A few years ago, G. Oberdieck conjectured a multiple cover formula that determines the number of curves of fixed genus and degree passing through a configuration of points in an abelian surface. This formula was proved by the author using tropical techniques and Nishinou’s correspondence theorem. Using the same techniques, we give a much shorter proof of the multiple cover formula for point insertions, relying on the same geometrical idea, but avoiding any kind of tropical enumeration.

We prove a general likely intersections theorem, a counterpart to the Zilber-Pink conjectures, under the assumption that the Ax-Schanuel property and some mild additional conditions are known to hold for a given category of complex quotient spaces definable in some fixed o-minimal expansion of the ordered field of real numbers.

For an instance of our general result, consider the case of subvarieties of Shimura varieties. Let S be a Shimura variety. Let $\pi :D \to \Gamma \backslash D = S$ realize S as a quotient of D, a homogeneous space for the action of a real algebraic group G, by the action of $\Gamma < G$, an arithmetic subgroup. Let $S' \subseteq S$ be a special subvariety of S realized as $\pi (D')$ for $D' \subseteq D$ a homogeneous space for an algebraic subgroup of G. Let $X \subseteq S$ be an irreducible subvariety of S not contained in any proper weakly special subvariety of S. Assume that the intersection of X with $\pi (gD')$ is persistently likely as g ranges through G with $\pi (gD')$ a special subvariety of S, meaning that whenever $\zeta :S_1 \to S$ and $\xi :S_1 \to S_2$ are maps of Shimura varieties (regular maps of varieties induced by maps of the corresponding Shimura data) with $\zeta $ finite, $\dim \xi \zeta ^{-1} X + \dim \xi \zeta ^{-1} \pi (gD') \geq \dim \xi S_1$. Then $X \cap \bigcup _{g \in G, \pi (g D') \text { is special }} \pi (g D')$ is dense in X for the Euclidean topology.

We give a presentation of the torus-equivariant (small) quantum K-ring of flag manifolds of type C as an explicit quotient of a Laurent polynomial ring; our presentation can be thought of as a quantization of the classical Borel presentation of the ordinary K-ring of flag manifolds. Also, we give an explicit Laurent polynomial representative for each special Schubert class in our Borel-type presentation of the quantum K-ring.

For each of the four particle processes given by Dieker and Warren, we show the n-step transition kernels are given by the (dual) (weak) refined symmetric Grothendieck functions up to a simple overall factor. We do so by encoding the particle dynamics as the basis of free fermions first introduced by the first author, which we translate into deformed Schur operators acting on partitions. We provide a direct combinatorial proof of this relationship in each case, where the defining tableaux naturally describe the particle motions.

We compute the integral Chow rings of $\overline {\mathcal {M}}_{1,n}$ for $n=3,4$. The alternative compactifications introduced by Smyth – and studied further by Lekili and Polishchuk – present each of these stacks as a sequence of weighted blow-ups and blow-downs from a weighted projective space. We compute all the integral Chow rings by repeated application of the blow-up formula.

We define a generic rigidity matroid for k-volumes of a simplicial complex in $\mathbb {R}^d$ and prove that for $2\leq k \leq d-1$ it has the same rank as the classical generic d-rigidity matroid on the same vertex set (namely, the case $k=1$). This is in contrast with the $k=d$ case, previously studied by Lubetzky and Peled, which presents a different behavior. We conjecture a characterization for the bases of this matroid in terms of d-rigidity of the $1$-skeleton of the complex and a combinatorial Hall condition on incidences of edges in k-faces.

In this paper, we study the class of polytopes which can be obtained by taking the convex hull of some subset of the points $\{e_i-e_j \ \vert \ i \neq j\} \cup \{\pm e_i\}$ in $\mathbb {R}^n$, where $e_1,\dots ,e_n$ is the standard basis of $\mathbb {R}^n$. Such a polytope can be encoded by a quiver Q with vertices $V \subseteq \{{\upsilon }_1,\dots ,{\upsilon }_n\} \cup \{\star \}$, where each edge ${\upsilon }_j\to {\upsilon }_i$ or $\star \to {\upsilon }_i$ or ${\upsilon }_i\to \star $ gives rise to the point $e_i-e_j$ or $e_i$ or $-e_i$, respectively; we denote the corresponding polytope as $\operatorname {Root}(Q)$. These polytopes have been studied extensively under names such as edge polytope and root polytope. We show that if the quiver Q is strongly-connected, then the root polytope $\operatorname {Root}(Q)$ is reflexive and terminal; we moreover give a combinatorial description of the facets of $\operatorname {Root}(Q)$. We also show that if Q is planar, then $\operatorname {Root}(Q)$ is (integrally equivalent to) the polar dual of the flow polytope of the planar dual quiver $Q^{\vee }$. Finally, we consider the case that Q comes from the Hasse diagram of a finite ranked poset P and show in this case that $\operatorname {Root}(Q)$ is polar dual to (a translation of) a marked order polytope. We then go on to study the toric variety $Y(\mathcal {F}_Q)$ associated to the face fan $\mathcal {F}_Q$ of $\operatorname {Root}(Q)$. If Q comes from a ranked poset P, we give a combinatorial description of the Picard group of $Y(\mathcal {F}_Q)$, in terms of a new canonical ranked extension of P, and we show that $Y(\mathcal {F}_Q)$ is a small partial desingularisation of the Hibi projective toric variety $Y_{\mathcal {O}(P)}$ of the order polytope $\mathcal {O}(P)$. We show that $Y(\mathcal {F}_Q)$ has a small crepant toric resolution of singularities $Y(\widehat {\mathcal {F}}_Q)$ and, as a consequence that the Hibi toric variety $Y_{\mathcal {O}(P)}$ has a small resolution of singularities for any ranked poset P. These results have applications to mirror symmetry [61].

Let $\pi :X\rightarrow Z$ be a Fano type fibration with $\dim X-\dim Z=d$ and let $(X,B)$ be an $\epsilon $-lc pair with $K_X+B\sim _{\mathbb {R}} 0/Z$. The canonical bundle formula gives $(Z,B_Z+M_Z)$ where $B_Z$ is the discriminant divisor and $M_Z$ is the moduli divisor which is determined up to $\mathbb {R}$-linear equivalence. Shokurov conjectured that one can choose $M_Z\geqslant 0$ such that $(Z,B_Z+M_Z)$ is $\delta $-lc where $\delta $ only depends on $d,\epsilon $. Very recently, this conjecture was proved by Birkar [8]. For $d=1$ and $\epsilon =1$, Han, Jiang, and Luo [13] gave the optimal value of $\delta =1/2$. In this paper, we give the optimal value of $\delta $ for $d=1$ and arbitrary $0<\epsilon \leqslant 1$.

A landmark result of Erdős, Ginzburg, and Ziv (EGZ) states that any sequence of $2n-1$ elements in ${\mathbb {Z}}/n$ contains a zero-sum subsequence of length n. While algebraic techniques have predominated in deriving many deep generalizations of this theorem over the past sixty years, here we introduce topological approaches to zero-sum problems which have proven fruitful in other combinatorial contexts. Our main result is a topological criterion for determining when any ${\mathbb {Z}}/n$-coloring of an n-uniform hypergraph contains a zero-sum hyperedge. In addition to applications for Kneser hypergraphs, for complete hypergraphs our methods recover Olson’s generalization of the EGZ theorem for arbitrary finite groups. Furthermore, we give a fractional generalization of the EGZ theorem with applications to balanced set families and provide a constrained EGZ theorem which imposes combinatorial restrictions on zero-sum sequences in the original result.

In this paper we compute the r-point Seshadri constant on $\mathbb {P}^1\times \mathbb {P}^1$ for those line bundles where the answer might be expected to be governed by $(-1)$-curves. As a consequence we obtain explicit formulas for the symplectic packing problem for $\mathbb {P}^1\times \mathbb {P}^1$. Some exact values for the r-point Seshadri constant outside the region governed by Mori’s cone theorem are also given. These latter results use a useful new “reflection method”.

In the analysis there is a striking difference between the cases when r is odd and when r is even. When r is even the problem admits an infinite order automorphism, and there are infinitely many $(-1)$-curves to consider. In contrast, when r is odd only a finite number (usually four) types of $(-1)$-curves are relevant to our answer.

We study the energy-critical half-wave maps equation:

$$\begin{align*}\partial_t \mathbf{u} = \mathbf{u} \times |D| \mathbf{u} \end{align*}$$

for $\mathbf {u} : [0, T) \times \mathbb {R} \to \mathbb {S}^2$. Our main result establishes the global existence and uniqueness of solutions for all rational initial data $\mathbf {u}_0 : \mathbb {R} \to \mathbb {S}^2$. This demonstrates global well-posedness for a dense subset within the scaling-critical energy space $ \dot {H}^{1/2}(\mathbb {R}; \mathbb {S}^2) $. Furthermore, we prove soliton resolution for a dense subset of initial data in the energy space with uniform bounds for all higher Sobolev norms $\dot {H}^s$ for $s> 0$.

Our analysis utilizes the Lax pair structure of the half-wave maps equation on Hardy spaces in combination with an explicit flow formula. Extending these results, we establish global well-posedness for rational initial data (along with a soliton resolution result) for a generalized class of matrix-valued half-wave maps equations with target spaces in the complex Grassmannians $ \mathsf {Gr}_k(\mathbb {C}^d) $. Notably, this includes the complex projective spaces $ \mathbb {CP}^{d-1} \cong \mathsf {Gr}_1(\mathbb {C}^d) $ thereby extending the classical case of the target $\mathbb {S}^2 \cong \mathbb {CP}^1 $.

We prove that the family of graphs containing no cycle with exactly k-chords is $\chi $-bounded, for k large enough or of form $\ell (\ell -2)$ with $\ell \ge 3$ an integer. This verifies (up to a finite number of values k) a conjecture of Aboulker and Bousquet (2015).

A complete classification of unimodular valuations on the set of lattice polygons with values in the spaces of polynomials and formal power series, respectively, is established. The valuations are classified in terms of their behavior with respect to dilation using extensions to unbounded polyhedra and basic invariant theory.

We extend the theoretical framework of proof mining by establishing general logical metatheorems that allow for the extraction of the computational content of theorems with prima facie “noncomputational” proofs from probability theory, thereby unlocking a major branch of mathematics as a new area of application for these methods. Concretely, we first devise proof-theoretically tame logical systems that allow for the formalization of proofs involving algebras of sets together with probability contents, that is probability measures which are only assumed to be finitely additive. Based on these systems, we provide extensions for the tame treatment of Lebesgue integrals on probability contents as well as $\sigma $-algebras and associated probability measures, all via intensional approaches. All these systems are then shown to be amenable to proof-theoretic metatheorems in the style of proof mining which guarantee the extractability of effective and tame bounds from large classes of ineffective existence proofs in probability theory. Moreover, these extractable bounds are guaranteed to be highly uniform in the sense that they will be independent of all parameters relating to the underlying probability space, particularly regarding events or measures of them. As such, these results in particular provide the first logical explanation for the success and the observed uniformities of the previous ad hoc case studies of proof mining in these areas and further illustrate their extent. Lastly, we establish a general proof-theoretic transfer principle that allows for the lift of quantitative information on a relationship between different modes of convergence for sequences of real numbers to sequences of random variables.

We construct pathological examples of MMP singularities in every positive characteristic using quotients by $\alpha _p$-actions. In particular, we obtain non-$S_3$ terminal singularities, as well as locally stable (respectively stable) families whose general fibers are smooth (respectively klt, Cohen–Macaulay, and F-injective) and whose special fibers are non-$S_2$. The dimensions of these examples are bounded below by a linear function of the characteristic.

We introduce a framework to prove integral rigidity results for the Seiberg–Witten invariants of a closed $4$-manifold X containing a nonseparating hypersurface Y satisfying suitable (chain-level) Floer theoretic conditions. As a concrete application, we show that if X has the homology of a four-torus, and it contains a nonseparating three-torus, then the sum of all Seiberg–Witten invariants of X is determined in purely cohomological terms.

Our results can be interpreted as $(3+1)$-dimensional versions of Donaldson’s TQFT approach to the formula of Meng–Taubes, and build upon a subtle interplay between irreducible solutions to the Seiberg–Witten equations on X and reducible ones on Y and its complement. Along the way, we provide a concrete description of the associated graded map (for a suitable filtration) of the map on $\overline {\mathit {HM}}_*$ induced by a negative-definite cobordism between three-manifolds, which might be of independent interest.

We consider the Cauchy problem to the 3D fractional Schrödinger equation with quadratic interaction of $u\bar u$ type. We prove the global existence of solutions and scattering properties for small initial data. For the proof, one novelty is that we combine the normal form methods and the space-time resonance methods. Using the normal form transform enables us to have more flexibility in designing the resolution spaces so that we can control various interactions. It is also convenient for the final data problem.

In this paper we unveil novel monotonicity conditions applicable to Mean Field Games through the exploration of finite dimensional canonical transformations. Our findings contribute to establishing new global well-posedness results for the associated master equations, also in the case of potentially degenerate idiosyncratic noise. Additionally, we show that recent advancements in global well-posedness results, specifically those related to displacement semi-monotone and anti-monotone data, can be easily obtained as a consequence of our main results.

The purpose of this paper is to expose and investigate natural phase-space formulations of two longstanding problems in the restriction theory of the Fourier transform. These problems, often referred to as the Stein and Mizohata–Takeuchi conjectures, assert that Fourier extension operators associated with rather general (codimension 1) submanifolds of Euclidean space may be effectively controlled by the classical X-ray transform via weighted $L^2$ inequalities. Our phase-space formulations, which have their origins in recent work of Dendrinos, Mustata and Vitturi expose close connections with a conjecture of Flandrin from time-frequency analysis, and rest on the identification of an explicit ‘geometric’ Wigner transform associated with an arbitrary (smooth strictly convex) submanifold S of $\mathbb {R}^n$. Our main results are certain natural ‘Sobolev variants’ of the Stein and Mizohata–Takeuchi conjectures and involve estimating the Sobolev norms of such Wigner transforms by geometric forms of classical bilinear fractional integrals. Our broad geometric framework allows us to explore the role of the curvature of the submanifold in these problems, and in particular we obtain bounds that are independent of any lower bound on the curvature; a feature that is uncommon in the wider restriction theory of the Fourier transform. Finally, we provide a further illustration of the effectiveness of our analysis by establishing a form of Flandrin’s conjecture in the plane with an $\varepsilon $-loss. While our perspective comes primarily from Euclidean harmonic analysis, the procedure used for constructing phase-space representations of extension operators is well-known in optics.