Tautological relations via r-spin structures

Relations among tautological classes on the moduli space of stable curves are obtained via the study of Witten's r-spin theory for higher r. In order to calculate the quantum product, a new formula relating the r-spin correlators in genus 0 to the representation theory of sl2 is proven. The Givental-Teleman classification of CohFTs is used at two special semisimple points of the associated Frobenius manifold. At the first semisimple point, the R-matrix is exactly solved in terms of hypergeometric series. As a result, an explicit formula for Witten's r-spin class is obtained (along with tautological relations in higher degrees). As an application, the r=4 relations are used to bound the Betti numbers of the tautological ring of the moduli of nonsingular curves. At the second semisimple point, the form of the R-matrix implies a polynomiality property in r of Witten's r-spin class. In the Appendix (with F. Janda), a conjecture relating the r=0 limit of Witten's r-spin class to the class of the moduli space of holomorphic differentials is presented.Comment: Corrected powers of phi in the analysis of the second shift. Appendix on the moduli of holomorphic differentials by F. Janda, R. Pandharipande, A. Pixton, and D.Zvonkine. Final versio

Fuzzy splicing systems

In this paper we introduce a new variant of splicing systems, called fuzzy splicing systems, and establish some basic properties of language families generated by this type of splicing systems. We study the “fuzzy effect” on splicing operations, and show that the “fuzzification” of splicing systems can increase and decrease the computational power of splicing systems with finite components with respect to fuzzy operations and cut-points chosen for threshold languages

Holomorphic anomaly equations and the Igusa cusp form conjecture

Let $S$ be a K3 surface and let $E$ be an elliptic curve. We solve the reduced Gromov-Witten theory of the Calabi-Yau threefold $S \times E$ for all curve classes which are primitive in the K3 factor. In particular, we deduce the Igusa cusp form conjecture. The proof relies on new results in the Gromov-Witten theory of elliptic curves and K3 surfaces. We show the generating series of Gromov-Witten classes of an elliptic curve are cycle-valued quasimodular forms and satisfy a holomorphic anomaly equation. The quasimodularity generalizes a result by Okounkov and Pandharipande, and the holomorphic anomaly equation proves a conjecture of Milanov, Ruan and Shen. We further conjecture quasimodularity and holomorphic anomaly equations for the cycle-valued Gromov-Witten theory of every elliptic fibration with section. The conjecture generalizes the holomorphic anomaly equations for ellliptic Calabi-Yau threefolds predicted by Bershadsky, Cecotti, Ooguri, and Vafa. We show a modified conjecture holds numerically for the reduced Gromov-Witten theory of K3 surfaces in primitive classes.Comment: 68 page

On two-dimensional surface attractors and repellers on 3-manifolds

We show that if $f: M^3\to M^3$ is an $A$-diffeomorphism with a surface two-dimensional attractor or repeller $\mathcal B$ and $M^2_ \mathcal B$ is a supporting surface for $\mathcal B$, then $\mathcal B = M^2_{\mathcal B}$ and there is $k\geq 1$ such that: 1) $M^2_{\mathcal B}$ is a union $M^2_1\cup...\cup M^2_k$ of disjoint tame surfaces such that every $M^2_i$ is homeomorphic to the 2-torus $T^2$. 2) the restriction of $f^k$ to $M^2_i$ $(i\in\{1,...,k\})$ is conjugate to Anosov automorphism of $T^2$

Drip and Mate Operations Acting in Test Tube Systems and Tissue-like P systems

The operations drip and mate considered in (mem)brane computing resemble the operations cut and recombination well known from DNA computing. We here consider sets of vesicles with multisets of objects on their outside membrane interacting by drip and mate in two different setups: in test tube systems, the vesicles may pass from one tube to another one provided they fulfill specific constraints; in tissue-like P systems, the vesicles are immediately passed to specified cells after having undergone a drip or mate operation. In both variants, computational completeness can be obtained, yet with different constraints for the drip and mate operations

Accepting splicing systems with permitting and forbidding words

Abstract: In this paper we propose a generalization of the accepting splicingsystems introduced in Mitrana et al. (Theor Comput Sci 411:2414?2422,2010). More precisely, the input word is accepted as soon as a permittingword is obtained provided that no forbidding word has been obtained sofar, otherwise it is rejected. Note that in the new variant of acceptingsplicing system the input word is rejected if either no permitting word isever generated (like in Mitrana et al. in Theor Comput Sci 411:2414?2422,2010) or a forbidding word has been generated and no permitting wordhad been generated before. We investigate the computational power ofthe new variants of accepting splicing systems and the interrelationshipsamong them. We show that the new condition strictly increases thecomputational power of accepting splicing systems. Although there areregular languages that cannot be accepted by any of the splicing systemsconsidered here, the new variants can accept non-regular and even non-context-free languages, a situation that is not very common in the case of(extended) finite splicing systems without additional restrictions. We alsoshow that the smallest class of languages out of the four classes definedby accepting splicing systems is strictly included in the class of context-free languages. Solutions to a few decidability problems are immediatelyderived from the proof of this result

The sperm factor: paternal impact beyond genes

The fact that sperm carry more than the paternal DNA has only been discovered just over a decade ago. With this discovery, the idea that the paternal condition may have direct implications for the fitness of the offspring had to be revisited. While this idea is still highly debated, empirical evidence for paternal effects is accumulating. Male condition not only affects male fertility but also offspring early development and performance later in life. Several factors have been identified as possible carriers of non-genetic information, but we still know little about their origin and function and even less about their causation. I consider four possible non-mutually exclusive adaptive and non-adaptive explanations for the existence of paternal effects in an evolutionary context. In addition, I provide a brief overview of the main non-genetic components found in sperm including DNA methylation, chromatin modifications, RNAs and proteins. I discuss their putative functions and present currently available examples for their role in transferring non-genetic information from the father to the offspring. Finally, I identify some of the most important open questions and present possible future research avenues