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String theory is a proposal for a description of nature on the smallest length scales, where both quantum mechanics and gravity are expected to play an essential role. A curious feature of the theory is that its basic building blocks – one-dimensional strings – behave as if they were moving in nine-dimensional space, as opposed to the three-dimensional space we are familiar with. String theory proposes that the six additional dimensions are curled up into extremely small sizes, in a process called “compactification”. Importantly, what happens in these hidden six dimensions has an enormous influence on the physics we observe at larger length scales, including the strength of the interactions between elementary particles and the value of the cosmological constant. An intriguing aspect of string theory is that there are countless possible ways to compactify these extra dimensions, resulting in a vast "landscape" of potential universes, each with distinct physical properties. In a beautiful interplay of physics and mathematics, the physical characteristics of these universes are intricately linked to the geometric properties of the internal six-dimensional space. The main aim of this thesis is to study the latter using the sophisticated mathematical framework of asymptotic Hodge theory. One of the central results is that one can obtain a good approximation of numerous physical observables by studying the allowed singularities of the six-dimensional internal space. In accordance with some fundamental theorems in the field of algebraic geometry, these can be classified in great generality and reveal intriguing underlying structures. In particular, this abstract approach leads to a comprehensive, algorithmic method for calculating physical observables in string compactifications. In this way, we obtain a very general characterization of universes that can be constructed in string theory and find that they are in fact finite in number. Additionally, based on recent advances

There are two ways that certain Fano fourfolds (for example, cubic fourfoldsand Gushel-Mukai fourfolds) can be associated with K3 surfaces. On the one hand, we can associate a K3 surface to the fourfold Hodge-theoretically, meaning the middle cohomology of the fourfold contains the middle cohomology of the K3 as a sub-Hodge structure; on the other hand, we can homologically associate a K3 to the fourfold, which is to require the Kuznetsov component of the fourfolds to be equivalent to the bounded derived of the K3. Conjecturally, one expect such K3 associations detect rationality of the fourfolds. It has been proved that for cubic fourfolds, these two types of K3 associations are equivalent, whereas for Gushel-Mukai fourfolds, Hodge-association of a K3 strictly implies homological association of a K3. We continue this line of study, instead of comparing fourfolds- K3 association we consider fourfolds-fourfolds association. We prove that, at least for a generic Gushel-Mukai fourfold in the Hodge-special loci, if it admits Hodgeassociated cubic fourfolds, then it admits a homological-associated one, and vice versa.

The Hodge conjecture is one of the seven millennium problems, and is framed within differential geometry and algebraic geometry. It was proposed by William Hodge in 1950 and is currently a stimulus for the development of several theories based on geometry, analysis, and mathematical physics. It proposes a natural condition for the existence of complex submanifolds within a complex manifold. Manifolds are the spaces in which geometric objects can be considered. In complex manifolds, the structure of the space is based on complex numbers, instead of the most intuitive structure of geometry, based on real numbers., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

The thesis applies asymptotic Hodge theory of the internal Calabi-Yau space in string compactification to study the four-dimensional physics near various corners of the field space. This approach sets up some universal understandings of the swampland in quantum gravity. The first chapter introduces the basic concepts used in this thesis, which are string compactification, swampland, and Hodge theory. The second chapter uses Hodge theory to study the swampland distance conjecture. The setting is type IIB string theory on Calabi-Yau threefolds, and we examine the distance conjecture in the complex structure moduli space of the Calabi-Yau's. With Hodge theory, we equip every limit in the moduli space with an algebraic structure, and classify all possible structures that can appear within a Calabi-Yau. When the Calabi-Yau admits multiple-stage degenerations, all possible transitions between the corresponding algebraic structures are classified. Thus, an intricate network of the algebraic structures associated with infinite distance limits is discovered. Using this network of algebraic structures, we find a light tower of states in the distance conjecture in most infinite distance limits. The approach is exemplified in two-parameter models. The third chapter uses Hodge theory to study the swampland de Sitter conjecture near the corners in the scalar field space. The setting is F-theory on Calabi-Yau fourfolds with four-form flux, and we work in the complex structure moduli space of the fourfolds. There are again algebraic structures associated to every asymptotic limit in the complex structure moduli space, and these algebraic structures can be classified. Using the classification, we further exhaust the possible forms of the scalar potentials in the asymptotic limits. We carry out this program for two-parameter families of Calabi-Yau fourfolds, and study the vacuum structure of these theories near the limits. We confirm the de Sitter conjecture near the limits. Via dual

Let G be a finite connected graph, and let T be a spanning tree of G chosen uniformly at random. The work of Kirchhoff on electrical networks can be used to show that the events e1 E T and e2 E T are negatively correlated for any distinct edges e1 and e2. What can be said for such events when the underlying matroid is not necessarily graphic? We use Hodge theory for matroids to bound the correlation between the events e E B, where B is a randomly chosen basis of a matroid. As an application, we prove Mason's conjecture that the number of k-element independent sets of a matroid forms an ultra-log-concave sequence in k., QC 20220331

We show that the André motive of a hyper-Kähler variety X over a field K⊂ C with b2(X) > 6 is governed by its component in degree 2. More precisely, we prove that if X1 and X2 are deformation equivalent hyper-Kähler varieties with b2(Xi) > 6 and if there exists a Hodge isometry f: H2(X1, Q) → H2(X2, Q) , then the André motives of X1 and X2 are isomorphic after a finite extension of K, up to an additional technical assumption in presence of non-trivial odd cohomology. As a consequence, the Galois representations on the étale cohomology of X1 and X2 are isomorphic as well. We prove a similar result for varieties over a finite field which can be lifted to hyper-Kähler varieties for which the Mumford–Tate conjecture is true.

"The 3rd Japan-Taiwan Joint Conference on Differential Geometry" was held on November 1-3, 2021. This volume records the abstracts and the slides of talks presented in this conference.

We study the Mumford–Tate conjecture for hyperkähler varieties. We show that the full conjecture holds for all varieties deformation equivalent to either an Hilbert scheme of points on a K3 surface or to O’Grady’s ten dimensional example, and all of their self-products. For an arbitrary hyperkähler variety whose second Betti number is not 3, we prove the Mumford–Tate conjecture in every codimension under the assumption that the Künneth components in even degree of its André motive are abelian. Our results extend a theorem of André.

We use Hodge theory and functional analysis to develop a clean approach to heat flows and intrinsic harmonic analysis on Riemannian manifolds with boundary. We also introduce heatable currents as the natural analogue to tempered distributions and justify their importance in Hodge theory. As an application, we prove Onsager's conjecture (energy conservation of ideal fluids), where the weak solution lies in the trace-critical Besov space $B_{3,1}^{\frac{1}{3}}$.In the second half of the thesis, by applying techniques from geometric microlocal analysis to construct the Hodge-Neumann heat kernel, we obtain off-diagonal decay and local Bernstein estimates, and then use them to extend the result to the Besov space $\widehat{B}_{3,V}^{\frac{1}{3}}$, which generalizes both the space $\widehat{B}_{3,c(\mathbb{N})}^{1/3}$ from (Isett-Oh 2014) and the space $\underline{B}_{3,\text{VMO}}^{1/3}$ from (Bardos et al. 2019, Nguyen-Nguyen 2020) --- the best known function space where Onsager's conjecture holds on flat backgrounds.

It is known that some GIT compactifications associated to moduli spaces of either points in the projective line or cubic surfaces are isomorphic to Baily-Borel compactifications of appropriate ball quotients. In this paper, we show that their respective toroidal compactifications are isomorphic to moduli spaces of stable pairs as defined in the context of the MMP. Moreover, we give a precise mixed-Hodge-theoretic interpretation of this isomorphism for the case of eight labeled points in the projective line. (c) 2021 Elsevier Inc. All rights reserved., QC 20210421

Kazuya Kato, Chikara Nakayama, Sampei Usui. Toroidal compactifications and Borel-Serre compactifications. https://arxiv.org/abs/2107.10999

Kazuya Kato, Chikara Nakayama, Sampei Usui. Classifying spaces of degenerating mixed Hodge structures, V: Extended period domains and algebraic groups. https://arxiv.org/abs/2107.03561.

We use tools from generalized complex geometry to develop the theory of SKT (a.k.a. pluriclosed Hermitian) manifolds and more generally manifolds with special holonomy with respect to a metric connection with closed skew-symmetric torsion. We develop Hodge theory on such manifolds showing how the reduction of the holonomy group causes a decomposition of the twisted cohomology. For SKT manifolds this decomposition is accompanied by an identity between different Laplacian operators and equates different cohomologies defined in terms of the SKT structure. We illustrate our theory with examples based on Calabi–Eckmann manifolds, instantons, Hopf surfaces and Lie groups.

In this thesis, we study the geometry of Teichmuller space of punctured Riemann surfaces.We use L2 Hodge theory to describe the deformation theory for punctured Riemann surfaces,in which we defined Weil-Petersson metric, Hodge metric and Kodaira-Spencer map. Wealso give a new proof of Wolpert's curvature formula by computing the expansion of volumeform and the Kodaira-Spencer map. We use Wolpert's formula to estimate upper bound forvarious curvature tensor. We construct an extension of pluricanonical form and compare itto the expansion of the Kodaira-Spencer map under Hodge metric.

We prove that an admissible normal function over a surface and the zero section simultaneously extend to sections of a log N ́eron model. This gives a new proof of the surface base case of the algebraicity of zero loci of admissible normal functions.

HodgeRank on random graphs is proposed recently as an effective framework for multimedia quality assessment problem based on paired comparison methods. With a random design on graphs, it is particularly suitable for large scale crowdsourcing experiments on the Internet. However, there still lacks a systematic study about online schemes to deal with the rising streaming and massive data in crowdsourceable scenarios. To fill in this gap, we propose in this paper an online ranking/rating scheme based on stochastic approximation of HodgeRank on random graphs for Quality of Experience (QoE) evaluation, where assessors and rating pairs enter the system in a sequential or streaming way. The scheme is shown in both theory and experiments to be efficient in obtaining global ranking by exhibiting the same asymptotic performance as batch HodgeRank under a general edge-independent sampling process. Moreover, the proposed framework enables us to monitor topological changement and triangular inconsistency in real time. Among a wide spectrum of choices, two particular types of random graphs are studied in detail, i.e., Erdös-Rényi random graph and preferential attachment random graph. The former is the simplest I.I.D. (independent and identically distributed) sampling and the latter may achieve more efficient performance in ranking the top-k items due to its Rich-get-Richer property. We demonstrate the effectiveness of the proposed framework on LIVE and IVC databases. © 1999-2012 IEEE.

Presented at the AFOSR Spring Review 2013, 4-8 March, Arlington, VA.

Given a singular scheme X over a field k, we consider the problem of resolving the singularities of X by an algebraic stack. When X is a toroidal embedding or is etale locally the quotient of a smooth scheme by a linearly reductive group scheme, we show that such &ldquo stacky resolutions &rdquo exist. Moreover, these resolutions are canonical and easily understandable in terms of the singularities of X.We give three applications of our stacky resolution theorems: various generalizations of the Chevalley-Shephard-Todd Theorem, a Hodge decomposition in characteristic p, and a theory of toric Artin stacks extending the work of Borisov-Chen-Smith. While these applications are seemingly different, they are all related by the common theme of using stacky resolutions to study singular schemes.

Bekannte Theoreme von Carlson und Griffiths gestatten es, die Variation von Hodgestrukturen assoziiert zu einer Familie von glatten Hyperflächen sowie das Cupprodukt auf der mittleren Kohomologie explizit zu beschreiben. Wir benutzen M. Saitos Theorie der gemischten Hodgemoduln, um diesen Kalkül auf die Variation der Hodgestruktur der Schnittkohomologie von Familien nodaler Hyperflächen zu verallgemeinern., Well known theorems of Carlson and Griffiths provide an explicit description of the variation of Hodge structures associated to a family of smooth hypersurfaces together with the cupproduct pairing on the middle cohomology. We give a generalization to families of nodal hypersurfaces using M. Saitos theory of mixed Hodge modules.

by Chow Ha Tak., Thesis (M.Phil.)--Chinese University of Hong Kong, 1998., Includes bibliographical references (leaves 96-97)., also in Chinese., Chapter 1 --- The Laplacian on a Riemannian Manifold --- p.5, Chapter 1.1 --- Riemannian metrics --- p.5, Chapter 1.2 --- L2 Spaces of Functions and Forms --- p.6, Chapter 1.3 --- The Laplacian on Functions and Forms --- p.8, Chapter 2 --- Hodge Theory for Functions and Forms --- p.14, Chapter 2.1 --- Analytic Preliminaries --- p.14, Chapter 2.2 --- The Hodge Theorem for Functions --- p.20, Chapter 2.3 --- The Hodge Theorem for Forms --- p.27, Chapter 2.4 --- Regularity Results --- p.29, Chapter 2.5 --- The Kernel of the Laplacian on Forms --- p.33, Chapter 3 --- Fermion Calculus and Weitzenbock Formula --- p.36, Chapter 3.1 --- The Levi-Civita Connection --- p.36, Chapter 3.2 --- Fermion calculus --- p.39, Chapter 3.3 --- "Weitzenbock Formula, Bochner Formula and Garding's Inequality" --- p.53, Chapter 3.4 --- The Laplacian in Exponential Coordinates --- p.59, Chapter 4 --- The Construction of the Heat Kernel --- p.63, Chapter 4.1 --- Preliminary Results for the Heat Kernel --- p.63, Chapter 4.2 --- Construction of the Heat Kernel --- p.66, Chapter 4.2.1 --- Construction of the Parametrix --- p.66, Chapter 4.2.2 --- The Heat Kernel for Functions --- p.70, Chapter 4.2.3 --- The Heat Kernel for Forms --- p.76, Chapter 4.3 --- The Asymptotics of the Heat Kernel --- p.77, Chapter 5 --- The Heat Equation Approach to the Chern-Gauss- Bonnet Theorem --- p.82, Chapter 5.1 --- The Heat Equation Approach --- p.82, Chapter 5.2 --- Proof of the Chern-Gauss-Bonnet Theorem --- p.85, Chapter 5.3 --- Introduction to Atiyah-Singer Index Theorem --- p.87, Chapter 5.3.1 --- A Survey of Characteristic Forms --- p.87, Chapter 5.3.2 --- The Hirzenbruch Signature Theorem --- p.90, Chapter 5.3.3 --- The Atiyah-Singer Index Theorem --- p.93, Bibliography, Notation index, http://library.cuhk.edu.hk/record=b5889724, Use of this resource is governed by the terms and conditions of the Creative Commons “Attribution-NonCommercial-NoDerivatives 4.0 International” License (http://creativecommons.org/licenses/by-nc-nd/4.0/)

Kazuya Kato, Chikara Nakayama, Sampei Usui. Toroidal compactifications and Borel-Serre compactifications. https://arxiv.org/abs/2107.10999

Kazuya Kato, Chikara Nakayama, Sampei Usui. Classifying spaces of degenerating mixed Hodge structures, V: Extended period domains and algebraic groups. https://arxiv.org/abs/2107.03561.

Kazuya Kato, Chikara Nakayama, Sampei Usui. Toroidal compactifications and Borel-Serre compactifications. https://arxiv.org/abs/2107.10999

Kazuya Kato, Chikara Nakayama, Sampei Usui. Classifying spaces of degenerating mixed Hodge structures, V: Extended period domains and algebraic groups. https://arxiv.org/abs/2107.03561.

In a joint work with Kazuya Kato and Chikara Nakayama, log higher Albanese manifolds were constructed as an application of log mixed Hodge theory with group action. In this framework, we describe a work of Deligne on some nilpotent quotients of the fundamental group of the projective line minus three points, where polylogarithms appear. As a result, we have q-expansions of higher Albanese maps at boundary points, i.e., log higher Albanese maps over the boundary.

In a joint work with Kazuya Kato and Chikara Nakayama, log higher Albanese manifolds were constructed as an application of log mixed Hodge theory with group action. In this framework, we describe a work of Deligne on some nilpotent quotients of the fundamental group of the projective line minus three points, where polylogarithms appear. As a result, we have q-expansions of higher Albanese maps at boundary points, i.e., log higher Albanese maps over the boundary.