Alternating links, rational balls, and cube tilings

When does the double cover of the three-sphere branched along an alternating link bound a rational homology ball? Heegaard Floer homology generates a necessary condition for it to bound: the link's chessboard lattice must be cubiquitous, implying that its normalized determinant is less than or equal to one. We conjecture that the converse holds and prove it when the normalized determinant equals one. The proof involves flows on planar graphs and the Haj\'os-Minkowski theorem that a lattice tiling of Euclidean space by cubes contains a pair of cubes which touch along an entire facet. We extend our main results to the study of ribbon cobordism and ribbon concordance

Categorification and applications in topology and representation theory

This thesis splits into two major parts. The connection between the two parts is the notion of "categorification" which we shortly explain/recall in the introduction. In the first part of this thesis we extend Bar-Natan's cobordism based categorification of the Jones polynomial to virtual links. Our topological complex allows a direct extension of the classical Khovanov complex (h=t=0), the variant of Lee (h=0,t=1) and other classical link homologies. We show that our construction allows, over rings of characteristic 2, extensions with no classical analogon, e.g. Bar-Natan's Z/2Z-link homology can be extended in two non-equivalent ways. Our construction is computable in the sense that one can write a computer program to perform calculations, e.g. we have written a MATHEMATICA based program. Moreover, we give a classification of all unoriented TQFTs which can be used to define virtual link homologies from our topological construction. Furthermore, we prove that our extension is combinatorial and has semi-local properties. We use the semi-local properties to prove an application, i.e. we give a discussion of Lee's degeneration of virtual homology. In the second part of this thesis (which is based on joint work with Mackaay and Pan) we use Kuperberg's sl3 webs and Khovanov's sl3 foams to define a new algebra K_S, which we call the sl3 web algebra. It is the sl3 analogue of Khovanov's arc algebra H_n. We prove that K_S is a graded symmetric Frobenius algebra. Furthermore, we categorify an instance of q-skew Howe duality, which allows us to prove that K_S is Morita equivalent to a certain cyclotomic KLR-algebra. This allows us to determine the split Grothendieck group K_0(K_S), to show that its center is isomorphic to the cohomology ring of a certain Spaltenstein variety, and to prove that K_S is a graded cellular algebra.Comment: 189 pages, many figures, Ph.D. thesi

Introduction to the Gopakumar-Vafa Large N Duality

Gopakumar-Vafa large N duality is a correspondence between Chern-Simons invariants of a link in a 3-manifold and relative Gromov-Witten invariants of a 6-dimensional symplectic manifold relative to a Lagrangian submanifold. We address the correspondence between the Chern-Simons free energy of S^3 with no link and the Gromov-Witten invariant of the resolved conifold in great detail. This case avoids mathematical difficulties in formulating a definition of relative Gromov-Witten invariants, but includes all of the important ideas. There is a vast amount of background material related to this duality. We make a point of collecting all of the background material required to check this duality in the case of the 3-sphere, and we have tried to present the material in a way complementary to the existing literature. This paper contains a large section on Gromov-Witten theory and a large section on quantum invariants of 3-manifolds. It also includes some physical motivation, but for the most part it avoids physical terminology.Comment: This is the version published by Geometry & Topology Monographs on 21 September 200

International Congress of Mathematicians: 2022 July 6–14: Proceedings of the ICM 2022

Following the long and illustrious tradition of the International Congress of Mathematicians, these proceedings include contributions based on the invited talks that were presented at the Congress in 2022. Published with the support of the International Mathematical Union and edited by Dmitry Beliaev and Stanislav Smirnov, these seven volumes present the most important developments in all fields of mathematics and its applications in the past four years. In particular, they include laudations and presentations of the 2022 Fields Medal winners and of the other prestigious prizes awarded at the Congress. The proceedings of the International Congress of Mathematicians provide an authoritative documentation of contemporary research in all branches of mathematics, and are an indispensable part of every mathematical library

Lagrangian Floer theory for trivalent graphs and homological mirror symmetry for curves

Mirror symmetry for higher genus curves is usually formulated and studied in terms of Landau-Ginzburg models; however the critical locus of the superpotential is arguably of greater intrinsic relevance to mirror symmetry than the whole Landau-Ginzburg model. Accordingly, we propose a new approach to the A-model of the mirror, viewed as a trivalent configuration of rational curves together with some extra data at the nodal points. In this context, we introduce a version of Lagrangian Floer theory and the Fukaya category for trivalent graphs, and show that homological mirror symmetry holds, namely, that the Fukaya category of a trivalent configuration of rational curves is equivalent to the derived category of a non-Archimedean generalized Tate curve.Comment: 52 pages, 6 figure

Local spectral universality for random matrices with independent entries

We consider the local eigenvalue statistics of large self-adjoint $N \times N$ - random matrices, $\mathbf{H}=\mathbf{H}^*$, with centred independent entries. In contrast to previous works the matrix of variances, $s_{i j}= \mathbb{E}\,|h_{i j}|^2$, is not assumed to be stochastic. Hence the density of states is not the Wigner semicircle law. In this work we prove that as $N$ tends to infinity the $k$ - point correlation function of finitely many eigenvalues becomes universal, i.e., it depends only on the symmetry class of the underlying random matrix ensemble and not on the distributions of its entries. The proof consists of three major steps. In the first step we analyse the solution, $\mathbf{m}(z)=(m_1(z), \dots, m_N(z))$, of the quadratic vector equation (QVE), $-1/m_i(z)= z+ \sum_j s_{i j}m_j(z)$, for any complex number $z$. We show that the entries, $m_i$, can be represented as Stieltjes transforms of probability densities on the real line. We characterise these densities in terms of their singularities, which are algebraic of degree at most three. We present a complete stability analysis of the QVE everywhere, including the vicinity of the singularities. This stability analysis is used in the second step. Here we prove that the diagonal elements of the resolvent, $\mathbf{G} = (\mathbf{H}-z)^{-1}$, satisfy the perturbed QVE, $-1/G_{ii}(z)= z+ \sum_j s_{i j}G_{jj}(z)+d_i(z)$, with a random noise vector $\mathbf{d}$. We show that as $N$ grows the noise vanishes and the resolvent is close to the deterministic diagonal matrix $\text{diag}(m_1, \dots, m_N)$. This result is shown with a precision down to the finest spectral scale, just above the typical eigenvalue spacing. It thus implies the local law and rigidity of the eigenvalue positions for this random matrix model. In the third and final step, we use the Dyson-Brownian-motion to establish universality of the local eigenvalue statistics.Wir analysieren die lokale Eigenwertstatistik gro{\ss}er selbstadjungierter $N\times N$ - Zufallsmatrizen, $\mathbf{H}=\mathbf{H}^*$, mit unabh\"angigen und zentrierten Eintr\"agen. Anders als in vorangegangenen Arbeiten nehmen wir nicht an, dass die Matrix der Varianzen, $s_{i j}= \mathbb{E}\,|h_{i j}|^2$, stochastisch ist. Insbesondere ist somit auch die globale Eigenwertdichte nicht durch Wigners Halbkreisverteilung gegeben. Wir beweisen in dieser Arbeit, dass mit wachsender Gr\"o{\ss}e $N$ der Zufallsmatrix die $k$-Punktfunktion endlich vieler Eigenwerte einem universellen Limes entgegen strebt. Dieser ist ausschlie{\ss}lich durch die Symmetrieklasse des zugrundeliegenden Matrixensembles bestimmt und von den Details der Verteilung der individuellen Eintr\"age unabh\"angig. Der Beweis wird in drei Schritten gef\"uhrt. Im ersten Schritt analysieren wir die L\"osung, $\mathbf{m}(z)=(m_1(z), \dots, m_N(z))$, der quadratischen Vektorgleichung (QVE), $-1/m_i(z)= z+ \sum_j s_{i j}m_j(z)$, in der $z$ eine komplexe Zahl ist. Wir zeigen, dass die Komponenten, $m_i$, der L\"osung als Stieltjes-Transformation gewisser Wahrscheinlichkeitsdichten auf der reellen Achse dargestellt werden k\"onnen. Wir charakterisieren diese Dichten anhand ihres Singularit\"atsverhaltens und zeigen dass dieses h\"ochstens von algebraischer Ordnung drei ist. Wir f\"uhren eine vollst\"andige Stabilit\"atsanalyse der QVE durch, welche auch die Umgebung der Singularit\"aten einschlie{\ss}t. Diese wird im zweiten Schritt des Beweises verwendet, in welchem wir zeigen, dass die Diagonaleintr\"age der Resolvente, $\mathbf{G} = (\mathbf{H}-z)^{-1}$, die gest\"orte QVE, $-1/G_{ii}(z)= z+ \sum_j s_{i j}G_{jj}(z)+d_i(z)$, mit einer zuf\"alligen vektorwertigen St\"orung, $\mathbf{d}$, erf\"ullen. Da mit wachsendem $N$ die St\"orung gegen Null konvergiert, n\"ahert sich die Resolvente im Limes der deterministischen Diagonalmatrix $\text{diag}(m_1, \dots, m_N)$ an. Dieses Resultat wird mit einer optimalen spektralen Aufl\"osung gezeigt, welche knapp \"uber dem typischen Abstand der Eigenwerte liegt. Als Konsequenz sehen wir, dass die Fluktuation der Eigenwerte die durch diese Aufl\"osung gegebene Gr\"o{\ss}enordung nicht \"ubersteigt. Im dritten und letzten Schritt nutzen wir den von Dyson eingef\"uhrten Prozess der Dyson-Brownschen Bewegung der Eigenwerte und die K\"urze seine lokalen Relaxationszeit um die Universalit\"at der lokale Eigenwertstatistik zu beweisen

Discrete Geometry

[no abstract available