There exists a homomorphism from the affine super Yangian to the completion of the universal enveloping algebra of $\widehat{\mathfrak{gl}}(m|n)$, called the evaluation map. In this paper, we show that the image of this homomorphism is dense. Via this homomorphism, we obtain irreducible representations of the affine super Yangian.

Using gauge theory, we classify $\text{SU}(2)$-equivariant holomorphic embeddings from $\mathbf CP^1$ with the Fubini-Study metric into Grassmann manifold $\mathit{Gr}_{N-2}(\mathbf C^N)$. It is shown that the moduli spaces of those embeddings are identified with the gauge equivalence classes of non-flat invariant connections satisfying semi-positivity on the vector bundles given by \textit{extensions} of line bundles. A topology on the moduli is obtained by means of $L^2$-inner product on Dolbeault cohomology group to which the extension class belongs. The compactification of the moduli is provided with geometric meaning from viewpoint of maps.

Let $M_\phi$ be a surface bundle over a circle with monodromy $\phi:S \rightarrow S$. We study deformations of certain reducible representations of $\pi_1(M_\phi)$ into $\operatorname{SL}(n,\mathbb{C})$, obtained by composing a reducible representation into $\operatorname{SL}(2,\mathbb{C})$ with the irreducible representation $\operatorname{SL}(2,\mathbb{C}) \rightarrow \operatorname{SL}(n,\mathbb{C})$. In particular, we show that under certain conditions on the eigenvalues of $\phi^*$, the reducible representation is contained in a $(n+1+k)(n-1)$ dimensional component of the representation variety, where $k$ is the number of components of $\partial M_\phi$. This result applies to mapping tori of pseudo-Anosov maps with orientable invariant foliations whenever 1 is not an eigenvalue of the induced map on homology, where the reducible representation is also a limit of irreducible representations.

We classify the $3$-manifolds obtained as the preimages of arcs on the plane for simplified genus-$2$ trisection maps, which we call vertical $3$-manifolds. Vertical $3$-manifolds of $(2, 1)$- and $(2, 2)$-trisection maps can be easily classified. A vertical $3$-manifold of a $(2, 0)$-trisection map is a connected sum of finite copies of $6$-tuple of vertical $3$-manifolds over specific $6$ arcs and $S^1 \times S^2$. Consequently, we show that each of the $6$-tuples determines the source $4$-manifold uniquely up to orientation reversing diffeomorphisms except for the trivial case. We also show that, in contrast to the fact that summands of vertical $3$-manifolds of simplified $(2, 0)$-trisection maps are lens spaces, there exist infinitely many simplified $(2, 0)$-$4$-section maps that admit hyperbolic vertical $3$-manifolds.

We study locally standard $T^k$-manifolds $M$. In particular, we study the case where there is a continuous section to the orbit map $\pi:M\rightarrow M/T$. We give a classification of $T^k$-manifolds satisfying these conditions up to equivariant diffeomorphism.

We introduce the notion of para-real forms of para-Hermitian symmetric spaces and classify para-real forms of absolutely simple para-Hermitian symmetric spaces of hyperbolic orbit type.

Let $G$ be a finite group of Lie type. In order to determine the character table of $G$, Lusztig developed the theory of character sheaves. In this framework, one has to find the transformation between two bases for the space of class functions on $G$, one of them being the irreducible characters of $G$, the other one consisting of characteristic functions associated to character sheaves. In principle, this has been achieved by Lusztig and Shoji, but the underlying process involves some scalars which are still unknown in many cases. The problem of specifying these scalars can be reduced to considering cuspidal character sheaves. We will deal with the latter for the specific case where $G=E_7(q)$, and $q$ is a power of the bad prime $p=2$ for $E_7$.

The paper develops the theory of quandle rings introduced by the authors in a recent work. Orderability of quandles is investigated and many interesting examples of such quandles are given. It is proved that quandle rings of left or right orderable quandles which are semi-latin have no zero-divisors. Idempotents in quandle rings of certain interesting quandles are computed and used to determine sets of maximal quandles in these rings. Understanding of idempotents is further applied to determine automorphism groups of these quandle rings. Also, commutator width of quandle rings is introduced and computed in a few cases.

In this article, we give a description of the closed cone of curves of the projective bundle $\mathbb{P}(E)$ over a smooth projective variety $X$. Using duality, we then calculate the nef cone of divisors in $\mathbb{P}(E)$ over some special surfaces $X$ and for some special bundles on $X$. As an application, we also calculate the Seshadri constants of semistable ample vector bundles with vanishing discriminant on some special ruled surfaces at special points.

We provide an account for the existence and uniqueness of solutions to rough differential equations in infinite dimensions under the framework of controlled rough paths. The case when the driving path is $\alpha$-Hölder continuous for $\alpha>1/3$ is widely available in the literature. In its extension to the case when $\alpha\leqslant1/3,$ the main challenge and missing ingredient is to show that controlled rough paths are closed under composition with Lipschitz transformations. Establishing such a property precisely, which has a strong algebraic nature, is a main purpose of the present article.

We establish a quandle version of the twisted Alexander polynomial. We also develop a theory that reduces the size of a twisted Alexander matrix with column relations. The reduced matrix can be used to refine invariants derived from the twisted Alexander matrix.

The theory of polarized supersingular abelian varieties $(A,\lambda)$ is often essentially related to the theory of quaternion hermitian lattices. In this paper, we add one more such relation by giving an adelic description of supersingular loci of low dimensions. Katsura, Li and Oort have shown that the supersingular locus in the moduli of principally polarized abelian varieties is not irreducible and that the number of its irreducible components is equal to the class number of certain maximal quaternion hermitian lattices. Now the locus has certain natural algebraic subsets consisting of $A$ with fixed \textit{$a$-numbers} that are defined as the dimensions of embeddings from $\alpha_p$ to $A$. For low dimensional cases when $\dim A\leq 3$, we describe configuration of these subsets in the locus by intersection properties of some adelic subgroups of quaternion hermitian groups corresponding to parahoric subgroups locally at characteristic $p$. In particular, we describe which components each superspecial point lies on. This is proved by using certain liftability property of isogenies of abelian varieties, where the isogenies are interpreted to cosets of $GL_n$ and of parahoric subgroups of the quaternion hermitian groups acting on quaternion hermitian matrices.