The amoeba of an affine algebraic variety $V\subset ({\mathbb{C}}^{*}){}^r$ is the image of $V$ under the map $(z_1,\dots ,z_r)\to (\log |z_1|,\dots ,\log |z_r|)$. We give a characterisation of the amoeba, based on the triangle inequality, which we call testing for lopsidedness. We show that if a point is outside the amoeba of $V$, there is an element of the defining ideal which witnesses this fact by being lopsided. This condition is necessary and sufficient for amoebas of arbitrary codimension as well as for compactifications of amoebas inside any toric variety. Our approach naturally leads to methods for approximating hypersurface amoebas and their spines by systems of linear inequalities. Finally, we remark that our main result can be seen as a precise analogue of a Nullstellensatz statement for tropical varieties

In this article, we introduce a new construction of endoscopic lifting in classical groups. To do that, we study a certain small representation and use it as a kernel function to construct the liftings. As an application of the construction, we study the relations of poles of tensor $L$-function with certain liftings and certain period integrals

We prove that the dual or any quotient of a separable reflexive Banach space with the unconditional tree property (UTP) has the UTP. This is used to prove that a separable reflexive Banach space with the UTP embeds into a reflexive Banach space with an unconditional basis. This solves several longstanding open problems. In particular, it yields that a quotient of a reflexive Banach space with an unconditional finite-dimensional decomposition (UFDD) embeds into a reflexive Banach space with an unconditional basis

We study the degree growth of iterates of meromorphic self-maps of compact Kähler surfaces. Using cohomology classes on the Riemann-Zariski space, we show that the degrees grow similarly to those of mappings that are algebraically stable on some bimeromorphic model

The classical Painlevé theorem tells us that sets of zero length are removable for bounded analytic functions, while (some) sets of positive length are not. For general $K$-quasiregular mappings in planar domains, the corresponding critical dimension is $2/(K+1)$. We show that when $K>1$, unexpectedly one has improved removability. More precisely, we prove that sets $E$ of $\sigma$-finite Hausdorff $(2/(K+1))$-measure are removable for bounded $K$-quasiregular mappings. On the other hand, $\dim (E)=2/(K+1)$ is not enough to guarantee this property.

We also study absolute continuity properties of pullbacks of Hausdorff measures under $K$-quasiconformal mappings: in particular, at the relevant dimensions $1$ and $2/(K+1)$. For general Hausdorff measures $H^t$, $0<t<2$, we reduce the absolute continuity properties to an open question on conformal mappings (see Conjecture 2.3)

For each oriented surface $\Sigma$ of genus $g$, we study a limit of quantum representations of the mapping class group arising in topological quantum field theory (TQFT) derived from the Kauffman bracket. We determine that these representations converge in the Fell topology to the representation of the mapping class group on $H(\Sigma )$, the space of regular functions on the $\mathrm{SL}(2,\mathbb{C})$-representation variety with its Hermitian structure coming from the symplectic structure of the $\mathrm{SU}(2)$-representation variety. As a corollary, we give a new proof of the asymptotic faithfulness of quantum representations