Strengthening Hadwiger’s conjecture, Gerards and Seymour conjectured in 1995 that every graph with no odd $K_t$-minor is properly $(t-1)$-colourable. This is known as the Odd Hadwiger’s conjecture. We prove a relaxation of the above conjecture, namely we show that every graph with no odd $K_t$-minor admits a vertex $(2t-2)$-colouring such that all monochromatic components have size at most $\lceil \frac{1}{2}(t-2) \rceil$. The bound on the number of colours is optimal up to a factor of $2$, improves previous bounds for the same problem by Kawarabayashi (2008, Combin. Probab. Comput. 17 815–821), Kang and Oum (2019, Combin. Probab. Comput. 28 740–754), Liu and Wood (2021, arXiv preprint, arXiv:1905.09495), and strengthens a result by van den Heuvel and Wood (2018, J. Lond. Math. Soc. 98 129–148), who showed that the above conclusion holds under the more restrictive assumption that the graph is $K_t$-minor-free. In addition, the bound on the component-size in our result is much smaller than those of previous results, in which the dependency on $t$ was given by a function arising from the graph minor structure theorem of Robertson and Seymour. Our short proof combines the method by van den Heuvel and Wood for $K_t$-minor-free graphs with some additional ideas, which make the extension to odd $K_t$-minor-free graphs possible.

We define cohomological complexes of locally compact abelian groups associated with varieties over p-adic fields and prove a duality theorem under some assumption. Our duality takes the form of Pontryagin duality between locally compact motivic cohomology groups.

In this work we study the Cauchy problem in Gevrey spaces for a generalized class of equations that contains the case $b=0$ of the $b$-equation. For the generalized equation, we prove that it is locally well-posed for initial data in Gevrey spaces. Moreover, as we move to global well-posedness, we show that for a particular choice of the parameter in the equation the local solution is global analytic in both time and spatial variables.

In this paper, we construct uncountably many examples of multiparameter CCR flows, which are not pullbacks of $1$-parameter CCR flows, with any given index. Moreover, the constructed CCR flows are type I in the sense that the associated product system is the smallest subsystem containing its units.

In this paper, we study asymmetric Ramsey properties of the random graph $G_{n,p}$. Let $r \in \mathbb{N}$ and $H_1, \ldots, H_r$ be graphs. We write $G_{n,p} \to (H_1, \ldots, H_r)$ to denote the property that whenever we colour the edges of $G_{n,p}$ with colours from the set $[r] \,{:\!=}\, \{1, \ldots, r\}$ there exists $i \in [r]$ and a copy of $H_i$ in $G_{n,p}$ monochromatic in colour $i$. There has been much interest in determining the asymptotic threshold function for this property. In several papers, Rödl and Ruciński determined a threshold function for the general symmetric case; that is, when $H_1 = \cdots = H_r$. A conjecture of Kohayakawa and Kreuter from 1997, if true, would fully resolve the asymmetric problem. Recently, the $1$-statement of this conjecture was confirmed by Mousset, Nenadov and Samotij.

Building on work of Marciniszyn, Skokan, Spöhel and Steger from 2009, we reduce the $0$-statement of Kohayakawa and Kreuter’s conjecture to a certain deterministic subproblem. To demonstrate the potential of this approach, we show this subproblem can be resolved for almost all pairs of regular graphs. This therefore resolves the $0$-statement for all such pairs of graphs.

The method of equivariant moving frames is employed to construct and completely classify the differential invariants for the action of the projective group on functions defined on the two-dimensional projective plane. While there are four independent differential invariants of order $\leq 3$, it is proved that the algebra of differential invariants is generated by just two of them through invariant differentiation. The projective differential invariants are, in particular, of importance in image processing applications.

We study the compatibility of the action of the DAHA of type GL with two inverse systems of polynomial rings obtained from the standard Laurent polynomial representations. In both cases, the crucial analysis is that of the compatibility of the action of the Cherednik operators. Each case leads to a representation of a limit structure (the +/– stable limit DAHA) on a space of almost symmetric polynomials in infinitely many variables (the standard representation). As an application, we show that the defining representation of the double Dyck path algebra arises from the standard representation of the +stable limit DAHA.

We suggest two related conjectures dealing with the existence of spanning irregular subgraphs of graphs. The first asserts that any $d$-regular graph on $n$ vertices contains a spanning subgraph in which the number of vertices of each degree between $0$ and $d$ deviates from $\frac{n}{d+1}$ by at most $2$. The second is that every graph on $n$ vertices with minimum degree $\delta$ contains a spanning subgraph in which the number of vertices of each degree does not exceed $\frac{n}{\delta +1}+2$. Both conjectures remain open, but we prove several asymptotic relaxations for graphs with a large number of vertices $n$. In particular we show that if $d^3 \log n \leq o(n)$ then every $d$-regular graph with $n$ vertices contains a spanning subgraph in which the number of vertices of each degree between $0$ and $d$ is $(1+o(1))\frac{n}{d+1}$. We also prove that any graph with $n$ vertices and minimum degree $\delta$ contains a spanning subgraph in which no degree is repeated more than $(1+o(1))\frac{n}{\delta +1}+2$ times.

We present sufficient conditions under which a given linear nonautonomous system and its nonlinear perturbation are topologically conjugated. Our conditions are of a very general form and provided that the nonlinear perturbations are well-behaved, we do not assume any asymptotic behaviour of the linear system. Moreover, the control on the nonlinear perturbations may differ along finitely many mutually complementary directions. We consider both the cases of one-sided discrete and continuous dynamics.

Let $G$ be a compact connected simple Lie group of type $(n_{1},\,\ldots,\,n_{l})$, where $n_{1}<\cdots < n_{l}$. Let $\mathcal {G}_k$ be the gauge group of the principal $G$-bundle over $S^{4}$ corresponding to $k\in \pi _3(G)\cong \mathbb {Z}$. We calculate the mod-$p$ homology of the classifying space $B\mathcal {G}_k$ provided that $n_{l}< p-1$.

For $N\geq 2$, a bounded smooth domain $\Omega$ in $\mathbb {R}^{N}$, and $g_0,\, V_0 \in L^{1}_{loc}(\Omega )$, we study the optimization of the first eigenvalue for the following weighted eigenvalue problem:

\[ -\Delta_p \phi + V |\phi|^{p-2}\phi = \lambda g |\phi|^{p-2}\phi \text{ in } \Omega, \quad \phi=0 \text{ on } \partial \Omega, \]

$g$

$V$

$g_0$

$V_0$

$(\underline {g},\,\underline {V})$

$(\overline {g},\,\overline {V})$

$g_0$

$V_0$

$p=2$

We show that the $4$-state anti-ferromagnetic Potts model with interaction parameter $w\in (0,1)$ on the infinite $(d+1)$-regular tree has a unique Gibbs measure if $w\geq 1-\dfrac{4}{d+1_{_{\;}}}$ for all $d\geq 4$. This is tight since it is known that there are multiple Gibbs measures when $0\leq w\lt 1-\dfrac{4}{d+1}$ and $d\geq 4$. We moreover give a new proof of the uniqueness of the Gibbs measure for the $3$-state Potts model on the $(d+1)$-regular tree for $w\geq 1-\dfrac{3}{d+1}$ when $d\geq 3$ and for $w\in (0,1)$ when $d=2$.

We study ribbons of vanishing Gaussian curvature, i.e. flat ribbons, constructed along a curve in $\mathbb {R}^{3}$. In particular, we first investigate to which extent the ruled structure determines a flat ribbon: in other words, we ask whether for a given curve $\gamma$ and ruling angle (angle between the ruling line and the curve's tangent) there exists a well-defined flat ribbon. It turns out that the answer is positive only up to an initial condition, expressed by a choice of normal vector at a point. We then study the set of infinitely narrow flat ribbons along a fixed curve $\gamma$ in terms of energy. By extending a well-known formula for the bending energy of the rectifying developable, introduced in the literature by Sadowsky in 1930, we obtain an upper bound for the difference between the bending energies of two solutions of the initial value problem. We finally draw further conclusions under some additional assumptions on the ruling angle and the curve $\gamma$.

For subsets in the standard symplectic space $(\mathbb {R}^{2n},\omega _0)$ whose closures are intersecting with coisotropic subspace $\mathbb {R}^{n,k}$ we construct relative versions of the Ekeland–Hofer capacities of the subsets with respect to $\mathbb {R}^{n,k}$, establish representation formulas for such capacities of bounded convex domains intersecting with $\mathbb {R}^{n,k}$. We also prove a product formula and a fact that the value of this capacity on a hypersurface $\mathcal {S}$ of restricted contact type containing the origin is equal to the action of a generalized leafwise chord on $\mathcal {S}$.

The paper proves transportation inequalities for probability measures on spheres for the Wasserstein metrics with respect to cost functions that are powers of the geodesic distance. Let $\mu$ be a probability measure on the sphere ${\bf S}^n$ of the form $d\mu =e^{-U(x)}{\rm d}x$ where ${\rm d}x$ is the rotation invariant probability measure, and $(n-1)I+{\hbox {Hess}}\,U\geq {\kappa _U}I$, where $\kappa _U>0$. Then any probability measure $\nu$ of finite relative entropy with respect to $\mu$ satisfies ${\hbox {Ent}}(\nu \mid \mu ) \geq (\kappa _U/2)W_2(\nu,\, \mu )^2$. The proof uses an explicit formula for the relative entropy which is also valid on connected and compact $C^\infty$ smooth Riemannian manifolds without boundary. A variation of this entropy formula gives the Lichnérowicz integral.

In this note, we prove two monotonicity formulas for solutions of $\Delta _H f = c$ and $\Delta _H f - \partial _t f = c$ in Carnot groups. Such formulas involve the right-invariant carré du champ of a function and they are false for the left-invariant one. The main results, theorems 1.1 and 1.2, display a resemblance with two deep monotonicity formulas respectively due to Alt–Caffarelli–Friedman for the standard Laplacian, and to Caffarelli for the heat equation. In connection with this aspect we ask the question whether an ‘almost monotonicity’ formula be possible. In the last section, we discuss the failure of the nondecreasing monotonicity of an Almgren type functional.

We prove stronger variants of a multiplier theorem of Kislyakov. The key ingredients are based on ideas of Kislyakov and the Kahane–Salem–Zygmund inequality. As a by-product, we show various multiplier theorems for spaces of trigonometric polynomials on the n-dimensional torus $\mathbb {T}^n$ or Boolean cubes $\{-1,1\}^N$. Our more abstract approach based on local Banach space theory has the advantage that it allows to consider more general compact abelian groups instead of only the multidimensional torus. As an application, we show that various recent $\ell _1$-multiplier theorems for trigonometric polynomials in several variables or ordinary Dirichlet series may be proved without the Kahane–Salem–Zygmund inequality.