Current developments in mathematics

Greg Muller, David E Speyer · Proceedings of the London Mathematical Society · 2017

The purpose of this document is to connect two maps related to certain graphs embedded in the disc. The first is Postnikov's boundary measurement map, which combines partition functions of matchings in the graph into a map from an algebraic torus to an open positroid variety in a Grassmannian. The second is a rational map from the open positroid variety…

Bethany Marsh, Konstanze Rietsch · Advances in Mathematics · 2020

Ulrich Görtz, Xuhua He, Sian Nie · Peking Mathematical Journal · 2019

Pavel Galashin, Steven N. Karp, Thomas Lam · Advances in Mathematics · 2021

We prove that three spaces of importance in topological combinatorics are homeomorphic to closed balls: the totally nonnegative Grassmannian, the compactification of the space of electrical networks, and the cyclically symmetric amplituhedron.

Steven N. Karp · Journal of Combinatorial Theory Series A · 2016

Pavel Galashin, Steven N. Karp, Thomas Lam · Journal of the American Mathematical Society · 2021

We show that the totally nonnegative part of a partial flag variety G / P G/P (in the sense of Lusztig) is a regular CW complex, confirming a conjecture of Williams. In particular, the closure of each positroid cell inside the totally nonnegative Grassmannian is homeomorphic to a ball, confirming a conjecture of Postnikov.

Sergio Caracciolo A, Gabriele Sicuro · Physical Review E · 2015

We analyze the random Euclidean bipartite matching problem on the hypertorus in d dimensions with quadratic cost and we derive the two-point correlation function for the optimal matching, using a proper ansatz introduced by Caracciolo et al. [Phys. Rev. E 90, 012118 (2014)] to evaluate the average optimal matching cost. We consider both the grid-Poisson matching problem and the Poisson-Poisson…

Martin Hairer, Weijun Xu · Communications on Pure and Applied Mathematics · 2018

Abstract We study a class of three‐dimensional continuous phase coexistence models, and show that, under different symmetry assumptions on the potential, the large‐scale behavior of such models near a bifurcation point is described by the dynamical models for p ∊ {2,3,4}. This result is specific to space dimension 3 and does not hold in dimension 2. © 2018 Wiley Periodicals,…

P.G. Linares, Félix Otto, Markus Tempelmayr, Pavlos Tsatsoulis · Inventiones mathematicae · 2024

Abstract In this paper, we explore the version of Hairer’s regularity structures based on a greedier index set than trees, as introduced in (Otto et al. in A priori bounds for quasi-linear SPDEs in the full sub-critical regime, 2021, arXiv: 2103.11039 ) and algebraically characterized in (Linares et al. in Comm. Am. Math. Soc. 3:1–64, 2023). More precisely, we construct…

Valentin Blomer, Péter Maga · International Mathematics Research Notices · 2014

Let F be an L2-normalized Hecke–Maaß cusp form for the group |$\Gamma = {\rm SL}_{4}(\mathbb {Z})$| with Laplace eigenvalue λF. Assume that F satisfies the Ramanujan conjecture at |$\infty $| (this is satisfied by almost all cusp forms). If Ω is a compact subset of |$\Gamma \backslash \mathcal {H}_4$|⁠, we show the bound |$\| F|_{\Omega }\|_{\infty } \ll \lambda _F^{3/2…

Niklas Christoph Affolter, Max Glick, Pavlo Pylyavskyy, Sanjay Ramassamy · Selecta Mathematica · 2023

Elizabeth Milićević, Eva Viehmann · Forum of Mathematics Sigma · 2020

Abstract We consider the Newton stratification on Iwahori-double cosets in the loop group of a reductive group. We describe a group-theoretic condition on the generic Newton point, called cordiality, under which the Newton poset (that is, the index set for non-empty Newton strata) is saturated and Grothendieck’s conjecture on closures of the Newton strata holds. Finally, we give several large…

Christoph Weiß, S. A. Gardiner, Heinz‐Peter Breuer · Physical Review A · 2015

Brownian motion is ballistic on short time scales and diffusive on long time scales. Our theoretical investigations indicate that one can observe the exact opposite---an ``anomalous diffusion process'' where initially diffusive motion becomes ballistic on longer time scales---in an ultracold atomic system with a size comparable to macromolecules. This system is the center-of-mass motion of a quantum matter-wave bright soliton…

Justin Hilburn, Sam Raskin · Journal of the American Mathematical Society · 2022

We calculate the category of D D -modules on the loop space of the affine line in coherent terms. Specifically, we find that this category is derived equivalent to the category of ind-coherent sheaves on the moduli space of rank one de Rham local systems with a flat section. Our result establishes a conjecture coming out of the 3 d…

Xuhua He · Forum of Mathematics Pi · 2021

Abstract The affine Deligne–Lusztig variety $X_w(b)$ in the affine flag variety of a reductive group ${\mathbf G}$ depends on two parameters: the $\sigma $ -conjugacy class $[b]$ and the elementwin the Iwahori–Weyl group $\tilde {W}$ of ${\mathbf G}$ . In this paper, for any given $\sigma $ -conjugacy class $[b]$ , we determine the nonemptiness pattern and the dimension formula…

Shirshendu Ganguly · Notices of the American Mathematical Society · 2021

This article reviews recent and past advances in understanding random planar metric spaces with a focus on geodesic geometry.

Abdelmalek Abdesselam · P-Adic Numbers Ultrametric Analysis and Applications · 2018

In this outline of a program, based on rigorous renormalization group theory, we introduce new definitions which allow one to formulate precise mathematical conjectures related to conformal invariance as studied by physicists in the area known as higher-dimensional conformal bootstrap which has developed at a breathtaking pace over the last few years. We also explore a second theme, intimately tied…

Pavel Galashin · Communications in Mathematical Physics · 2023

Aleksander Doan, Thomas Walpuski · Advances in Mathematics · 2021

Anthony M. Bloch, Steven N. Karp · Advances in Mathematics · 2023