We explain how to construct a quantum deformation of a spectral curve associated to a tau-function of the KP hierarchy. This construction is applied to Witten–Kontsevich tau-function to give a natural explanation of some earlier work. We also apply it to higher Weil–Petersson volumes and Witten’s r-spin intersection numbers.

We introduce the notion of ungraded matrix factorization as a mirror of non-orientable Lagrangian submanifolds. An ungraded matrix factorization of a polynomial $W$, with coefficients in a field of characteristic 2, is a square matrix $Q$ of polynomial entries satisfying $Q^2 = W \cdot \mathrm{Id}$. We then show that non-orientable Lagrangians correspond to ungraded matrix factorizations under the localized mirror functor and illustrate this construction in a few instances. Our main example is the Lagrangian submanifold $\mathbb{R}P^2 \subset \mathbb{C}P^2$ and its mirror ungraded matrix factorization, which we construct and study. In particular, we prove a version of Homological Mirror Symmetry in this setting.

We construct the quantum curve for the Baker-Akhiezer function of the orbifold Gromov-Witten theory of the weighted projective line $\mathbb P[r]$. Furthermore, we deduce the explicit bilinear Fermionic formula for the (stationary) Gromov-Witten potential via the lifting operator contructed from the Baker-Akhiezer function.

This review is devoted to one of the most interesting and actively developing fields in condensed matter physics—theory of topological insulators. Apart from its importance for theoretical physics, this theory enjoys numerous connections with modern mathematics, in particular, with topology and homotopy theory, Clifford algebras, K-theory and non-commutative geometry. From the physical point of view topological invariance is equivalent to adiabatic stability. Topological insulators are characterized by the broad energy gap, stable under small deformations, which motivates application of topological methods. A key role in the study of topological objects in the solid state physics is played by their symmetry groups. There are three main types of symmetries—time reversion symmetry, preservation of the number of particles (charge symmetry) and PH-symmetry (particle-hole symmetry). Based on the study of symmetry groups and representation theory of Clifford algebras Kitaev proposed a classification of topological objects in solid state physics. In this review we pay special attention to the topological insulators invariant under time reversion.

Our previous papers introduce topological notions of normal crossings symplectic divisor and variety, show that they are equivalent, in a suitable sense, to the corresponding geometric notions, and establish a topological smoothability criterion for normal crossings symplectic varieties. The present paper constructs a blowup, a complex line bundle, and a logarithmic tangent bundle naturally associated with a normal crossings symplectic divisor and determines the Chern class of the last bundle. These structures have applications in constructions and analysis of various moduli spaces. As a corollary of the Chern class formula for the logarithmic tangent bundle, we refine Aluffi’s formula for the Chern class of the tangent bundle of the blowup at a complete intersection to account for the torsion and extend it to the blowup at the deepest stratum of an arbitrary normal crossings divisor.

In the present paper, we study partial collapsing degeneration of Hamiltonian-perturbed Floer trajectories for an adiabatic ε-family and its reversal adiabatic gluing, as the prototype of the partial collapsing degeneration of 2-dimensional (perturbed) J-holomorphic maps to 1-dimensional gradient segments. We consider the case when the Floer equations are S¹-invariant on parts of their domains whose adiabatic limit has positive length as ε → 0, which we call thimble-flow-thimble configurations. The main gluing theorem we prove also applies to the case with Lagrangian boundaries such as in the problem of recovering holomorphic disks out of pearly configuration. In particular, our gluing theorem gives rise to a new direct proof of the chain isomorphism property between the Morse–Bott version of Lagrangian intersection Floer complex of L by Fukaya–Oh–Ohta–Ono and the pearly complex of L Lalonde and Biran–Cornea. It also provides another proof of the present authors’ earlier proof of the isomorphism property of the PSS map without involving the target rescaling and the scale-dependent gluing.

We provide an analytical construction of the gluing map for stable affine vortices over the upper half plane with the Lagrangian boundary condition. This result is a necessary ingredient in studies of the relation between gauged sigma model and nonlinear sigma model, such as the closed or open quantum Kirwan map.

We propose a conjecture relevant to Galkin’s lower bound conjecture, and verify it for the blow-ups of a four-dimensional quadric at a point or along a projective plane. We also show that Conjecture $\mathcal{O}$ holds in these two cases.

This paper, largely written in 2009/2010, fits Landau–Ginzburg models into the mirror symmetry program pursued by the last author jointly with Mark Gross since 2001. This point of view transparently brings in tropical disks of Maslov index 2 via the notion of broken lines, previously introduced in two dimensions by Mark Gross in his study of mirror symmetry for $\mathbb{P}^2$. A major insight is the equivalence of properness of the Landau–Ginzburg potential with smoothness of the anticanonical divisor on the mirror side. We obtain proper superpotentials which agree on an open part with those classically known for toric varieties. Examples include mirror LG models for non-singular and singular del Pezzo surfaces, Hirzebruch surfaces and some Fano threefolds.

Dubrovin establishes a certain relationship between the GUE partition function and the partition function of Gromov–Witten invariants of the complex projective line. In this paper, we give a direct proof of Dubrovin’s result. We also present in a diagram the recent progress on topological gravity and matrix gravity.

We discuss the mechanism of formation of singularities of solutions to the Novikov-Veselov, modified Novikov-Veselov, and Davey-Stewartson II (DSII) equations obtained by the Moutard type transformations. These equations admit the $L,A,B$-triple presentation, the generalization of the $L,A$-pairs for 2+1-soliton equations. We relate the blow-up of solutions to the non-conservation of the zero level of discrete spectrum of the $L$-operator. We also present a class of exact solutions, of the DSII system, which depend on two functional parameters, and show that all possible singularities of solutions to DSII equation obtained by the Moutard transformation are indeterminancies, i.e., points when approaching which in different spatial directions the solution has different limits.

We introduce a method to find differential equations for functions which define tables, such that associated billiard systems admit a local first integral. We illustrate this method in three situations: the case of (locally) integrable wire billiards, for finding surfaces in ${\mathbb R}^3$ with a first integral of degree one in velocities, and for finding a piece-wise smooth surface in ${\mathbb R}^3$ homeomorphic to a torus, being a table of a billiard admitting two additional first integrals.

Following Shokurov’s idea, we give a simple proof of the ACC conjecture for minimal log discrepancies for surfaces.

In this paper, we consider a critical Galton-Watson branching process with immigration stopped at zero W. Some precise estimation on the probability generating function of the n-th population are obtained, and the tail probability of the life period of W is studied. Based on above results, two conditional limit theorems for W are established.

In this paper, we investigate Sobolev mapping properties of the Bergman projection. The domain we focus on is defined by $$\Omega_{\frac{\mathbf{n}}{\mathbf{m}}}:= \{(z, w)\in \mathbb{C}^{1+N}: |w_1| < |z|^{\frac{n_1}{m_1}} < 1, \dots, |w_N| < |z|^{\frac{n_N}{m_N}} < 1\},$$ where $\mathbf{m}=(m_1,\dots,m_N)\in(\mathbb{Z}^+)^{N}, \mathbf{n}=(n_1,\dots,n_N)\in(\mathbb{Z}^+)^{N}, N\in\mathbb{Z}^{+}$. Sobolev irregularity of the Bergman projections on $\Omega_{\frac{\mathbf{n}}{\mathbf{m}}}$ is shown. We also prove some Sobolev regularity results of the Bergman projections on $\Omega_{\frac{\mathbf{n}}{\mathbf{m}}}$ for $\mathbf{m}=(1,\dots,1)$.

In this work, we give some criteria of the weakly compact sets and a representation theorem of Riesz’s type in Musielak sequence spaces using the ideas and techniques of sequence spaces and Musielak function. Finally, as an immediate consequence of the criteria considered in this paper, the criteria of the weakly compact sets of Orlicz sequence spaces are deduced.

The purpose of the paper is to describe the solution to the quantum Yang-Baxter equation associated with the $8$-dimensional spin representation $T_{\rm sp}$ of $U_q(\mathfrak{so}_7)$. The self-duality of $T_{\rm sp}$ and the symmetry of the corresponding braiding matrix are proved. Also, the minimal polynomial of the braiding $R$-matrix of $T_{\rm sp}$ is presented explicitly in an ingenious method by taking advantage of nice features of the spin representation $T_{\rm sp}$.

3D Burgers equation is an important model for turbulence. It is natural to expect the long-time behaviour for this hydrodynamics equation. However, there is no result about the long-time behaviour for this deterministic model. Surprisingly, if the system is perturbed by stochastic noise, we establish the existence and uniqueness of invariant measure for 3D stochastic Burgers equation.

We investigate the uniqueness problems of meromorphic functions and their difference operators by using a new method. It is proved that if a non-constant meromorphic function $f$ shares a non-zero constant and $\infty$ counting multiplicities with its difference operators $\Delta_c f(z)$ and $\Delta_c^2 f(z)$, then $\Delta_c f(z)\equiv \Delta_c^2 f(z)$. In particular, we give a difference analogue of a result of Jank-Mues-Volkmann. Our method has two distinct features: (i) It converts the relations between functions into the corresponding vectors. This makes it possible to deal with the uniqueness problem by linear algebra and combinatorics. (ii) It circumvents the obstacle of the difference logarithmic derivative lemma for meromorphic functions of infinite order, since this method does not depend on the growth of the functions. Furthermore, the idea in this paper can also be applied to the case for several variables.

Let $\mathcal{H}$ be a complex infinite dimensional Hilbert space and $\mathcal{B(H)}$ be the algebra of all bounded linear operators on $\mathcal{H}$. In this paper, we mainly study the operators that satisfy both a-Weyl's theorem and property $(R)$. Also, the operators whose functional calculus satisfies the two properties are also explored. We give the features for the operator or its functional calculus for which both a-Weyl's theorem and property $(R)$ hold.

Establishing the accurate relationship between fractional calculus and fractals is an important research content of fractional calculus theory. In the present paper, we investigate the relationship between fractional calculus and fractal functions, based only on fractal dimension considerations. Fractal dimension of the Riemann-Liouville fractional integral of continuous functions seems no more than fractal dimension of functions themselves. Meanwhile fractal dimension of the Riemann-Liouville fractional differential of continuous functions seems no less than fractal dimension of functions themselves when they exist. After further discussion, fractal dimension of the Riemann-Liouville fractional integral is at least linearly decreasing and fractal dimension of the Riemann-Liouville fractional differential is at most linearly increasing for the Hölder continuous functions. Investigation about other fractional calculus, such as the Weyl-Marchaud fractional derivative and the Weyl fractional integral has also been given elementary. This work is helpful to reveal the mechanism of fractional calculus on continuous functions. At the same time, it provides some theoretical basis for the rationality of the definition of fractional calculus. This is also helpful to reveal and explain the internal relationship between fractional calculus and fractals from the perspective of geometry.

A compact quantum metric space is a complete order unit space $A$ endowed with a Lip-norm $L$. We give some characterizations of almost periodic type group actions on a compact quantum metric space $(A,L)$ by means of several kinds of subsets of $A$, its induced equicontinuous actions on several important subsets of the dual Banach space $A^*$, and the Lip-norm $L$ with its induced metric space structures on the state space $\mathcal{S}(A)$ of $A$.

This paper first shows that for any $p\in(1,2)$ there exists a continuously differentiable function $f$ on $l^p$ (and $L^p$) such that the proximal subdifferential of $f$ is empty everywhere, and hence it is not suitable to develop theory on proximal subdifferential in the classical Banach spaces $l^p$ and $L^P$ with $p\in(1,2)$. On the other hand, this paper establishes variational analysis based on the proximal subdifferential in the framework of smooth Banach spaces of power type 2, which conclude all Hilbert spaces and all the classical spaces $l^p$ and $L^p$ with $p\in(2,+\infty)$. In particular, in such a smooth space, we provide the proximal subdifferential rules for sum functions, product functions, composite functions and supremum functions, which extend the basic results on the proximal subdifferential established in the framework of Hilbert spaces. Some of our main results are new even in the Hilbert space case. As applications, we provide KKT-like conditions for nonsmooth optimization problems in terms of proximal subdifferential.

Gel'fand-Dorfman bialgebra, which is both a Lie algebra and a Novikov algebra with some compatibility condition, appeared in the study of Hamiltonian pairs in completely integrable systems. They also emerged in the description of a class special Lie conformal algebras called quadratic Lie conformal algebras. In this paper, we investigate the extending structures problem for Gel'fand-Dorfman bialgebras, which is equivalent to some extending structures problem of quadratic Lie conformal algebras. Explicitly, given a Gel'fand-Dorfman bialgebra $(A, \circ, [\cdot,\cdot])$, this problem is how to describe and classify all Gel'fand-Dorfman bialgebra structures on a vector space $E$ $(A\subset E$) such that $(A, \circ, [\cdot,\cdot])$ is a subalgebra of $E$ up to an isomorphism whose restriction on $A$ is the identity map. Motivated by the theories of extending structures for Lie algebras and Novikov algebras, we construct an object $\mathcal{GH}^2(V,A)$ to answer the extending structures problem by introducing the notion of a unified product for Gel'fand-Dorfman bialgebras, where $V$ is a complement of $A$ in $E$. In particular, we investigate the special case when $\text{dim}(V)=1$ in detail.

In this article, by studying the Bernstein degrees and Goldie rank polynomials, we establish a comparison between the irreducible representations of $G=\text{GL}_n(\mathbb{C})$ possessing the minimal Gelfand-Kirillov dimension and those induced from finite-dimensional representations of the maximal parabolic subgroup of $G$ of type $(n-1,1)$. We give the transition matrix between the two bases for the corresponding coherent families.

Let $\mathfrak{g}$ be a classical complex simple Lie algebra and $\mathfrak{q}$ be a parabolic subalgebra. Let $M$ be a generalized Verma module induced from a one dimensional representation of $\mathfrak{q}$. Such $M$ is called a scalar generalized Verma module. In this paper, we will determine the reducibility of scalar generalized Verma modules associated to maximal parabolic subalgebras by computing explicitly the Gelfand-Kirillov dimension of the corresponding highest weight modules.

Let $G$ be a special linear group over the real, the complex or the quaternion, or a special unitary group. In this note, we determine all special unipotent representations of $G$ in the sense of Arthur and Barbasch-Vogan, and show in particular that all of them are unitarizable.

In this paper we consider the theta correspondence over a non-Archimedean local field. Using the homological method and the theory of derivatives, we show that under a mild condition the big theta lift is irreducible.

We prove a converse theorem for split even special orthogonal groups over finite fields. This is the only case left on converse theorems of classical groups and the difficulty is the existence of the outer automorphism. In this paper, we develop new ideas and overcome this difficulty.

We study the transfer between small special unipotent representations for all equal rank real forms of type $E_6$ and $E_7$. As a consequence, one can verify these modules are unitarity using the results of Wallach and Zhu. Moreover, the $K$-spectra of these modules can be obtained explicitly.

We establish an explicit embedding of a quantum affine $\mathfrak{sl}_n$ into a quantum affine $\mathfrak{sl}_{n+1}$. This embedding serves as a common generalization of two natural, but seemingly unrelated, embeddings, one on the quantum affine Schur algebra level and the other on the non-quantum level. The embedding on the quantum affine Schur algebras is used extensively in the analysis of canonical bases of quantum affine $\mathfrak{sl}_n$ and $\mathfrak{gl}_n$. The embedding on the non-quantum level is used crucially in a work of Riche and Williamson on the study of modular representation theory of general linear groups over a finite field. The same embedding is also used in a work of Maksimau on the categorical representations of affine general linear algebras. We further provide a more natural compatibility statement of the embedding on the idempotent version with that on the quantum affine Schur algebra level. A ${\hat{\mathfrak{gl}}}_n$-variant of the embedding is also established.

We extend the $\imath$Hall algebra realization of $\imath$quantum groups arising from quantum symmetric pairs, which establishes an injective homomorphism from the universal $\imath$quantum group of Kac-Moody type to the $\imath$Hall algebra associated to an arbitrary $\imath$quiver (not necessarily virtually acyclic). This generalizes Lu-Wang's result.

Let W be a real symplectic space and (G, G') an irreducible dual pair in Sp(W), in the sense of Howe, with G compact. Let $\widetilde {\text{G}}$ be the preimage of G in the metaplectic group $\widetilde {\text{Sp}}$(W). Given an irreducible unitary representation $\Pi$ of $\widetilde {\text{G}}$ that occurs in the restriction of the Weil representation to $\widetilde {\text{G}}$, let $\Theta_\Pi$ denote its character. We prove that, for a suitable embedding $T$ of $\widetilde {\text{Sp}}$(W) in the space of tempered distributions on W, the distribution $T(\check\Theta_\Pi)$ admits an asymptotic limit, and the limit is a nilpotent orbital integral. As an application, we compute the wave front set of $\Pi'$, the representation of $\widetilde {\text{G'}}$ dual to $\Pi$, by elementary means.

In this note we study character sheaves for graded Lie algebras arising from inner automorphisms of special linear groups and Vinberg’s type II classical graded Lie algebras.

In a previous paper, the author and his collaborator studied the problem of lifting Hamiltonian group actions on symplectic varieties and Lagrangian subvarieties to their graded deformation quantizations and apply the general results to coadjoint orbit method for semisimple Lie groups. Only even quantizations were considered there. In this paper, these results are generalized to the case of general quantizations with arbitrary periods. The key step is to introduce an enhanced version of the (truncated) period map defined by Bezrukavnikov and Kaledin for quantizations of any smooth symplectic variety $X$, with values in the space of Picard Lie algebroid over $X$. As an application, we study quantizations of nilpotent orbits of real semisimple groups satisfying certain codimension condition.

In this article, we construct free centroid hom-associative algebras and free centroid hom-Lie algebras. We also construct some other relatively free centroid hom-associative algebras by applying the Gröbner–Shirshov basis theory for (unital) centroid hom-associative algebras. Finally, we prove that the "Poincaré-Birkhoff-Witt theorem" holds for certain type of centroid hom-Lie algebras over a field of characteristic 0, namely, every centroid hom-Lie algebra such that the eigenvectors of the map $\beta$ linearly generates the whole algebra can be embedded into its universal enveloping centroid hom-associative algebra, and the linear basis of the universal enveloping algebra does not depend on the multiplication table of the centroid hom-Lie algebra under consideration.

In this paper, the dynamics (including shadowing property, expansiveness, topological stability and entropy) of several types of upper semi-continuous set-valued maps are mainly considered from differentiable dynamical systems points of view. It is shown that (1) if $f$ is a hyperbolic endomorphism then for each $\varepsilon>0$ there exists a $C^1$-neighborhood $\mathcal{U}$ of $f$ such that the induced set-valued map $F_{f,\mathcal{U}}$ has the $\varepsilon$-shadowing property, and moreover, if $f$ is an expanding endomorphism then there exists a $C^1$-neighborhood $\mathcal{U}$ of $f$ such that the induced set-valued map $F_{f,\mathcal{U}}$ has the Lipschitz shadowing property; (2) when a set-valued map $F$ is generated by finite expanding endomorphisms, it has the shadowing property, and moreover, if the collection of the generators has no coincidence point then $F$ is expansive and hence is topologically stable; (3) if $f$ is an expanding endomorphism then for each $\varepsilon>0$ there exists a $C^1$-neighborhood $\mathcal{U}$ of $f$ such that $h(F_{f,\mathcal{U}}, \varepsilon)=h(f)$; (4) when $F$ is generated by finite expanding endomorphisms with no coincidence point, the entropy formula of $F$ is given. Furthermore, the dynamics of the set-valued maps based on discontinuous maps on the interval are also considered.

Recurrent event data are commonly encountered in many scientific fields, including biomedical studies, clinical trials and epidemiological surveys, and many statistical methods have been proposed for their analysis. In this paper, we consider to use a weighted composite endpoint of recurrent and terminal events, which is weighted by the severity of each event, to assess the overall effects of covariates on the two types of events. A flexible additive-multiplicative model incorporating both multiplicative and additive effects on the rate function is proposed to analyze such weighted composite event process, and more importantly, the dependence structure among the recurrent and terminal events is left unspecified. For the estimation, we construct the unbiased estimating equations by virtue of the inverse probability weighting technique, and the resulting estimators are shown to be consistent and asymptotically normal under some mild regularity conditions. We evaluate the finite-sample performance of the proposed method via simulation studies and apply the proposed method to a set of real data arising from a bladder cancer study.

Let $R$ be a unitary ring and $a, b\in R$ with $ab=0$. We find the $2/3$ property of Drazin invertibility: if any two of $a, b$ and $a+b$ are Drazin invertible, then so is the third one. Then, we combine the $2/3$ property of Drazin invertibility to characterize the existence of generalized inverses by means of units. As applications, the need for two invertible morphisms used by You and Chen to characterize the group invertibility of a sum of morphisms is reduced to that for one invertible morphism, and the existence and expression of the inverse along a product of two regular elements are obtained, which generalizes the main result of Mary and Patrício (2016) about the group inverse of a product.