Ratio coordinates for higher Teichm\"uller spaces

We define new coordinates for Fock-Goncharov's higher Teichm\"uller spaces for a surface with holes, which are the moduli spaces of representations of the fundamental group into a reductive Lie group $G$. Some additional data on the boundary leads to two closely related moduli spaces, the $\mathscr{X}$-space and the $\mathscr{A}$-space, forming a cluster ensemble. Fock and Goncharov gave nice descriptions of the coordinates of these spaces in the cases of $G = PGL_m$ and $G=SL_m$, together with Poisson structures. We consider new coordinates for higher Teichm\"uller spaces given as ratios of the coordinates of the $\mathscr{A}$-space for $G=SL_m$, which are generalizations of Kashaev's ratio coordinates in the case $m=2$. Using Kashaev's quantization for $m=2$, we suggest a quantization of the system of these new ratio coordinates, which may lead to a new family of projective representations of mapping class groups. These ratio coordinates depend on the choice of an ideal triangulation decorated with a distinguished corner at each triangle, and the key point of the quantization is to guarantee certain consistency under a change of such choices. We prove this consistency for $m=3$, and for completeness we also give a full proof of the presentation of Kashaev's groupoid of decorated ideal triangulations.Comment: 42 pages, 6 figure

Improving the Performance and Energy Efficiency of GPGPU Computing through Adaptive Cache and Memory Management Techniques

Department of Computer Science and EngineeringAs the performance and energy efficiency requirement of GPGPUs have risen, memory management techniques of GPGPUs have improved to meet the requirements by employing hardware caches and utilizing heterogeneous memory. These techniques can improve GPGPUs by providing lower latency and higher bandwidth of the memory. However, these methods do not always guarantee improved performance and energy efficiency due to the small cache size and heterogeneity of the memory nodes. While prior works have proposed various techniques to address this issue, relatively little work has been done to investigate holistic support for memory management techniques. In this dissertation, we analyze performance pathologies and propose various techniques to improve memory management techniques. First, we investigate the effectiveness of advanced cache indexing (ACI) for high-performance and energy-efficient GPGPU computing. Specifically, we discuss the designs of various static and adaptive cache indexing schemes and present implementation for GPGPUs. We then quantify and analyze the effectiveness of the ACI schemes based on a cycle-accurate GPGPU simulator. Our quantitative evaluation shows that ACI schemes achieve significant performance and energy-efficiency gains over baseline conventional indexing scheme. We also analyze the performance sensitivity of ACI to key architectural parameters (i.e., capacity, associativity, and ICN bandwidth) and the cache indexing latency. We also demonstrate that ACI continues to achieve high performance in various settings. Second, we propose IACM, integrated adaptive cache management for high-performance and energy-efficient GPGPU computing. Based on the performance pathology analysis of GPGPUs, we integrate state-of-the-art adaptive cache management techniques (i.e., cache indexing, bypassing, and warp limiting) in a unified architectural framework to eliminate performance pathologies. Our quantitative evaluation demonstrates that IACM significantly improves the performance and energy efficiency of various GPGPU workloads over the baseline architecture (i.e., 98.1% and 61.9% on average, respectively) and achieves considerably higher performance than the state-of-the-art technique (i.e., 361.4% at maximum and 7.7% on average). Furthermore, IACM delivers significant performance and energy efficiency gains over the baseline GPGPU architecture even when enhanced with advanced architectural technologies (e.g., higher capacity, associativity). Third, we propose bandwidth- and latency-aware page placement (BLPP) for GPGPUs with heterogeneous memory. BLPP analyzes the characteristics of a application and determines the optimal page allocation ratio between the GPU and CPU memory. Based on the optimal page allocation ratio, BLPP dynamically allocate pages across the heterogeneous memory nodes. Our experimental results show that BLPP considerably outperforms the baseline and state-of-the-art technique (i.e., 13.4% and 16.7%) and performs similar to the static-best version (i.e., 1.2% difference), which requires extensive offline profiling.clos

Selective permeability in gels: Beyond the solution-diffusion model

Permeability, a measure of potential transport of macromolecules through crowded media such as hydrogels, determines important control parameters in bio-soft functional material applications, e.g., for filtration, drug release, and transport of reactants in responsive nano-reactors. Tuning permeability is thus of great importance since it enables selective barrier crossings in molecular transport. We develop a model of semi- flexible cross-linked polymer gel networks by means of extensive coarse-grained simulations and scaling theories. The gel system consists of randomly formed tetra- functional network regions and also bulk regions where the macromolecular cosolutes diffuse in both regions, enabling a quantitative study of partitioning, diffusivity, and permeability. The gel undergoes a sharp volume transition upon changing inter- and intra-particle interactions, yielding a rich topology of the partitioning phase landscape which is highly correlated with the cosolute diffusivity. Moreover, we find that resultant permeability is largely maximized or minimized at an optimal gel density and inter-particle couplings between the networks and the cosolutes. This nontrivial phenomenon is triggered by a competition between partitioning and diffusion, resulting in a large anti-correlation. It is revealed that permeability can be highly selective by tuning the coupling interactions and the solvent quality. By applying external driving forces, we show this selectiveness of permeability beyond the linear response regime based on the solution-diffusion model. Finally we present scaling theories for partitioning, diffusion and thus permeability in crowded systems.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech. Vicerrectorado de Investigación de la UM

The dilogarithmic central extension of the Ptolemy-Thompson group via the Kashaev quantization

Quantization of universal Teichm\"uller space provides projective representations of the Ptolemy-Thompson group, which is isomorphic to the Thompson group $T$. This yields certain central extensions of $T$ by $\mathbb{Z}$, called dilogarithmic central extensions. We compute a presentation of the dilogarithmic central extension $\hat{T}^{Kash}$ of $T$ resulting from the Kashaev quantization, and show that it corresponds to $6$ times the Euler class in $H^2(T;\mathbb{Z})$. Meanwhile, the braided Ptolemy-Thompson groups $T^*$, $T^\sharp$ of Funar-Kapoudjian are extensions of $T$ by the infinite braid group $B_\infty$, and by abelianizing the kernel $B_\infty$ one constructs central extensions $T^*_{ab}$, $T^\sharp_{ab}$ of $T$ by $\mathbb{Z}$, which are of topological nature. We show $\hat{T}^{Kash}\cong T^\sharp_{ab}$. Our result is analogous to that of Funar and Sergiescu, who computed a presentation of another dilogarithmic central extension $\hat{T}^{CF}$ of $T$ resulting from the Chekhov-Fock(-Goncharov) quantization and thus showed that it corresponds to $12$ times the Euler class and that $\hat{T}^{CF} \cong T^*_{ab}$. In addition, we suggest a natural relationship between the two quantizations in the level of projective representations.Comment: 43 pages, 15 figures. v2: substantially revised from the first version, and the author affiliation changed. // v3: Groups M and T are shown to be anti-isomorphic (new Prop.2.32), which makes the whole construction more natural. And some minor changes // v4: reflects all changes made for journal publication (to appear in Adv. Math.

How a single stretched polymer responds coherently to a minute oscillation in fluctuating environments: An entropic stochastic resonance

Within the cell, biopolymers are often situated in constrained, fluid environments, e.g., cytoskeletal networks, stretched DNAs in chromatin. It is of paramount importance to understand quantitatively how they, utilizing their flexibility, optimally respond to a minute signal, which is, in general, temporally fluctuating far away from equilibrium. To this end, we analytically study viscoelastic response and associated stochastic resonance in a stretched single semi-flexible chain to an oscillatory force or electric field. Including hydrodynamic interactions between chain segments, we evaluate dynamics of the polymer extension in coherent response to the force or field. We find power amplification factor of the response at a noise-strength (temperature) can attain the maximum that grows as the chain length increases, indicative of an entropic stochastic resonance (ESR). In particular for a charged chain under an electric field, we find that the maximum also occurs at an optimal chain length, a new feature of ESR. The hydrodynamic interaction is found to enhance the power amplification, representing unique polymer cooperativity which the fluid background imparts despite its overdamping nature. For the slow oscillatory force, the resonance behavior is explained by the chain undulation of the longest wavelength. This novel ESR phenomenon suggests how a biopolymer self-organizes in an overdamping environment, utilizing its flexibility and thermal fluctuations