Extends Borel invariant to measurable cocycles of 3-manifold groups.
problem Defining and analyzing Borel invariant for measurable cocycles.
method Introducing pullback along measurable cocycles and extending Borel invariant.
result Maximal cocycles are trivializable to irreducible representations.
Study of ω-Borel invariant for representations into SL(n,Cω).
problem Characterizing representations into SL(n,Cω) using ω-Borel invariant. method Defined ω-Borel invariant βnω(ρω) for representations ρω:ΓightarrowSL(n,Cω), studied sequences of ω-bounded representations and their limits. result If a sequence of representations ρl into SL(2,C) determines a reducible action on the asymptotic cone Cω(H3,d/λl,O), then β2ω(ρω)=0. Maximal and Borel Anosov representations in Sp(4,R) are proven to be Hitchin.
problem Characterizing representations of surface groups into Sp(4,R) that are Borel Anosov and maximal. method Proving representations are Hitchin if they have maximal Toledo invariant and are Borel Anosov.
result Maximal and Borel Anosov representations in Sp(4,R) are Hitchin. Mathematical structures link Gromov-Witten to Donaldson-Thomas invariants.
problem Understanding non-perturbative topological string theory.
method Borel summation of Gromov-Witten potential and analysis of Stokes phenomena.
result Stokes phenomena encode Donaldson-Thomas invariants of the resolved conifold.
Let S be a non-exceptional oriented surface of finite type. We discuss the action of subgroups of the mapping class group of S on the CAT(0)-boundary of the completion of Teichmueller space with respect to the Weil-Petersson metric. We show that the set of invariant Borel probability measures for the Weil-Petersson flo…
In this paper, we show that the Euler characteristic of an even dimensional closed projectively flat manifold is equal to the total measure which is induced from a probability Borel measure on RP^n invariant under the holonomy action, and then discuss its consequences and applications. As an application, we show that t…
The paper connects arithmetic invariants of hyperbolic 3-manifolds.
problem Understanding the arithmetic properties of hyperbolic 3-manifolds.
method Analyzes profinite completions and algebraic invariants of fundamental groups.
result Uniform lattices with isomorphic profinite completions have identical arithmetic properties.
Extends equivariant contact structure results to mod p L-spaces.
problem Equivariant contact structures on minimal L-spaces.
method Uses Serre spectral sequence of Borel Floer cohomology.
result Introduces two new numerical invariants.
Quantum dilogarithm function proven from a linear difference equation.
problem Proving Faddeev's quantum dilogarithm from a linear difference equation.
method Proved Faddeev's quantum dilogarithm using Borel summation of a formal power series solution of a linear difference equation.
result Borel summation of a formal power series solution produces Faddeev's quantum dilogarithm.
For a hyperbolic link complement with a triangulation, there are hyperbolicity equations of the triangulation, which guarantee the hyperbolic structure of the link complement. In this paper, we explain that the number of the essential solutions of the equations is equal to or bigger than the extension degree of the inv…
Two 4-manifolds are stably diffeomorphic if they become diffeomorphic after connected sum with S^2 x S^2's. This paper shows that two closed, orientable, homotopy equivalent, smooth 4-manifolds are stably diffeomorphic, provided a certain map from the second homology of the fundamental group with coefficients in Z/2 to…
Novel construction of Bauer--Furuta invariant using sheaves of spectra.
problem Constructing the Bauer--Furuta invariant without finite-dimensional approximations.
method Using sheaves of spectra and Borel--Moore homology, avoiding approximations.
result Defines the shriek functors and Thom spectra for index calculations.
If Γ<PSL(2,C) is a lattice, we define an invariant of a representation Γ→PSL(n,C) using the Borel class β(n)∈Hc3(PSL(n,C),R). We show that the invariant is bounded and its maximal value is attained by conjugation of t…
New invariant recovers known contact element and considers finite coverings.
problem Defining and studying new contact invariants in Seiberg-Witten Floer spectra.
method Cohomotopy set of Seiberg-Witten Floer spectrum, equivariant Borel cohomology.
result New invariant recovers known contact element and considers finite coverings.
We study and classify topologically invariant σ-ideals with a Borel base on the Hilbert cube and evaluate their cardinal characteristics. One of the results of this paper solves (positively) a known problem whether the minimal cardinalities of the families of Cantor sets covering the unit interval and the Hilbert cub…
Classifies manifolds and discrete subgroups of Lie groups using descriptive set theory.
problem Classifying manifolds and discrete subgroups of Lie groups.
method Descriptive set theory and Borel complexity computations.
result Complexity of homeomorphism problems for manifolds and conjugacy relations for subgroups.
Let X be a locally symmetric space associated to a reductive algebraic group G defined over Q. L-modules are a combinatorial analogue of constructible sheaves on the reductive Borel-Serre compactification of X; they were introduced in [math.RT/0112251]. That paper also introduced the micro-support of an L-module, a com…
The study shows how certain ODEs and integrals are regular under Borel summation.
problem Analyzing the regularity of solutions to ODEs and integration problems.
method Using geometric perspective on Laplace and Borel transforms, the study examines level 1 ODEs and exponential period integrals over Lefschetz thimbles.
result Solutions of certain ODEs and integration problems are Borel regular.
New findings on cusped Borel Anosov representations and their properties.
problem Characterizing and understanding cusped Borel Anosov representations.
method Analyzing representations of lattices in PGL2(R) to PGLd(R). result Cusped Borel Anosov representations with specific properties are Hitchin representations.
The BPS decomposition theorem splits cohomology of symmetric stacks into invariant parts.
problem Decomposing the cohomology of smooth symmetric stacks into invariant parts.
method Using cohomological Hall induction and intersection cohomology of moduli spaces.
result Establishes the BPS decomposition theorem for various symplectic stacks.
Study shows Roller compactification's median graph has limited asymptotic dimension.
problem Understanding the asymptotic dimension of Roller compactifications.
method Proved using finite dimensional CAT(0) cube complexes and Borel median graph.
result Borel asymptotic dimension is bounded by the complex's dimension.
We show, up to h-cobordism, that the existence and uniqueness of connected sum decompositions of oriented 4-dimensional manifolds is an invariant of homotopy equivalence, assuming that the fundamental group of each summand is "good" in the sense of Freedman and Quinn. On a separate note, we observe that the Borel Conje…
We survey the recent results and current issues on the topological rigidity problem for closed aspherical manifolds, i.e., connected closed manifolds whose universal coverings are contractible. A number of open problems and conjectures are presented during the course of the discussion. We also review the status and app…
The 2-rank of a compact Lie group G is the maximal possible rank of the elementary 2-subgroup Z2×...Z2 of G. The study of 2-ranks (and p-rank for any prime p) of compact Lie groups was initiated in 1953 by A. Borel and J.-P. Serre. Since then the 2-ranks of compact Lie groups h…
Quantum invariant constructed for sutured 3-manifolds using Hopf superalgebra.
problem Quantum invariants for balanced sutured 3-manifolds with Spinc structure. method Involutive Hopf superalgebra H and Fox calculus to compute the invariant. result Invariant is a normalization of Reidemeister torsion when H is Borel subalgebra of Uq(gl(1∣1)). In this project we further investigate the idea of reducing the dimensionality of datasets using a Borel isomorphism with the purpose of subsequently applying supervised learning algorithms, as originally suggested by my supervisor V. Pestov (in 2011 Dagstuhl preprint). Any consistent learning algorithm, for example kN…
Study stabilizes Morse-Bott cohomology for equivariant manifolds.
problem Equivariant cohomology of manifolds with group actions.
method Stabilization technique to construct Morse-Bott functions.
result Realization of equivariant transversality and orientability.
Proves resurgent nature of a series solution to deformed Painlevé I equation.
problem Analyzing the resurgent nature of a series solution to the deformed Painlevé I equation.
method Proves resurgent nature through formal ℏ-power series solution and Borel summability. result Borel transform defines a global multivalued holomorphic function on a Fermat quintic surface.
Short proof shows Borel class stability under Cartan involution for 3-manifold groups.
problem Stability of Borel class under Cartan involution for 3-manifold representations.
method Direct proof for PGL(3,C) representations, leveraging a more general result for PGL(n,C).
result Borel class is preserved under Cartan involution up to sign for 3-manifold groups.
Geometrically proves WKB solutions of Schrödinger equations are resurgent.
problem Understanding resurgent behavior of WKB solutions on Riemann surfaces.
method Purely geometric approach using holomorphic Lie groupoids and spectral curves.
result Formal WKB solutions are Borel summable in almost all directions.
We describe an explicit semi-algebraic partition for the complement of a real hyperplane arrangement such that each piece is contractible and so that the pieces form a basis of Borel-Moore homology. We also give an explicit correspondence between the de Rham cohomology and the Borel-Moore homology.
Proves Borel Conjecture for certain 3D spaces.
problem Characterizing fundamental groups of 3D Alexandrov spaces.
method Analyzes properties of Alexandrov 3-spaces.
result Proves Borel Conjecture for specific types of spaces.
We introduce systems of objects and operators in linear monoidal categories called Ψ^-systems. A Ψ^-system satisfying several additional assumptions gives rise to a topological invariant of triples (a closed oriented 3-manifold M, a principal bundle over M, a link in M). This construction generalizes …
Classifies actions of groups on hyperbolic spaces, proving dichotomy.
problem Classifying actions of groups on hyperbolic spaces.
method Formalization using Borel equivalence relations, focusing on non-elementary actions without fixed points at infinity.
result For every countable group G, either all general type actions can be classified by an explicit invariant or they are unclassifiable in a strong sense. Study non-perturbative quantum geometry of string theories using finite difference equations and resurgence analysis.
problem Non-perturbative quantum geometry of open and closed topological string on the resolved conifold.
method Finite difference equations, resurgence analysis, exact WKB techniques.
result Identify 5d BPS states and relate spectral problems to quantum integrable systems.
We explain how the Harish-Chandra Plancherel Theorem and results in relative Lie algebra cohomology can be used in order to compute in a uniform way the L2-Betti numbers, the Novikov-Shubin invariants, and the L2-torsion of compact locally symmetric spaces thus completing results previously obtained by Borel, Lot…
The paper discusses rigidity results for inequalities on weighted Riemannian manifolds.
problem Rigidity of inequalities on weighted Riemannian manifolds.
method Theorems of rigidity on curvature and measure for the Borell-Brascamp-Lieb inequality, generalizing a theorem by Balogh and Kristály.
result A generalization of the curvature rigidity theorem to the weighted setting.
Research examines coamenable subgroups in higher rank groups.
problem Investigates coamenable normal subgroups in higher rank groups.
method Analyzes three complementary phenomena in higher rank groups.
result Growth indicators of coamenable subgroups are not preserved but the Riemannian critical exponent remains rigid.
Link homology compared with geometric link invariants using Bott-Samelson varieties.
problem Comparing different link homology theories with geometric link invariants.
method Using Khovanov-Rozansky homology and equivariant cohomology applied to Bott-Samelson varieties.
result Equivariant integral sl(n) link homology with specialized or universal potential.
Study proves hyperbolic groups have specific subgroup properties.
problem Characterizing subgroups of word hyperbolic groups.
method Analyzes Anosov representations into PGL(4q+2,R). result Affirmative answer to Sambarino's question for Borel Anosov representations.
A unified approach to geometric, symbol and deformation quantizations on a generalized flag manifold endowed with an invariant pseudo-Kaehler structure is proposed. The Hilbert space of states is realized via the Bott-Borel-Weil theorem in the sheaf cohomology of the geometric quantization line bundle. The correspondin…
We prove the Borel Conjecture for a class of groups containing word-hyperbolic groups and groups acting properly, isometrically and cocompactly on a finite dimensional CAT(0)-space.
We consider the family of harmonic measures on a lamination L of a compact space X by locally symmetric spaces L of noncompact type, i.e. L≃ΓL\G/K. We establish a natural bijection between these measures and the measures on an associated lamination foliated by G-orbits, $\hat{\mathc…
Study on Borel Anosov subgroups in SL(d,R) for d≠5,8k±1.
problem Characterizing Borel Anosov subgroups in SL(d,R).
method Analysis of antipodal subsets and quasi-isometric embeddings.
result Borel Anosov subgroups are virtually free or hyperbolic surface groups.
Quantum invariants of Seifert fibered homology spheres are resummated and related to classical Chern-Simons values.
problem Resummation and classification of quantum invariants for Seifert fibered homology spheres.
method Analysis of q-series and Ohtsuki series, asymptotic expansions, Borel transform, and classification of moduli spaces.
result The limit of the resummation equals the WRT quantum invariant and classifies components of the moduli space.
Proves summability of state integrals for specific hyperbolic knots.
problem Summability of perturbative series for hyperbolic knots.
method Algorithm to compute Borel-Laplace resummation as state integrals.
result Complete description of resurgent structure and explicit computations of Stokes constants.
Let M=G/Γ be a compact nilmanifold endowed with an invariant complex structure. We prove that, on an open set of any connected component of the moduli space C(g) of invariant complex structures on M, the Dolbeault cohomology of M is isomorphic to the one of the differential bigraded algebra ass…
The Kashaev invariants of 3-manifolds are based on 6j-symbols from the representation theory of the Weyl algebra, a Hopf algebra corresponding to the Borel subalgebra of $U_q(sl(2,\C))$. In this paper, we show that Kashaev's 6j-symbols are intertwining operators of local representations of quantum Teichmüller space…