Let $P$ and $I$ be a projective and an injective representations of a Dynkin quiver. We consider quiver Grassmannians of subrepresentations of dimension $\dim P$ inside representations of dimension $\dim P + \dim I$. Based on extensive computer experiments, we formulate several conjectures about the algebro-geometric properties of these quiver Grassmannians.

In this paper, we construct recollements and ladders for Brieskorn-Pham singularities via reduction/insertion functors, and study the singularity categories of the Brieskorn-Pham singularities using these ladders. In particular, we construct a class of tilting objects, called the extended tilting $n$-cuboids, whose endomorphism algebras are $n$-fold tensor products of certain Nakayama algebras. Moreover, we show that such an endomorphism algebra is derived equivalent to a certain replicated algebra. This generalizes the Happel-Seidel symmetry to the context of Brieskorn-Pham singularities.

We describe germs of mappings $(\mathbb{C}^2,0) \to (\mathbb{C}^2,0)$ ramified along a germ of irreducible curve whose image is of the form $x^p=y^q$.

In this work we establish a connection between copositivity, that is, nonnegativity on the positive orthant, of sparse real Laurent polynomials and discriminants. Specifically, we consider Laurent polynomials in the positive orthant with fixed support and fixed coefficient signs. We provide a criterion to decide whether a given polynomial is copositive that is based in determining the intersection points of the signed discriminant and a path going through the coefficients of the polynomial. If the signed support satisfies a combinatorial condition termed nonseparability, we show additionally that this intersection consists of one point, and that tracking one path in homotopy continuation methods suffices to decide upon copositivity. Building on these results, we show that any copositive polynomial with nonseparable signed support can be decomposed into a sum of nonnegative circuit polynomials, generalising thereby previously known supports having this property.

We adapt Caro's notion of overholonomicity to give a definition of holonomic D-cap-modules on rigid analytic spaces. We prove stability under five of the six operations (both inverse image functors, duality, and both direct image functors for projective morphisms), as well as base change results. Up to the open problem of stability under tensor products, we obtain an analogue of the usual six-functor formalism for holonomic D-modules.

We give a bimeromorphic classification of compact Kähler manifolds of Kodaira codimension one that admit a holomorphic one form without zeros.

We define several versions of a class of varieties $X_{\mathfrak{g}}$ attached to a complex reductive Lie algebra $\mathfrak{g}$, generalizing the Hilbert scheme of points on the plane. These include trigonometric and elliptic versions attached to the corresponding groups. We also define the corresponding isospectral varieties $Y_{\mathfrak{g}}$. We prove a Gordon-Stafford localization theorem for $X_{\mathfrak{g}}$ and the corresponding equal-parameter rational Cherednik algebras, relate these varieties to the affine Springer fiber-sheaf correspondence of arXiv:2204.00303, and discuss examples. We conjecture that the torus-fixed points of our varieties are in bijection with two-sided cells in the finite Weyl group and prove this in types $ABC$. We relate these results to known results about Calogero-Moser spaces.

We introduce a general theory of homological Milnor-Witt cycle modules over an excellent base scheme equipped with a dimension function, extending both Rost's cycle modules and Feld's theory over fields. To any such module we associate a Rost-Schmid type complex whose homology defines a Borel-Moore intersection theory with quadratic coefficients, satisfying homotopy invariance, localization, proper pushforwards, smooth pullbacks, and Gysin morphisms for essentially smoothable lci morphisms. Using duality data induced by pinning structures, we define cohomological Milnor-Witt modules and establish a duality equivalence between homological and cohomological theories. As applications, we extend Chow-Witt groups to schemes over general (possibly singular or arithmetic) bases, prove generalized Bloch formulas and representability results, and compute graded Chow-Witt groups over Dedekind schemes of finite type over the integers. In particular, we obtain finiteness results for Chow-Witt and related Milnor-Witt invariants in dimension at most one.

These are lecture notes from a mini-course taught at Winterbraids XIII (Montpellier, 2024). The main character of these notes are curves in the complex projective plane, viewed from a topological perspective.

In this extended abstract, we study special tropical prevarieties which we call Coxeter Dressians. They arise from equations capturing a generalization of valuated symmetric basis exchange for Coxeter matroids. In particular, we study subdivisions of the associated Coxeter matroid polytopes. We show that the subdivisions induced by points of the Coxeter Dressian consist of cells which are strong Coxeter matroidal. This generalizes well-known results in type $A$ to other Lie types. Finally, we implement explicit computations of Coxeter Dressians in OSCAR.

For $C/\mathbb{Q}_p$ a complete algebraically closed field, we construct a collection of non-isomorphic rank two $\mathbb{Q}_p$-local systems on $\mathbb{P}^1_C$ indexed by $C$. This implies that the de Jong fundamental group $\pi_{1,\mathrm{dJ}}(\mathbb{P}^1_C)$ depends on $C$ and, if $C$ has cardinality $>2^{\mathbb{N}}$, that $\pi_{1,\mathrm{dJ}}(\mathbb{P}^1_C)$ is not topologically countably generated. The argument in fact applies to any connected rigid analytic variety over $C$ with a non-constant function to $\mathbb{P}^1_C$.

We show the de Jong fundamental group of any non-trivial abelian variety over a complete algebraically closed extension of $\mathbb{Q}_p$ is non-abelian.

In this monograph, we lay the foundations for a new theory that generalizes real algebraic geometry. Let $R|K$ be a field extension, where $R$ is a real closed field and $K$ is an ordered subfield of $R$. The main objective is to study $K$-algebraic subsets of $R^n$, i.e., those subsets of $R^n$ that are the zero loci of polynomials with coefficients in $K$. Real algebraic geometry already covers the case when $K$ is also a real closed field. Our goal is to extend real algebraic geometry to the case when $K$ is not real closed, for example when $K$ is the field $\mathbb{Q}$ of rational numbers. Several new geometric phenomena appear. There is no complex counterpart to this generalized real algebraic geometry. The reason is as follows. If $C|K$ is a field extension with $C$ algebraically closed and $X$ is a $K$-algebraic subset of $C^n$, then Hilbert's Nullstellensatz implies that the ideal of polynomials with coefficients in $C$ that vanish on~$X$ is generated by the ideal of polynomials with coefficients in $K$ that vanish on $X$. In the real realm, this is false in general, for example when we consider field extensions $R|K$ with $R$ real closed and $K=\mathbb{Q}$. This monograph also presents some applications of the theory developed. Here is an example. The celebrated Nash-Tognoli theorem states that every compact smooth manifold $M$ is diffeomorphic to a nonsingular real algebraic set $M'$, called algebraic model of $M$. The theory developed here provides the theoretical basis to prove that the algebraic model $M'$ of $M$ can be chosen to be $\mathbb{Q}$-algebraic and $\mathbb{Q}$-nonsingular. This guarantees for the first time that, up to smooth diffeomorphisms, every compact smooth manifold can be encoded both globally and locally involving only finitely many exact data.

We construct canonical semi-orthogonal decompositions for derived categories of smooth projective surfaces. These decompositions are compatible with the operations in the minimal model program, such as blow-ups and conic bundles. Therefore our construction confirms a conjecture of Kontsevich in dimension two. We work in the G-equivariant setting and over an arbitrary perfect field, and canonical decompositions are consistent with group change and algebraic field extensions. Our method is based on the G-minimal model program for surfaces and on the Sarkisov link factorisation of birational maps between Mori fibre spaces. We characterise rationality of surfaces, and in certain cases, birationality between surfaces in terms of the pieces of these decompositions, which we call atoms.

Let $f$ be a polarized endomorphism of an abelian variety $A$. Kollár and Zhuang asked whether the essential dimension $\mathrm{ed}(f)$ equals $\mathrm{dim}(A)$. We provide counterexamples to this question. Instead, we prove that, under the hypothesis that every subtorus of $A$ is $f$-preperiodic up to translation (a condition arising from the dynamical Manin--Mumford conjecture), we have $\mathrm{ed}(f^s)=\mathrm{dim}(A)$ for some integer $s>0$. Our examples also show the necessity of both the hypothesis and iteration. We also give an affirmative answer to Kollár and Zhuang's original question when $A$ is a simple abelian surface and $f$ is not $2$-polarized.

In this paper we study the arithmetic invariants of Euclidean lattice in the context of Arakelov geometry. We regard a Euclidean lattice as a hermitian vector bundle $\bar E$ on ${\rm Spec}(\mathbb{Z})$ and consider two typical arithmetic analogues of the dimension of the space of global sections of a vector bundle on an algebraic curve. One is $$h^0_{\rm Ar}(\bar E):=\log \vert E\cap B_1 \vert$$ where $B_1$ is the unit ball, and the other is $$h^0_{\theta}(\bar{E}):=\log\sum_{v\in E}e^{-\pi\Vert v\Vert^2}$$ where $\sum_{v\in E}e^{-\pi\Vert v\Vert^2}$ is the theta function of $\bar E$. In this paper, we shall prove the following three statements: (i) the fact that one can not reach an absolute Riemann-Roch theorem for $h^0_{\rm Ar}(\bar E)$ is an instance of the Heissenberg uncertainty principle; (ii) the finiteness of equivalence classes in the genus of a positive quadratic form defined over $\mathbb{Z}$ is equivalent to the finiteness of certain isometry classes of hermitian vector bundles on ${\rm Spec}(\mathbb{Z})$, and it can be deduced from a finiteness theorem in Arakelov theory of ${\rm Spec}(\mathbb{Z})$; (iii) for any smooth function $f$ on $\mathbb{R}_{+}$ such that $f>0$ and that $f\circ {\rm exp}$ is a Schwartz function on $\mathbb{R}$, the Mellin transform of $f$ can be written as an integral over the Arakelov divisor class group of ${\rm Spec}(\mathbb{Z})$.

We define genus zero open Gromov-Witten invariants with boundary and interior constraints for a Lagrangian submanifold of arbitrary even dimension. The definition relies on constructing a canonical family of bounding cochains that satisfy the point-like condition of the second author and Tukachinsky. Since the Lagrangian is even dimensional, the parameter of the family is odd. Thus, to avoid the vanishing of invariants with more than one boundary constraint, the parameter must be non-commutative. The invariants are defined either when the Lagrangian is a rational cohomology sphere or when the Lagrangian is fixed by an anti-symplectic involution, has dimension $2$ modulo $4$, and its cohomology is that of a sphere aside from degree $1$ modulo $4$. In dimension $2$, these invariants recover Welschinger's invariants. We develop an obstruction theory for the existence and uniqueness of bounding cochains in a Fukaya $A_\infty$ algebra with non-commutative coefficients. The obstruction classes belong to twisted cohomology groups of the Lagrangian instead of the de Rham cohomology of the commutative setting. A spectral sequence is constructed to compute the twisted cohomology groups. The extension of scalars of an $A_\infty$ algebra by a non-commutative ring is treated in detail. A theory of pseudo-completeness is introduced to guarantee the convergence of the Maurer-Cartan equation, which defines bounding cochains, even though the non-commutative parameter is given zero filtration.

We use derived methods to study the Gauss-Manin connection in Hochschild homology, infinitesimal cohomology, and derived de Rham cohomology. As applications, we give new approaches to nilinvariance, the Quillen spectral sequence, and the HKR filtration. We extend the results of Bhatt's work on de Rham cohomology in characteristic zero to infinitesimal cohomology in mixed characteristic and show that the comparison to Hartshorne's algebraic de Rham complex "is" the Gauss-Manin connection. Finally, we explain the main features of prismatic cohomology in characteristic zero via the Gauss-Manin connection.

We recast quantum entanglement as a cohomological obstruction to reconstructing a global quantum state from locally compatible information. We address this by considering presheaf cohomologies of states and entanglement witnesses. Sheafification erases the global-from-local signature while leaving within-patch multipartite structure, captured by local entanglement groups introduced here. For smooth parameter families, the obstruction admits a differential-geometric representative obtained by pairing an appropriate witness field with the curvature of a natural unitary connection on the associated bundle of amplitudes. We also introduce a Quantum Entanglement Index (QEI) as an index-theoretic invariant of entangled states and explain its behavior. Finally, we outline a theoretical physics approach to probe these ideas in quantum many-body systems and suggest a possible entanglement-induced correction as an experimental target.