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## Geometrical semantics for linear logic (multiplicative fragment)

Theoretical Computer Science. 2006. Vol. 357. No. 1-3. P. 215-229.

Slavnov S. A., Annals of Pure and Applied Logic 2005 Vol. 131 No. 1-3 P. 177-225

Added: March 4, 2013

Sergey Slavnov, Annals of Pure and Applied Logic 2014 Vol. 165 No. 1 P. 357-370

Just as intuitionistic proofs can be modeled by functions, linear logic proofs, being symmetric in the inputs and outputs, can be modeled by relations (for example, cliques in coherence spaces). However generic relations do not establish any functional dependence between the arguments, and therefore it is questionable whether they can be thought as reasonable generalizations ...

Added: October 7, 2013

Sergey Slavnov, Mathematical Structures in Computer Science 2021 Vol. 31 No. 5 P. 495-534

Ehrhard et al. (2018. Proceedings of the ACM on Programming Languages, POPL 2, Article 59.) proposed a model of probabilistic functional programming in a category of normed positive cones and stable measurable cone maps, which can be seen as a coordinate-free generalization of probabilistic coherence spaces (PCSs). However, unlike the case of PCSs, it remained unclear ...

Added: November 16, 2021

Blute R., Panangaden P., Slavnov S. A., Applied Categorical Structures 2012 Vol. 20 No. 3 P. 209-228

This paper proposes a definition of categorical model of the deep inference system BV, defined by Guglielmi. Deep inference introduces the idea of performing a deduction in the interior of a formula, at any depth. Traditional sequent calculus rules only see the roots of formulae. However in these new systems, one can rewrite at any ...

Added: December 28, 2012

Galkin S., Golyshev V., Iritani H., Duke Mathematical Journal 2016 Vol. 165 No. 11 P. 2005-2077

We propose Gamma Conjectures for Fano manifolds which can be thought of as a square root of the index theorem. Studying the exponential asymptotics of solutions to the quantum differential equation, we associate a principal asymptotic class A_F to a Fano manifold F. We say that F satisfies Gamma Conjecture I if A_F equals the ...

Added: November 18, 2014

Entov M., Verbitsky M., Full symplectic packing for tori and hyperkahler manifolds / Cornell University. Series math "arxiv.org". 2014.

Let M be a closed symplectic manifold of volume V. We say that M admits a full symplectic packing by balls if any collection of symplectic balls of total volume less than V admits a symplectic embedding to M. In 1994 McDuff and Polterovich proved that symplectic packings of Kahler manifolds can be characterized in ...

Added: February 5, 2015

Verbitsky M., Communications in Mathematical Physics 2013 Vol. 324 No. 1 P. 173-177

Let M be an almost complex manifold equipped with a Hermitian form such that its de Rham differential has Hodge type (3,0)+(0,3), for example a nearly Kahler manifold. We prove that any connected component of the moduli space of pseudoholomorphic curves on M is compact. This can be used to study pseudoholomorphic curves on a ...

Added: February 16, 2013

Springer, 2013

Vladimir Arnold was one of the great mathematical scientists of our time. He is famous for both the breadth and the depth of his work. At the same time he is one of the most prolific and outstanding mathematical authors. This second volume of his Collected Works focuses on hydrodynamics, bifurcation theory, and algebraic geometry. ...

Added: February 20, 2013

Abouzaid M., Auroux D., Efimov Alexander I. et al., Journal of the American Mathematical Society 2013 Vol. 26 No. 4 P. 1051-1083

We prove that the wrapped Fukaya category of a punctured sphere ($ S^{2}$ with an arbitrary number of points removed) is equivalent to the triangulated category of singularities of a mirror Landau-Ginzburg model, proving one side of the homological mirror symmetry conjecture in this case. By investigating fractional gradings on these categories, we conclude that ...

Added: October 31, 2013

Slavnov S. A., Mathematical Structures in Computer Science 2005 Vol. 15 No. 6 P. 1151-1178

Added: March 3, 2013

Galkin S., Golyshev V., Iritani H., Gamma classes and quantum cohomology of Fano manifolds: Gamma conjectures / Cornell University. Series math "arxiv.org". 2014. No. 1404.6407.

We propose Gamma Conjectures for Fano manifolds which can be thought of as a square root of the index theorem. Studying the exponential asymptotics of solutions to the quantum differential equation, we associate a principal asymptotic class A_F to a Fano manifold F. We say that F satisfies Gamma Conjecture I if A_F equals the ...

Added: May 4, 2014

V. V. Shevchishin, Izvestiya. Mathematics 2009 Vol. 73 No. 4 P. 797-859

In this paper we prove the non-existence of Lagrangian embeddings of the Klein bottle K in R4 and CP2. We exploit the existence of a special embedding of K in a symplectic Lefschetz pencil pr:X→S2 and study its monodromy. As the main technical tool, we develop the combinatorial theory of mapping class groups. The results ...

Added: March 18, 2013

Sergey Slavnov, On partial traces and compactification of *-autonomous Mix-categories / Cornell University. Series arXiv "math". 2016.

We study the question when a $*$-autonomous Mix-category has a representation as a $*$-autonomous Mix-subcategory of a compact one. We define certain partial trace-like operation on morphisms of a Mix-category, which we call a mixed trace, and show that any structure preserving embedding of a Mix-category into a compact one induces a mixed trace on ...

Added: November 23, 2016

49606783, Mathematical notes 2014 Vol. 96 No. 6 P. 977-982

We give a geometric interpretation of the thermodynamic potential, free and internal energy, and enthalpy in terms of a Lagrangian manifold in the phase space of pairs (T, -S), (-mu, N), and (P,V) of intensive and extensive variables.
The Lagrangian manifold is viewed as the dequantization of the tunnel canonical operator. With this approach, the critical point ...

Added: December 24, 2014

Sergey Slavnov, Journal of Cognitive Science 2021 Vol. 22 No. 2 P. 68-91

We propose a concrete surface representation of abstract categorial grammars in the category of word cobordisms or cowordisms
for short, which are certain bipartite graphs decorated withwords in a given alphabet, generalizing linear logic proof-nets.
We also introduce and study linear logic grammars, directly based on cobordisms and using classical multiplicative linear logic
as a typing system. ...

Added: October 2, 2021

Slavnov S. A., Journal of Logic and Computation 2022 Vol. 32 No. 3 P. 479-517

We consider tensor grammars, which are an example of ‘commutative’ grammars, based on the classical (rather than intuitionistic) linear logic. They can be seen as a surface representation of abstract categorial grammars (ACG) in the sense that derivations of ACG translate to derivations of tensor grammars and this translation is isomorphic on the level of string ...

Added: October 21, 2021

49606783, Nazaikinskii V. E., Mathematical notes 2016 Vol. 100 No. 3 P. 421-428

For an arithmetic semigroup (G, ∂), we define entropy as a function on a naturally defined continuous semigroup Ĝ containing G. The construction is based on conditional maximization, which permits us to introduce the conjugate variables and the Lagrangian manifold corresponding to the semigroup (G, ∂). ...

Added: December 10, 2016

Vsevolod Shevchishin, Secondary Stiefel-Whitney class and diffeomorphisms of rational and ruled symplectic 4-manifolds / Cornell University. Series math "arxiv.org". 2010.

We introduce the secondary Stiefel-Whitney class $\tilde w_2$ of homotopically trivial diffeomorphisms and show that a homotopically trivial symplectomorphism of a ruled 4-manifold is isotopic to identity if and only if the class $\tilde w_2$ vanishes.
Using this, we give a detailed description of the combinatorial structure of the diffeotopy group of ruled symplectic 4-manifolds ...

Added: March 18, 2013

Pushkar P. E., Chekanov-type theorem for spherized cotangent bundles / Cornell University. Series arXiv "math". 2016. No. arXiv:1602.08743.

We prove a Chekanov-type theorem for the spherization of the cotangent bundle ST∗B of a closed manifold B. It claims that for Legendrian submanifolds in ST∗B the property "to be given by a generating family quadratic at infinity" persists under Legendrian isotopies. ...

Added: December 7, 2016

Pushkar P. E., On Hamiltonian and contact isotopy liftings / Cornell University. Series arXiv "math". 2016. No. arXiv:1602.07948.

We construct counterexamples to lifting properties of Hamiltonian and contact isotopies ...

Added: December 7, 2016

Kanovich M., Kuznetsov S., Scedrov A., Journal of Logic and Computation 2020 Vol. 30 No. 1 P. 239-256

The Lambek calculus can be considered as a version of non-commutative intuitionistic linear logic. One of the interesting features of the Lambek calculus is the so-called ‘Lambek’s restriction’, i.e. the antecedent of any provable sequent should be non-empty. In this paper, we discuss ways of extending the Lambek calculus with the linear logic exponential modality ...

Added: July 1, 2020

49606783, Mathematical notes 2017 Vol. 102 No. 4 P. 583-586

Added: November 17, 2018

Blute R., Panangaden P., Slavnov Sergey, Applied Categorical Structures 2012 Vol. 20 No. 3 P. 209-228

This paper proposes a definition of categorical model of the deep inference system BV, defined by Guglielmi. Deep inference introduces the idea of performing a deduction in the interior of a formula, at any depth. Traditional sequent calculus rules only see the roots of formulae. However in these new systems, one can rewrite at any ...

Added: February 18, 2013

Lekili Y., Polishchuk A., Compositio Mathematica 2020 Vol. 156 No. 7 P. 1310-1347

Using Auroux's description of Fukaya categories of symmetric products of punctured surfaces, we compute the partially wrapped Fukaya category of the complement of k+1 generic hyperplanes in ℂℙ^n, for k≥n, with respect to certain stops in terms of the endomorphism algebra of a generating set of objects. The stops are chosen so that the resulting algebra is formal. In ...

Added: August 12, 2020