This paper proposes a general duality framework for the problem of minimizing a convex integral functional over a space of stochastic processes adapted to a given filtration. The framework unifies many well-known duality frameworks from operations research and mathematical finance. The unification allows the extension …
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This dissertation explores T-duality between hyperkähler structures and branes on algebraic integrable systems.
problem Investigates T-duality between hyperkähler structures and branes on algebraic integrable systems.
method Uses techniques of generalized geometry and Fourier-Mukai transform to show T-duality between semi-flat hyperkähler structures and generalized branes.
result Shows T-duality between semi-flat hyperkähler structures and generalized branes on algebraic integrable systems.
Establishing criteria for top cell inertness in complexes.
problem Criteria for top cell inertness in Poincaré duality complexes.
method Algebraic intersection theory, homotopy fibrations, surgery, homogeneous spaces.
result Established various criteria for top cell inertness.
Paper offers a dual formulation for consumption problem with multiplicative habit.
problem Optimal consumption with multiplicative habit formation.
method Dual formulation using Fenchel's Duality Theorem.
result Strong duality result linking primal and dual controls.
We study generalized electric/magnetic duality in Abelian gauge theory by combining techniques from locally covariant quantum field theory and Cheeger-Simons differential cohomology on the category of globally hyperbolic Lorentzian manifolds. Our approach generalizes previous treatments using the Hamiltonian formalism …
We present a new approach to Morse and Novikov theories, based on the deRham Federer theory of currents, using the finite volume flow technique of Harvey and Lawson. In the Morse case, we construct a noncompact analogue of the Morse complex, relating a Morse function to the cohomology with compact forward supports of t…
We consider a financial market where stocks are available for dynamic trading, and European and American options are available for static trading (semi-static trading strategies). We assume that the American options are infinitely divisible, and can only be bought but not sold. In the first part of the paper, we work w…
This note provides a novel, simple analysis of the method of conjugate gradients for the minimization of convex quadratic functions. In contrast with standard arguments, our proof is entirely self-contained and does not rely on the existence of Chebyshev polynomials. Another advantage of our development is that it clar…
The paper extends collective arbitrage concepts to multi-agent markets with cooperation.
problem Understanding collective market completeness and pricing in multi-agent systems.
method Develops new techniques and theorems to establish collective pricing-hedging duality and collective replication.
result Established a Second Fundamental Theorem of Asset Pricing in cooperative multi-agent settings.
The paper extends Thurston's method to new variants of Mather-Thurston theorem.
problem Proving new variants of Mather-Thurston theorem for PL homeomorphisms and contactomorphisms.
method Generalizing Thurston's technique to prove new variants of Mather-Thurston theorem for PL homeomorphisms and contactomorphisms.
result The paper answers questions posed by Gelfand-Fuks and Greenberg on PL foliations and Rybicki on contactomorphisms.
Let be a finite group and $\Y$ a -gerbe over an orbifold $\B$. A disconnected orbifold $\hat{\Y}$ and a flat U(1)-gerbe on $\hat{\Y}$ is canonically constructed from $\Y$. Motivated by a proposal in physics, we study a mathematical duality between the geometry of the -gerbe $\Y$ and the geometry of $\hat{…
Investor optimizes wealth in a market with non-traded endowment, deriving expansions up to second order.
problem Optimizing wealth in an incomplete financial market with a non-traded endowment.
method Duality techniques and Kunita-Watanabe projections for deriving expansions up to second order.
result Derives expansions of the primal value function and optimal wealth process up to second order with respect to the non-traded endowment units.
We develop a dual-control method for approximating investment strategies in incomplete environments that emerge from the presence of trading constraints. Convex duality enables the approximate technology to generate lower and upper bounds on the optimal value function. The mechanism rests on closed-form expressions per…
Duality in Graphical Modelsmath.PR
Graphical models have proven to be powerful tools for representing high-dimensional systems of random variables. One example of such a model is the undirected graph, in which lack of an edge represents conditional independence between two random variables given the rest. Another example is the bidirected graph, in whic…
Several recently proposed architectures of neural networks such as ResNeXt, Inception, Xception, SqueezeNet and Wide ResNet are based on the designing idea of having multiple branches and have demonstrated improved performance in many applications. We show that one cause for such success is due to the fact that the mul…
Torsion sensitive intersection homology was introduced to unify several versions of Poincare duality for stratified spaces into a single theorem. This unified duality theorem holds with ground coefficients in an arbitrary PID and with no local cohomology conditions on the underlying space. In this paper we consider for…
Proves Poincaré duality for Hopf algebroids with bijective antipode.
problem Proving Poincaré duality for Hopf algebroids.
method Using twisted Poincaré duality and bijective antipode properties.
result Recovering and extending known Poincaré dualities for Hopf algebroids.
Study utility indifference pricing with delayed investment information in a Bachelier model.
problem Investment decisions based on delayed information in a Bachelier model.
method Developed discrete-time duality and used techniques from [7] to compute scaling limits.
result Utility indifference prices scaling limit for vanishing delay with quadratic penalty.
Fivebranes and Knotshep-th
We develop an approach to Khovanov homology of knots via gauge theory (previous physics-based approches involved other descriptions of the relevant spaces of BPS states). The starting point is a system of D3-branes ending on an NS5-brane with a nonzero theta-angle. On the one hand, this system can be related to a Chern…
We introduce the symplectic twistor operator in symplectic spin geometry, as a symplectic analogue of the twistor operator in Riemannian spin geometry. We focus on the real dimension 2 and compute the space of its solutions on . Our analysis is based on the techniques of metaplectic Howe duality.
The technique of generating families produces obstructions to the existence of embedded Lagrangian cobordisms between Legendrian submanifolds in the symplectizations of 1-jet bundles. In fact, generating families may be used to construct a TQFT-like theory that, in addition to giving the aforementioned obstructions, yi…
New proof of chain duality for simplicial complexes.
problem Proving the existence of chain duality for chain complexes over simplicial complexes.
method Geometric and conceptual treatment of chain duality.
result Fundamental for Ranicki's surgery exact sequence.
Introduces Kähler duality between domains in complex space.
problem None explicitly stated; focuses on concept introduction.
method None explicitly stated; focuses on concept introduction.
result Introduces Kähler duality between domains in complex space.
Research on dualities in geometric stereotypes.
problem Understanding dualities in geometric stereotypes.
method Continuation of previous research on stereotype spaces and algebras.
result New insights into geometric stereotype dualities.
Duality restored in gauge theory, gravity, and string theory models.
problem Restoring duality invariance in theories coupled to matter.
method Extending phase space to allow for violations of the algebraic Bianchi identity and considering the axion as the duality current.
result Duality current in NS-NS gravity is the divergence of the axion.
Unified proof of four Bavard dualities and new results on quasimorphisms.
problem Comparing stable commutator length and quasimorphisms on groups.
method Expository account and new strengthening of Bavard duality, providing complete proofs.
result Generalized mixed Bavard duality, recovering all previous dualities.
Our main objective is to demonstrate how homological perturbation theory (HPT) results over the last 40 years immediately or with little extra work give some of the Koszul duality results that have appeared in the last decade. Higher homotopies typically arise when a huge object, e. g. a chain complex defining various …
Verma Howe duality connects tensor products of Verma modules to LKB representations.
problem Understanding the relationship between tensor products of Verma modules and LKB representations.
method Established a quantized version of Verma Howe duality and used it to prove the simplicity of LKB representations.
result LKB representations arise from the quantized Verma Howe duality and are shown to be simple modules.
Cohomological and homological spectral sequences are shown to be isomorphic.
problem Cohomological and homological Atiyah-Hirzebruch spectral sequences are not always isomorphic.
method Spanier-Whitehead duality is used to establish an isomorphism between the two spectral sequences.
result Cohomological and homological Atiyah-Hirzebruch spectral sequences are isomorphic for finite spectra.
This work applies Double Field Theory to four-dimensional manifolds, revealing connections to integrability and twistor theory.
problem Understanding dualities in string theory and their geometrical structures.
method Generalized and para-Hermitian geometry applied to four-dimensional manifolds.
result Close relationship between para-Hermitian structures in Double Field Theory and algebraically special solutions to Einstein equations.
The paper proves T-duality and Hori formulae for winding loop spaces.
problem Realizing T-duality and Hori formulae for loop spaces.
method Proving T-duality and Hori formulae for winding q-loop spaces.
result T-duality and Hori formulae for winding q-loop spaces are proven.
The thesis examines stochastic calculus in option pricing with logistic models and numerical methods.
problem Exploring the application of stochastic calculus in option pricing.
method Monte-Carlo Simulation and machine learning algorithms.
result Insights from Peter Carr and Lorenzo Torricelli's convex duality in continuous models.
Optimal consumption from investment and random endowment in incomplete semimartingale marketsq-fin.PM
We consider the problem of maximizing expected utility from consumption in a constrained incomplete semimartingale market with a random endowment process, and establish a general existence and uniqueness result using techniques from convex duality. The notion of asymptotic elasticity of Kramkov and Schachermayer is ext…
Equivariant T-duality connects bundles with twists.
problem Establishing a relationship between bundles with twists.
method Formulating T-duality in equivariant K-theory for compact Lie group actions.
result T-duality is an isomorphism in equivariant K-theory for compact Lie group actions.
We consider a discrete time financial market with proportional transaction costs under model uncertainty, and study a numéraire-based semi-static utility maximization problem with an exponential utility preference. The randomization techniques recently developed in \cite{BDT17} allow us to transform the original proble…
We give the definition of a duality that is applicable to arbitrary -forms. The operator that defines the duality depends on a fixed form . Our definition extends in a very natural way the Hodge duality of -forms in dimensional spaces and the generalized duality of two-forms. We discuss the properties of …
The paper establishes T-duality for 2D σ-models with H-flux.
problem T-duality for 2D σ-models with H-flux.
method Localization and graded T-duality map (graded Hori morphism).
result Establishes the most general version of T-duality for Type II String Theory.
QP perspective on Poisson-Lie T-duality topology changes.
problem Understanding Poisson-Lie T-duality through QP manifolds.
method QP manifolds and canonical transformations for symplectic reductions.
result Canonical transformations mediate Poisson-Lie T-duality.
Unified framework for T-duality in both trivial and non-trivial topologies.
problem Unified description of T-duality for metrics and B-fields in non-trivial topology.
method Developed a new unifying framework for T-duality.
result Unified description of T-duality for metrics and B-fields in non-trivial topology.
Geometric duality connects graph isomorphism and knot equivalence.
problem Understanding the equivalence of graph isomorphism and knot equivalence.
method Observation of geometric duality in planar graphs and links.
result The equivalence relation defined by isomorphisms of checkerboard graphs is the same as 2-isomorphisms of checkerboard graphs.
Koszul duality for manifold modules proven.
problem Proving Koszul self duality of manifold modules.
method Using generalized Thom complexes and operads in Top.
result Koszul self duality of little disk modules proven.
One point compactifications of configuration spaces and the self duality of the little disks operadmath.AT
Maps self-duality in little disks operad to framed manifolds.
problem Self-duality of little disks operad.
method Configuration space level Pontryagin--Thom constructions.
result Existence of compatible self-duality map for framed manifolds.
-duality on nilmanifoldsmath.DG
We study generalized complex structures and -duality (in the sense of Bouwknegt, Evslin, Hannabuss and Mathai) on Lie algebras and construct the corresponding Cavalcanti and Gualtieri map. Such a construction is called "Infinitesimal -duality". As an application we deal with the problem of finding symplectic stru…
New spherical T-duality for higher degree forms in fiber bundles.
problem Extending T-duality to higher degree forms in fiber bundles.
method Generalizing T-duality to -bundles with closed odd forms of arbitrary degree.
result Existence and isomorphic twisted cohomology of -dual spaces.
The paper establishes a duality between non-compact and compact symmetric pairs.
problem Understanding the relationship between non-compact and compact symmetric pairs.
method Developed a duality theorem between non-compact pseudo-Riemannian semisimple symmetric pairs and commutative compact semisimple symmetric triads.
result Explicit description of a one-to-one correspondence between non-compact and compact symmetric pairs.
T-Duality in Superspacehep-th
We find the T-duality transformation rules for 2-dimensional (2,1) supersymmetric sigma-models in (2,1) superspace. Our results clarify certain aspects of the (2,1) sigma model geometry relevant to the discussion of T-duality. The complexified duality transformations we find are equivalent to the usual Buscher duality …
We study discrete group actions on coarse Poincare duality spaces, e.g. acyclic simplicial complexes which admit free cocompact group actions by Poincare duality groups. When G is an (n-1) dimensional duality group and X is a coarse Poincare duality space of formal dimension n, then a free simplicial action of G on X d…
The paper shows plentiful non-homotopy finite Poincaré duality spaces.
problem The existence of non-homotopy finite Poincaré duality spaces.
method Constructing a finitely dominated Poincaré space with a non-trivial 2-divisible element in the reduced Grothendieck group.
result The existence of finitely dominated Poincaré spaces that are not homotopy finite.