We establish large sets of Anderson localized states for the quasi-periodic nonlinear Schrödinger equation on $\mathbb Z^d$, thus extending Anderson localization from the linear (cf. Bourgain [Geom. Funct. Anal., 17(3):682–706, 2007]) to a nonlinear setting, and from the random (cf. Bourgain-Wang [J. Eur. Math. Soc., 10(1):1–45, 2008]) to a deterministic setting. Among the main ingredients are a new Diophantine estimate of quasi-periodic functions in arbitrary-dimensional phase space, and the application of Bourgain’s geometric lemma in [Geom. Funct. Anal., 17(3):682–706, 2007].

We systematically study the intersection flatness and Ohm-Rush properties for modules over a commutative ring, drawing inspiration from the work of Ohm and Rush and of Hochster and Jeffries. We establish new structural results for modules that are intersection flat/Ohm-Rush by exhibiting intimate connections between these notions and the seminal work of Raynaud and Gruson on Mittag-Leffler modules. In particular, we develop a theory of Ohm-Rush modules that is parallel to the theory of Mittag-Leffler modules. We also obtain descent and local-to-global results for intersection flat/Ohm-Rush modules. Our investigations reveal a particularly pleasing picture for flat modules over a complete local ring, in which case many otherwise distinct properties coincide.

The quantum duality principle (QDP) by Drinfeld predicts a connection between the quantized universal enveloping algebras and the quantized coordinate algebras, where the underlying classical objects are related by the duality in Poisson geometry. The current paper gives an explicit formulization of the QDP for quantum symmetric pairs.

Let $\mathfrak {g}$ be a complex semi-simple Lie algebra, equipped with the standard Lie bialgebra structure. Let $\theta $ be a Lie algebra involution on $\mathfrak {g}$ and denote by $\mathfrak {k}=\mathfrak {g}^\theta $ the fixed point subalgebra. The quantum symmetric pair $(\mathrm {U},\mathrm {U}^\imath )$ is originally defined to be a quantization of the symmetric pair of the universal enveloping algebras $(U(\mathfrak {g}),U(\mathfrak {k}))$. In this paper, we show that an explicit specialization of $(\mathrm {U},\mathrm {U}^\imath )$ gives rise to the pair of the coordinate algebras $(\mathcal {O}(G^*),\mathcal {O}(K^\perp \backslash G^*))$, where $G^*$ is the dual Poisson-Lie group with the Lie algebra $\mathfrak {g}^*$, and $K^\perp \backslash G^*$ is a $G^*$-Poisson homogeneous space. Here $K^\perp $ is the closed subgroup of $G^*$ associated to the complementary dual of $\mathfrak {k}$. Therefore $(\mathrm {U},\mathrm {U}^\imath )$ can be viewed as a pair of quantized coordinate algebras. This generalizes the result of De Concini–Procesi [14] that the quantum group $\mathrm {U}$ provides a quantization of the coordinate algebra of $G^*$.

Let G be a finite group and let p be a prime. In this paper, we prove a strengthened version of Brauer’s height zero conjecture for the principal p-block of G that takes the action of a certain group of Galois automorphisms into account. This answers a conjecture recently proposed by Malle, Moretó, Rizo and Schaeffer Fry. We then use this to obtain a structural result which can be seen as a Galois version of the Itô–Michler theorem.

We prove an analogue of Chebyshev’s alternation theorem for linearly independent discrete functions $\Phi _n=\{\varphi _k\}_{k=1}^n$ on the interval $[0,q]_{\scriptscriptstyle \mathbb {Z}}=[0,q]\cap \mathbb {Z}$. In particular, we establish that the polynomial of best uniform approximation of a discrete function admits a Chebyshev alternance set of length $n+1$ if and only if $\Phi _n$ is a Chebyshev $T_{\scriptscriptstyle \mathbb {Z}}$-system. We also obtain a discrete version of Sturm’s oscillation theorem, according to which the number of discrete zeros of the polynomial $\sum _{k=m}^{n}a_k\varphi _k$ is no less than $m-1$ and no more than $n-1$. This implies that $\Phi _n$ is a $T_{\scriptscriptstyle \mathbb {Z}}$-system and a discrete Sturm-Hurwitz spectral gap theorem is valid. As applications, we study the orthogonal polynomials with removed largest zeros. We establish the monotonicity property of coefficients in the Fourier expansions of such polynomials, thereby strengthening the results of H. Cohn and A. Kumar. We apply this to solve a Yudin-type extremal problem for polynomials with spectral gap.

We introduce action-driven flows for causal variational principles, being a class of nonconvex variational problems emanating from applications in fundamental physics. In the compact setting, Hölder continuous curves of measures are constructed by using the method of minimizing movements. As is illustrated in examples, these curves will in general not have a limit point, due to the nonconvexity of the action. This leads us to introducing a novel penalization which ensures the existence of a limit point, giving rise to approximative solutions of the Euler-Lagrange equations. The methods and results are adapted and generalized to the causal action principle in the finite-dimensional case. As an application, we construct a flow of measures for causal fermion systems in the infinite-dimensional situation.

Joyce vertex algebras are vertex algebra structures defined on the homology of certain $\mathbb {C}$-linear moduli stacks, and are used to express wall-crossing formulae for Joyce’s homological enumerative invariants. This paper studies the generalization of this construction to settings that come from nonlinear enumerative problems. In the special case of orthosymplectic enumerative geometry, we obtain twisted modules for Joyce vertex algebras.

We expect that our construction will be useful for formulating wall-crossing formulae for enumerative invariants for nonlinear moduli stacks. We include several variants of our construction that apply to different flavours of enumerative invariants, including Joyce’s homological invariants, DT4 invariants, and a version of K-theoretic enumerative invariants.

We extend the notion of T-duality to manifolds endowed with non-principal torus actions. The singularities of the torus action are controlled by a certain Lie algebroid, called the elliptic tangent bundle. Using this Lie algebroid, we explain how certain invariant generalized complex structures can be transported via T-duality. Along the way, we use the elliptic tangent bundle to define connections for these torus actions, and give new insight into the classification of such actions by Haefliger-Salem.

Let $H_g$ denote the 4-dimensional handlebody of genus g and $U_g$ its boundary. We show that for all $g \ge 0$ the map from $B\!\operatorname {\mathrm {Homeo}}(H_g)$ to $B\!\operatorname {\mathrm {Homeo}}(U_g)$ induced by restriction to the boundary admits a section.

We describe the K-moduli spaces of weighted hypersurfaces of degree $2(n+3)$ in $\mathbb {P}(1,2,n+2,n+3)$. We show that the K-polystable limits of these weighted hypersurfaces are also weighted hypersurfaces of the same degree in the same weighted projective space. This is achieved by an explicit study of the wall crossing for K-moduli spaces $M_w$ of certain log Fano pairs with coefficient w whose double cover gives the weighted hypersurface. Moreover, we show that the wall crossing of $M_w$ coincides with variation of GIT except at the last K-moduli wall which gives a divisorial contraction. Our K-moduli spaces provide new birational models for some natural loci in the moduli space of marked hyperelliptic curves.

Let $R \to S$ be a cyclically pure map of Noetherian $\mathbb {Q}$-algebras. In this paper, we show that if S has Du Bois singularities, then R has Du Bois singularities. Our result is new even when $R \to S$ is faithfully flat. Our proof also yields interesting results in prime characteristic and in mixed characteristic. As a consequence, we show that if $R \to S$ is a cyclically pure map of rings essentially of finite type over the complex numbers $\mathbb {C}$, S has log canonical type singularities, and $K_R$ is Cartier, then R has log canonical singularities. Along the way, we prove a version of the key injectivity theorem of Kovács and Schwede for Noetherian schemes of equal characteristic zero that have isolated non-Du Bois points. Throughout the paper, we use the characterization of the complex $\underline {\Omega }^0_X$ and of Du Bois singularities in terms of sheafification with respect to Grothendieck topologies.

We introduce and study different compactifications of the moduli space of n distinct weighted labeled points in a flag of affine spaces. We construct these spaces via the weighted and generalized Fulton-MacPherson compactifications of Routis and Kim-Sato. For certain weights, our compactifications are toric and isomorphic to the polypermutohedral and polystellahedral varieties, which arise in the work of Crowley-Huh-Larson-Simpson-Wang and Eur-Larson on polymatroids, a generalization of matroids. Moreover, we show that these toric compactifications have a fibration structure, with fibers isomorphic to the Losev-Manin space, and are related to each other via a geometric quotient.

Pólya trees are unlabeled rooted trees on n vertices. This paper gives a new way to generate Pólya trees, that conjecturally is very efficient. This allows comparing typical unlabeled and labeled tree statistics and comparing asymptotic theorems with “reality.”

Our method is an application of the Burnside process, alternating two steps: from a labeled rooted tree, produce a uniform permutation fixing it; and from a permutation, produce a uniform labeled rooted tree fixed by it. This last step is linked to a product formula, refining Cayley’s, for the number of rooted labeled trees preserved by a given permutation.

We consider the Brown measure of $a+\mathfrak {c}$, where a lies in a commutative tracial von Neumann algebra $\mathcal {B}$ and $\mathfrak {c}$ is a $\mathcal {B}$-valued circular element. Under certain regularity conditions on a and the covariance of $\mathfrak {c}$ this Brown measure has a density with respect to the Lebesgue measure on the complex plane which is real analytic apart from jump discontinuities at the boundary of its support. With the exception of finitely many singularities this one-dimensional spectral edge is real analytic. We provide a full description of all possible edge singularities as well as all points in the interior, where the density vanishes. The edge singularities are classified in terms of their local edge shape while internal zeros of the density are classified in terms of the shape of the density locally around these points. We also show that each of these countably infinitely many singularity types occurs for an appropriate choice of a when $\mathfrak {c}$ is a standard circular element. The Brown measure of $a+\mathfrak {c}$ arises as the empirical spectral distribution of a diagonally deformed non-Hermitian random matrix with independent entries when its dimension tends to infinity.

We consider a generalized model of random walk in dynamical random environment, and we show that the multiplicative-noise stochastic heat equation (SHE) describes the fluctuations of the quenched density at a certain precise spatial location in the tail called the critical scale. The distribution of transition kernels is fixed rather than changing under the diffusive rescaling of space-time, that is, there is no tuning of the model parameters needed to observe the stochastic PDE limit. The proof is done by pushing the methods developed in [DDP24a, DDP24b] to their maximum, substantially weakening the assumptions and obtaining fairly sharp conditions under which one expects to see the SHE arise in a wide variety of random walk models in random media. In particular we are able to get rid of conditions such as nearest-neighbor interaction as well as spatial independence of quenched transition kernels. Moreover, we observe an entire hierarchy of moderate deviation exponents at which the SHE can be found, confirming a physics prediction of [Has25] and mirroring a result from [HQ18] in the context of this model.

We provide a natural definition of an elliptic arrangement, extending the classical framework to an elliptic curve $\mathcal {E}$ with complex multiplication. We analyze the intersections of elements of the arrangement and their connected components as $\operatorname {End}(\mathcal {E})$-modules. Furthermore, we prove that the combinatorial data of elliptic arrangements define both an arithmetic matroid and a matroid over the ring $\operatorname {End}(\mathcal {E})$. In this way, we obtain a class of arithmetic matroids that is different from the class of arithmetic matroids realizable via toric arrangements. Finally, we show that the Euler characteristic of the complement is an evaluation of the arithmetic Tutte polynomial.

With a mere usage of well-established properties of paradifferential operators, the conjugacy equations in several model KAM problems are converted to parahomological equations solvable by standard fixed point argument. Such discovery greatly simplifies KAM proofs, renders the traditional KAM iteration steps unnecessary, and may suggest a systematic scheme of finding quasi-periodic solutions of realistic magnitude.

Liouville field theory has long been a cornerstone of two-dimensional quantum field theory and quantum gravity, which has attracted much recent attention in the mathematics literature. Timelike Liouville field theory is a version of Liouville field theory where the kinetic term in the action appears with a negative sign, which makes it closer to a theory of quantum gravity than ordinary (spacelike) Liouville field theory. Making sense of this “wrong sign” requires a theory of Gaussian random variables with negative variance. Such a theory is developed in this paper, and is used to prove the timelike DOZZ formula for the $3$-point correlation function when the parameters satisfy the so-called “charge neutrality condition.” Expressions are derived also for the k-point correlation functions for all $k\ge 3$, and it is shown that these functions approach the correct semiclassical limits as the coupling constant is sent to zero.

Saturated fusion systems are categories modeling properties of conjugacy of p-elements in finite groups. It was shown by Chermak that there are group-like structures called regular localities associated to saturated fusion systems. Both the theory of fusion systems and the theory of regular localities are developed in analogy to the theory of finite groups. In this paper we focus on a classical theorem of Wielandt, which states that any two subnormal subgroups of a finite group G generate a subnormal subgroup of G. We prove versions of this theorem for regular localities and for fusion systems. Along the way we prove also a purely group-theoretical result which may be of independent interest.

Given a polynomial $\sum _\nu a_\nu X^\nu $ of degree $<d$, bounded by one on the unit disk, how large can $ \left \lvert {a_0+a_1+\ldots +a_n} \right \rvert $ ($n<d$) get? This question dates back at least to the 1952 thesis work of H. S. Shapiro. In 1978, D. J. Newman gave an exact answer for $d=2(n+1)$, but there does not seem to have been further progress on the question since. We study variations on exact answers for some related coefficient sums, and answer the original question in an asymptotic sense, provided that n is ‘not too large’ in terms of d. The latter is achieved via a ‘quantitative’ Eneström–Kakeya theorem, while the former is based on certain identities for carefully selected Lagrange interpolators. From the interpolation approach we also obtain a general inequality for coefficient sums $ \left \lvert { t_0 a_0 + \ldots + t_{d-1} a_{d-1} } \right \rvert $ for arbitrary complex numbers $t_0,\ldots ,t_{d-1}$. This inequality fails to be sharp in general, yet it is in some cases and also yields non-trivial bounds for Shapiro’s problem for some choices of n and d.