

\begin{equation*}\lim _{N\to \infty }\frac {\big |\big \{(n_1,n_2): n_1 + n_2 = N, (n_1,n_2)\in A^2\big \}\big |}{N^{\varepsilon }} = 0.\end{equation*}

This answers a question of Erdős.

A popular method to perform adversarial attacks on neural networks is the so-called fast gradient sign method and its iterative variant. In this paper, we interpret this method as an explicit Euler discretization of a differential inclusion, where we also show convergence of the discretization to the associated gradient flow. To do so, we consider the concept of $p$-curves of maximal slope in the case $p=\infty$. We prove existence of $\infty$-curves of maximum slope and derive an alternative characterization via differential inclusions. Furthermore, we also consider Wasserstein gradient flows for potential energies, where we show that curves in the Wasserstein space can be characterized by a representing measure on the space of curves in the underlying Banach space, which fulfil the differential inclusion. The application of our theory to the finite-dimensional setting is twofold: On the one hand, we show that a whole class of normalized gradient descent methods (in particular, signed gradient descent) converge, up to subsequences, to the flow when sending the step size to zero. On the other hand, in the distributional setting, we show that the inner optimization task of adversarial training objective can be characterized via $\infty$-curves of maximum slope on an appropriate optimal transport space.

We provide a presymplectic characterization of Liouville sectors introduced by Ganatra–Pardon–Shende in [10, 12] in terms of the characteristic foliation of the boundary, which we call Liouville σ-sectors. We extend this definition to the case with corners using the presymplectic geometry of null foliations of the coisotropic intersections of transverse coisotropic collection of hypersurfaces, which appear in the definition of Liouville sectors with corners. We show that the set of Liouville σ-sectors with corners canonically forms a monoid that provides a natural framework for considering the Künneth-type functors in the wrapped Fukaya category. We identify its automorphism group that enables one to give a natural definition of bundles of Liouville sectors. As a byproduct, we affirmatively answer a question raised in [10, Question 2.6], which asks about the optimality of their definition of Liouville sectors in [10].

In the second half of the 19th century, Darboux obtained determinant formulae that provide the general solution for a linear hyperbolic second-order PDE with the finite Laplace series. These formulae played an important role in his study of the theory of surfaces and, in particular, in the theory of conjugate nets. During the last three decades, discrete analogues of conjugate nets (Q-nets) were actively studied. Laplace series can be defined also for hyperbolic difference operators. We prove discrete analogues of Darboux formulae for discrete and semi-discrete hyperbolic operators with finite Laplace series.

Jespers and Sun conjectured in [27] that if a finite group G has the property ND, i.e. for every nilpotent element n in the integral group ring $\mathbb{Z}G$ and every primitive central idempotent $e \in \mathbb{Q}G$ one still has $ne \in \mathbb{Z}G$, then at most one of the simple components of the group algebra $\mathbb{Q} G$ has reduced degree bigger than 1. With the exception of one very special series of groups we are able to answer their conjecture, showing that it is true—up to exactly one exception. To do so, we first classify groups with the so-called SN property which was introduced by Liu and Passman in their investigation of the Multiplicative Jordan Decomposition for integral group rings.

The conjecture of Jespers and Sun can also be formulated in terms of a group q(G) made from the group generated by the unipotent units, which is trivial if and only if the ND property holds for the group ring. We answer two more open questions about q(G) and notice that this notion allows to interpret the studied properties in the general context of linear semisimple algebraic groups. Here we show that q(G) is finite for lattices of big rank but can contain elements of infinite order in small rank cases.

We then study further two properties which appeared naturally in these investigations. A first which shows that property ND has a representation theoretical interpretation, while the other can be regarded as indicating that it might be hard to decide ND. Among others we show these two notions are equivalent for groups with SN.

Negami found an elegant splitting formula for the Tutte polynomial. We present an analogue of this for Bollobás and Riordan’s ribbon graph polynomial, and for the transition polynomial. From this we deduce a splitting formula for the Jones polynomial.

The evolution of a rotationally symmetric surface by a linear combination of its radii of curvature is considered. It is known that if the coefficients form certain integer ratios the flow is smooth and can be integrated explicitly. In this paper the non-integer case is considered for certain values of the coefficients and with mild analytic restrictions on the initial surface.

We prove that if the focal points at the north and south poles on the initial surface coincide, the flow converges to a round sphere. Otherwise the flow converges to a non-round Hopf sphere. Conditions on the fall-off of the astigmatism at the poles of the initial surface are also given that ensure the convergence of the flow.

The proof uses the spectral theory of singular Sturm-Liouville operators to construct an eigenbasis for an appropriate space in which the evolution is shown to converge.