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Oct 29, 2018
Number Theory
Algebraic Geometry

Let A(G) be the space of automorphic forms on G. In this section we show that the inclusion A(G)AG(R)o⊆C∞umg(G) also induces isomorphisms of (po,Ko)-cohomology with coefficients in V, hence in (4.1) we can replace the later by the former. This is pa…

Coherent cohomology of Shimura varieties and automorphic forms

We show that the cohomology of canonical extensions of automorphic vector bundles over toroidal compactifications of Shimura varieties can be compute…
Apr 30, 2018
Number Theory

In this section we conclude the proof of the results stated in the Introduction. The proof of Theorem B naturally splits into two cases according to the sign ϵ(π,K,12) (see the discussion at the beginning of \lx@sectionsign???), but in both cases it…

On the Iwasawa main conjectures for modular forms at non-ordinary pri…

In this paper, we prove under mild hypotheses the Iwasawa main conjectures of Lei--Loeffler--Zerbes for modular forms of weight $2$ at non-ordinary p…
Nov 27, 2016
Metric Geometry
Data Structures and Algorithms
Combinatorics
Functional Analysis

Given D⩾1 and a metric space (M,d), the definition in [LLR95] of the metric dimension dimD(M) can be naturally refined by restricting the potential target spaces into which we wish to embed M. Specifically, suppose that F is a family of finite-dimen…

A spectral gap precludes low-dimensional embeddings

We prove that there is a universal constant $C>0$ with the following property. Suppose that $n\in \mathbb{N}$ and that $\mathsf{A}=(a_{ij})\in M_n(\m…
Dec 29, 2016
Analysis of PDEs

AC, DC and JGS were partially supported by the grant MTM2014-59488-P (Spain) and ICMAT Severo Ochoa project SEV-2015-556. AC was partially supported by the Ramón y Cajal program RyC-2013-14317 and ERC grant 307179-GFTIPFD. JGS was partially supporte…

Uniformly rotating smooth solutions for the incompressible 2D Euler …

In this paper, we show the existence of a family of compactly supported smooth vorticities, which are solutions of the 2D incompressible Euler equati…
Mar 08, 2014
Symplectic Geometry
Geometric Topology

A contact invariant in sutured monopole homology

We define an invariant of contact 3-manifolds with convex boundary using Kronheimer and Mrowka's sutured monopole Floer homology theory (SHM). Our in…
Oct 07, 2009
Geometric Topology
Symplectic Geometry

Comultiplication in link Floer homology and transversely non-simple …

For a word w in the braid group on n-strands, we denote by T_w the corresponding transverse braid in the rotational symmetric tight contact structure…
Sep 28, 2009
Algebraic Geometry
Representation Theory

Sections 1-3 of the paper closely follow a three lecture course given by R.P. at the Geometry summer school in July 2009 at the Instituto Superior Técnico in Lisbon. The results of Section 4 and 5 were worked out together during the IST visit and af…

Quivers, curves, and the tropical vertex

Elements of the tropical vertex group are formal families of symplectomorphisms of the 2-dimensional algebraic torus. Commutators in the group are re…
Dec 17, 2011
Differential Geometry
Analysis of PDEs

Let us conclude by making precise how to use our methods to understand GNS inequalities (1.4) in the full range 1≤p≤q≤2nn−2. In particular, we will also give a variational characterization of the second family (1.6) of GNS inequalities whose sharp c…

Conformal invariants measuring the best constants for Gagliardo-Nir…

We introduce a family of conformal invariants associated to a smooth metric measure space which generalize the relationship between the Yamabe consta…
Sep 28, 2017
Number Theory

We thank N. Elkies, B. Howard, E. Howe, M. Stoll, A. Sutherland, and J. Wetherell for helpful comments and discussions. MB was partially supported by a Simons Investigator Grant and NSF Grant DMS-1001828. RJLO was partially supported by NSF grant DM…

Three-isogeny Selmer groups and ranks of abelian varieties in quadrat…

For an abelian variety $A$ over a number field $F$, we prove that the average rank of the quadratic twists of $A$ is bounded, under the assumption th…
Jan 25, 2019
Analysis of PDEs

Convex integration and phenomenologies in turbulence

In this review article we discuss a number of recent results concerning wild weak solutions of the incompressible Euler and Navier-Stokes equations. …
Jul 07, 2014
Geometric Topology

Concordance homomorphisms from knot Floer homology

We modify the construction of knot Floer homology to produce a one-parameter family of homologies for knots in the three-sphere. These invariants can…
May 30, 2017
Analysis of PDEs

In this section we prove the following:

Global solutions for the generalized SQG patch equation

We consider the inviscid generalized surface quasi-geostrophic equation (gSQG) in a patch setting, where the parameter $\alpha \in (1,2)$. The cases …
Dec 28, 2017
Analysis of PDEs

Logarithmic corrections in the asymptotic expansion for the radiation…

We obtain the second-order asymptotics for the radiation field of spherically symmetric solutions to the wave equation on spherically symmetric and a…
Jan 12, 2017
Geometric Topology

Link cobordisms and absolute gradings on link Floer homology

We show that the link cobordism maps defined by the author are graded and satisfy a grading change formula. Using the grading change formula, we prov…
Apr 25, 2017
Probability
Statistical Mechanics
Mathematical Physics

Invariant Measures for TASEP with a Slow Bond

Totally Asymmetric Simple Exclusion Process (TASEP) on $\mathbb{Z}$ is one of the classical exactly solvable models in the KPZ universality class. We…
Jun 15, 2009
Geometric Topology

Topics in Heegaard Floer homology

This is the author's PhD thesis, as submitted to the Princeton University. The results of this paper have already appeared in arXiv:math/0607777v4, a…
Oct 22, 2015
Symplectic Geometry
Geometric Topology

In this section we further investigate the topology of a filling (W,ω) of (Yg,ξg) which is not only exact but Stein; in this case we denote it by (W,J) to avoid confusion. In this case W has a handle decomposition consisting of only 0-, 1-, and 2-ha…

Fillings of unit cotangent bundles

We study the topology of exact and Stein fillings of the canonical contact structure on the unit cotangent bundle of a closed surface $\Sigma_g$, whe…
Oct 24, 2017
Functional Analysis
Computational Geometry
Metric Geometry

Impossibility of dimension reduction in the nuclear norm

Let $\mathsf{S}_1$ (the Schatten--von Neumann trace class) denote the Banach space of all compact linear operators $T:\ell_2\to \ell_2$ whose nuclear…
Apr 17, 2018
Geometric Topology

Connected Heegaard Floer homology of sums of Seifert fibrations

We compute the connected Heegaard Floer homology (defined by Hendricks, Hom, and Lidman) for a large class of 3-manifolds, including all linear combi…
Sep 02, 2011
Algebraic Geometry

The Geometry of Stable Quotients in Genus One

Stable quotient spaces provide an alternative to stable maps for compactifying spaces of maps. When the target is projective space and the domain cur…
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