Mixed network calculus

We show how to combine higgsed topological vertices introduced in [7] with conventional refined topological vertices. We demonstrate that the extended formalism describes very general interacting D5-NS5-D3 brane systems. In particular, we introduce new types of intertwining operators of Ding-Iohara-Miki algebra between different types of Fock representations corresponding to the crossings of NS5 and D5 branes. As a byproduct we obtain an algebraic description of the Hanany-Witten brane creation effect, give an efficient recipe to compute the brane factors in 3d$$ \mathcal{N} $$ N = 2 and $$ \mathcal{N} $$ N = 4 quiver gauge theories and demonstrate how 3d S-duality appears in our setup.

On PDE Characterization of Smooth Hierarchical Functions Computed by Neural Networks

Neural networks are versatile tools for computation, having the ability to approximate a broad range of functions. An important problem in the theory of deep neural networks is expressivity; that is, we want to understand the functions that are computable by a given network. We study real, infinitely differentiable (smooth) hierarchical functions implemented by feedforward neural networks via composing simpler functions in two cases: (1) each constituent function of the composition has fewer in puts than the resulting function and (2) constituent functions are in the more specific yet prevalent form of a nonlinear univariate function (e.g., tanh) applied to a linear multivariate function. We establish that in each of these regimes, there exist nontrivial algebraic partial differential equations (PDEs) that are satisfied by the computed functions. These PDEs are purely in terms of the partial derivatives and are dependent only on the topology of the network. Conversely, we conjecture that such PDE constraints, once accompanied by appropriate nonsingularity conditions and perhaps certain inequalities involving partial derivatives, guarantee that the smooth function under consideration can be represented by the network. The conjecture is verified in numerous examples, including the case of tree architectures, which are of neuroscientific interest. Our approach is a step toward formulating an algebraic description of functional spaces associated with specific neural networks, and may provide useful new tools for constructing neural networks.

Projective Metric Geometry and Clifford Algebras

Each vector space that is endowed with a quadratic form determines its Clifford algebra. This algebra, in turn, contains a distinguished group, known as the Lipschitz group. We show that only a quotient of this group remains meaningful in the context of projective metric geometry. This quotient of the Lipschitz group can be viewed as a point set in the projective space on the Clifford algebra and, under certain restrictions, leads to an algebraic description of so-called kinematic mappings.

Deformations of circle-valued Morse functions on 2-torus

In this paper we give an algebraic description of fundamental groups of orbits of circle-valued Morse functions on T2 with respect to the action of the group of diffeomorphisms of T2

Properties of implication in effect algebras

Effect algebras form a formal algebraic description of the structure of the so-called effects in a Hilbert space which serve as an event-state space for effects in quantum mechanics. This is why effect algebras are considered as logics of quantum mechanics, more precisely as an algebraic semantics of these logics. Because every productive logic is equipped with implication, we introduce here such a concept and demonstrate its properties. In particular, we show that this implication is connected with conjunction via a certain “unsharp” residuation which is formulated on the basis of a strict unsharp residuated poset. Though this structure is rather complicated, it can be converted back into an effect algebra and hence it is sound. Further, we study the Modus Ponens rule for this implication by means of so-called deductive systems and finally we study the contraposition law.

Twisted circle compactifications of 6d SCFTs

We study 6d superconformal field theories (SCFTs) compactified on a circle with arbitrary twists. The theories obtained after compactification, often referred to as 5d Kaluza-Klein (KK) theories, can be viewed as starting points for RG flows to 5d SCFTs. According to a conjecture, all 5d SCFTs can be obtained in this fashion. We compute the Coulomb branch prepotential for all 5d KK theories obtainable in this manner and associate to these theories a smooth local genus one fibered Calabi-Yau threefold in which is encoded information about all possible RG flows to 5d SCFTs. These Calabi-Yau threefolds provide hitherto unknown M-theory duals of F-theory configurations compactified on a circle with twists. For certain exceptional KK theories that do not admit a standard geometric description we propose an algebraic description that appears to retain the properties of the local Calabi-Yau threefolds necessary to determine RG flows to 5d SCFTs, along with other relevant physical data.

Homological epimorphisms, homotopy epimorphisms and acyclic maps

We show that the notions of homotopy epimorphism and homological epimorphism in the category of differential graded algebras are equivalent. As an application we obtain a characterization of acyclic maps of topological spaces in terms of induced maps of their chain algebras of based loop spaces. In the case of a universal acyclic map we obtain, for a wide class of spaces, an explicit algebraic description for these induced maps in terms of derived localization.

Design of Symmetric-Key Primitives for Advanced Cryptographic Protocols

While traditional symmetric algorithms like AES and SHA-3 are optimized for efficient hardware and software implementations, a range of emerging applications using advanced cryptographic protocols such as multi-party computation and zero knowledge proofs require optimization with respect to a different metric: arithmetic complexity.In this paper we study the design of secure cryptographic algorithms optimized to minimize this metric. We begin by identifying the differences in the design space between such arithmetization-oriented ciphers and traditional ones, with particular emphasis on the available tools, efficiency metrics, and relevant cryptanalysis. This discussion highlights a crucial point—the considerations for designing arithmetization-oriented ciphers are oftentimes different from the considerations arising in the design of software- and hardware-oriented ciphers.The natural next step is to identify sound principles to securely navigate this new terrain, and to materialize these principles into concrete designs. To this end, we present the Marvellous design strategy which provides a generic way to easily instantiate secure and efficient algorithms for this emerging domain. We then show two examples for families following this approach. These families — Vision and Rescue — are benchmarked with respect to three use cases: the ZK-STARK proof system, proof systems based on Rank-One Constraint Satisfaction (R1CS), and Multi-Party Computation (MPC). These benchmarks show that our algorithms achieve a highly compact algebraic description, and thus benefit the advanced cryptographic protocols that employ them.

Towards fast one-block quantifier elimination through generalised critical values

One-block quantifier elimination is comprised of computing a semi-algebraic description of the projection of a semi-algebraic set or of deciding the truth of a semi-algebraic formula with a single quantifier.