TY - JOUR AB - X and Y chromosomes can diverge when rearrangements block recombination between them. Here we present the first genomic view of a reciprocal translocation that causes two physically unconnected pairs of chromosomes to be coinherited as sex chromosomes. In a population of the common frog (Rana temporaria), both pairs of X and Y chromosomes show extensive sequence differentiation, but not degeneration of the Y chromosomes. A new method based on gene trees shows both chromosomes are sex‐linked. Furthermore, the gene trees from the two Y chromosomes have identical topologies, showing they have been coinherited since the reciprocal translocation occurred. Reciprocal translocations can thus reshape sex linkage on a much greater scale compared with inversions, the type of rearrangement that is much better known in sex chromosome evolution, and they can greatly amplify the power of sexually antagonistic selection to drive genomic rearrangement. Two more populations show evidence of other rearrangements, suggesting that this species has unprecedented structural polymorphism in its sex chromosomes. AU - Toups, Melissa A AU - Rodrigues, Nicolas AU - Perrin, Nicolas AU - Kirkpatrick, Mark ID - 7421 IS - 8 JF - Molecular Ecology SN - 0962-1083 TI - A reciprocal translocation radically reshapes sex‐linked inheritance in the common frog VL - 28 ER - TY - CONF AB - Proofs of sequential work (PoSW) are proof systems where a prover, upon receiving a statement χ and a time parameter T computes a proof ϕ(χ,T) which is efficiently and publicly verifiable. The proof can be computed in T sequential steps, but not much less, even by a malicious party having large parallelism. A PoSW thus serves as a proof that T units of time have passed since χ was received. PoSW were introduced by Mahmoody, Moran and Vadhan [MMV11], a simple and practical construction was only recently proposed by Cohen and Pietrzak [CP18]. In this work we construct a new simple PoSW in the random permutation model which is almost as simple and efficient as [CP18] but conceptually very different. Whereas the structure underlying [CP18] is a hash tree, our construction is based on skip lists and has the interesting property that computing the PoSW is a reversible computation. The fact that the construction is reversible can potentially be used for new applications like constructing proofs of replication. We also show how to “embed” the sloth function of Lenstra and Weselowski [LW17] into our PoSW to get a PoSW where one additionally can verify correctness of the output much more efficiently than recomputing it (though recent constructions of “verifiable delay functions” subsume most of the applications this construction was aiming at). AU - Abusalah, Hamza M AU - Kamath Hosdurg, Chethan AU - Klein, Karen AU - Pietrzak, Krzysztof Z AU - Walter, Michael ID - 7411 SN - 0302-9743 T2 - Advances in Cryptology – EUROCRYPT 2019 TI - Reversible proofs of sequential work VL - 11477 ER - TY - JOUR AB - Background Synaptic vesicles (SVs) are an integral part of the neurotransmission machinery, and isolation of SVs from their host neuron is necessary to reveal their most fundamental biochemical and functional properties in in vitro assays. Isolated SVs from neurons that have been genetically engineered, e.g. to introduce genetically encoded indicators, are not readily available but would permit new insights into SV structure and function. Furthermore, it is unclear if cultured neurons can provide sufficient starting material for SV isolation procedures. New method Here, we demonstrate an efficient ex vivo procedure to obtain functional SVs from cultured rat cortical neurons after genetic engineering with a lentivirus. Results We show that ∼108 plated cortical neurons allow isolation of suitable SV amounts for functional analysis and imaging. We found that SVs isolated from cultured neurons have neurotransmitter uptake comparable to that of SVs isolated from intact cortex. Using total internal reflection fluorescence (TIRF) microscopy, we visualized an exogenous SV-targeted marker protein and demonstrated the high efficiency of SV modification. Comparison with existing methods Obtaining SVs from genetically engineered neurons currently generally requires the availability of transgenic animals, which is constrained by technical (e.g. cost and time) and biological (e.g. developmental defects and lethality) limitations. Conclusions These results demonstrate the modification and isolation of functional SVs using cultured neurons and viral transduction. The ability to readily obtain SVs from genetically engineered neurons will permit linking in situ studies to in vitro experiments in a variety of genetic contexts. AU - Mckenzie, Catherine AU - Spanova, Miroslava AU - Johnson, Alexander J AU - Kainrath, Stephanie AU - Zheden, Vanessa AU - Sitte, Harald H. AU - Janovjak, Harald L ID - 7406 JF - Journal of Neuroscience Methods SN - 0165-0270 TI - Isolation of synaptic vesicles from genetically engineered cultured neurons VL - 312 ER - TY - CONF AB - Most of today's distributed machine learning systems assume reliable networks: whenever two machines exchange information (e.g., gradients or models), the network should guarantee the delivery of the message. At the same time, recent work exhibits the impressive tolerance of machine learning algorithms to errors or noise arising from relaxed communication or synchronization. In this paper, we connect these two trends, and consider the following question: Can we design machine learning systems that are tolerant to network unreliability during training? With this motivation, we focus on a theoretical problem of independent interest-given a standard distributed parameter server architecture, if every communication between the worker and the server has a non-zero probability p of being dropped, does there exist an algorithm that still converges, and at what speed? The technical contribution of this paper is a novel theoretical analysis proving that distributed learning over unreliable network can achieve comparable convergence rate to centralized or distributed learning over reliable networks. Further, we prove that the influence of the packet drop rate diminishes with the growth of the number of parameter servers. We map this theoretical result onto a real-world scenario, training deep neural networks over an unreliable network layer, and conduct network simulation to validate the system improvement by allowing the networks to be unreliable. AU - Yu, Chen AU - Tang, Hanlin AU - Renggli, Cedric AU - Kassing, Simon AU - Singla, Ankit AU - Alistarh, Dan-Adrian AU - Zhang, Ce AU - Liu, Ji ID - 7437 SN - 9781510886988 T2 - 36th International Conference on Machine Learning, ICML 2019 TI - Distributed learning over unreliable networks VL - 2019-June ER - TY - JOUR AB - We develop a framework for the rigorous analysis of focused stochastic local search algorithms. These algorithms search a state space by repeatedly selecting some constraint that is violated in the current state and moving to a random nearby state that addresses the violation, while (we hope) not introducing many new violations. An important class of focused local search algorithms with provable performance guarantees has recently arisen from algorithmizations of the Lovász local lemma (LLL), a nonconstructive tool for proving the existence of satisfying states by introducing a background measure on the state space. While powerful, the state transitions of algorithms in this class must be, in a precise sense, perfectly compatible with the background measure. In many applications this is a very restrictive requirement, and one needs to step outside the class. Here we introduce the notion of measure distortion and develop a framework for analyzing arbitrary focused stochastic local search algorithms, recovering LLL algorithmizations as the special case of no distortion. Our framework takes as input an arbitrary algorithm of such type and an arbitrary probability measure and shows how to use the measure as a yardstick of algorithmic progress, even for algorithms designed independently of the measure. AU - Achlioptas, Dimitris AU - Iliopoulos, Fotis AU - Kolmogorov, Vladimir ID - 7412 IS - 5 JF - SIAM Journal on Computing SN - 0097-5397 TI - A local lemma for focused stochastical algorithms VL - 48 ER - TY - JOUR AB - Multiple importance sampling (MIS) has become an indispensable tool in Monte Carlo rendering, widely accepted as a near-optimal solution for combining different sampling techniques. But an MIS combination, using the common balance or power heuristics, often results in an overly defensive estimator, leading to high variance. We show that by generalizing the MIS framework, variance can be substantially reduced. Specifically, we optimize one of the combined sampling techniques so as to decrease the overall variance of the resulting MIS estimator. We apply the approach to the computation of direct illumination due to an HDR environment map and to the computation of global illumination using a path guiding algorithm. The implementation can be as simple as subtracting a constant value from the tabulated sampling density done entirely in a preprocessing step. This produces a consistent noise reduction in all our tests with no negative influence on run time, no artifacts or bias, and no failure cases. AU - Karlík, Ondřej AU - Šik, Martin AU - Vévoda, Petr AU - Skrivan, Tomas AU - Křivánek, Jaroslav ID - 7418 IS - 6 JF - ACM Transactions on Graphics SN - 0730-0301 TI - MIS compensation: Optimizing sampling techniques in multiple importance sampling VL - 38 ER - TY - JOUR AB - We consider Bose gases consisting of N particles trapped in a box with volume one and interacting through a repulsive potential with scattering length of order N−1 (Gross–Pitaevskii regime). We determine the ground state energy and the low-energy excitation spectrum, up to errors vanishing as N→∞. Our results confirm Bogoliubov’s predictions. AU - Boccato, Chiara AU - Brennecke, Christian AU - Cenatiempo, Serena AU - Schlein, Benjamin ID - 7413 IS - 2 JF - Acta Mathematica SN - 0001-5962 TI - Bogoliubov theory in the Gross–Pitaevskii limit VL - 222 ER - TY - JOUR AB - The study of parallel ecological divergence provides important clues to the operation of natural selection. Parallel divergence often occurs in heterogeneous environments with different kinds of environmental gradients in different locations, but the genomic basis underlying this process is unknown. We investigated the genomics of rapid parallel adaptation in the marine snail Littorina saxatilis in response to two independent environmental axes (crab-predation versus wave-action and low-shore versus high-shore). Using pooled whole-genome resequencing, we show that sharing of genomic regions of high differentiation between environments is generally low but increases at smaller spatial scales. We identify different shared genomic regions of divergence for each environmental axis and show that most of these regions overlap with candidate chromosomal inversions. Several inversion regions are divergent and polymorphic across many localities. We argue that chromosomal inversions could store shared variation that fuels rapid parallel adaptation to heterogeneous environments, possibly as balanced polymorphism shared by adaptive gene flow. AU - Morales, Hernán E. AU - Faria, Rui AU - Johannesson, Kerstin AU - Larsson, Tomas AU - Panova, Marina AU - Westram, Anja M AU - Butlin, Roger K. ID - 7393 IS - 12 JF - Science Advances SN - 2375-2548 TI - Genomic architecture of parallel ecological divergence: Beyond a single environmental contrast VL - 5 ER - TY - JOUR AB - Polymer additives can substantially reduce the drag of turbulent flows and the upperlimit, the so called “maximum drag reduction” (MDR) asymptote is universal, i.e. inde-pendent of the type of polymer and solvent used. Until recently, the consensus was that,in this limit, flows are in a marginal state where only a minimal level of turbulence activ-ity persists. Observations in direct numerical simulations using minimal sized channelsappeared to support this view and reported long “hibernation” periods where turbu-lence is marginalized. In simulations of pipe flow we find that, indeed, with increasingWeissenberg number (Wi), turbulence expresses long periods of hibernation if the domainsize is small. However, with increasing pipe length, the temporal hibernation continuouslyalters to spatio-temporal intermittency and here the flow consists of turbulent puffs sur-rounded by laminar flow. Moreover, upon an increase in Wi, the flow fully relaminarises,in agreement with recent experiments. At even larger Wi, a different instability is en-countered causing a drag increase towards MDR. Our findings hence link earlier minimalflow unit simulations with recent experiments and confirm that the addition of polymersinitially suppresses Newtonian turbulence and leads to a reverse transition. The MDRstate on the other hand results from a separate instability and the underlying dynamicscorresponds to the recently proposed state of elasto-inertial-turbulence (EIT). AU - Lopez Alonso, Jose M AU - Choueiri, George H AU - Hof, Björn ID - 7397 JF - Journal of Fluid Mechanics SN - 0022-1120 TI - Dynamics of viscoelastic pipe flow at low Reynolds numbers in the maximum drag reduction limit VL - 874 ER - TY - JOUR AB - The order-k Voronoi tessellation of a locally finite set 𝑋⊆ℝ𝑛 decomposes ℝ𝑛 into convex domains whose points have the same k nearest neighbors in X. Assuming X is a stationary Poisson point process, we give explicit formulas for the expected number and total area of faces of a given dimension per unit volume of space. We also develop a relaxed version of discrete Morse theory and generalize by counting only faces, for which the k nearest points in X are within a given distance threshold. AU - Edelsbrunner, Herbert AU - Nikitenko, Anton ID - 5678 IS - 4 JF - Discrete and Computational Geometry SN - 01795376 TI - Poisson–Delaunay Mosaics of Order k VL - 62 ER - TY - JOUR AB - Hippocampus is needed for both spatial working and reference memories. Here, using a radial eight-arm maze, we examined how the combined demand on these memories influenced CA1 place cell assemblies while reference memories were partially updated. This was contrasted with control tasks requiring only working memory or the update of reference memory. Reference memory update led to the reward-directed place field shifts at newly rewarded arms and to the gradual strengthening of firing in passes between newly rewarded arms but not between those passes that included a familiar-rewarded arm. At the maze center, transient network synchronization periods preferentially replayed trajectories of the next chosen arm in reference memory tasks but the previously visited arm in the working memory task. Hence, reference memory demand was uniquely associated with a gradual, goal novelty-related reorganization of place cell assemblies and with trajectory replay that reflected the animal's decision of which arm to visit next. AU - Xu, Haibing AU - Baracskay, Peter AU - O'Neill, Joseph AU - Csicsvari, Jozsef L ID - 5828 IS - 1 JF - Neuron SN - 10974199 TI - Assembly responses of hippocampal CA1 place cells predict learned behavior in goal-directed spatial tasks on the radial eight-arm maze VL - 101 ER - TY - JOUR AB - We give a bound on the ground-state energy of a system of N non-interacting fermions in a three-dimensional cubic box interacting with an impurity particle via point interactions. We show that the change in energy compared to the system in the absence of the impurity is bounded in terms of the gas density and the scattering length of the interaction, independently of N. Our bound holds as long as the ratio of the mass of the impurity to the one of the gas particles is larger than a critical value m∗ ∗≈ 0.36 , which is the same regime for which we recently showed stability of the system. AU - Moser, Thomas AU - Seiringer, Robert ID - 5856 IS - 4 JF - Annales Henri Poincare SN - 14240637 TI - Energy contribution of a point-interacting impurity in a Fermi gas VL - 20 ER - TY - THES AB - In many shear flows like pipe flow, plane Couette flow, plane Poiseuille flow, etc. turbulence emerges subcritically. Here, when subjected to strong enough perturbations, the flow becomes turbulent in spite of the laminar base flow being linearly stable. The nature of this instability has puzzled the scientific community for decades. At onset, turbulence appears in localized patches and flows are spatio-temporally intermittent. In pipe flow the localized turbulent structures are referred to as puffs and in planar flows like plane Couette and channel flow, patches arise in the form of localized oblique bands. In this thesis, we study the onset of turbulence in channel flow in direct numerical simulations from a dynamical system theory perspective, as well as by performing experiments in a large aspect ratio channel. The aim of the experimental work is to determine the critical Reynolds number where turbulence first becomes sustained. Recently, the onset of turbulence has been described in analogy to absorbing state phase transition (i.e. directed percolation). In particular, it has been shown that the critical point can be estimated from the competition between spreading and decay processes. Here, by performing experiments, we identify the mechanisms underlying turbulence proliferation in channel flow and find the critical Reynolds number, above which turbulence becomes sustained. Above the critical point, the continuous growth at the tip of the stripes outweighs the stochastic shedding of turbulent patches at the tail and the stripes expand. For growing stripes, the probability to decay decreases while the probability of stripe splitting increases. Consequently, and unlike for the puffs in pipe flow, neither of these two processes is time-independent i.e. memoryless. Coupling between stripe expansion and creation of new stripes via splitting leads to a significantly lower critical point ($Re_c=670+/-10$) than most earlier studies suggest. While the above approach sheds light on how turbulence first becomes sustained, it provides no insight into the origin of the stripes themselves. In the numerical part of the thesis we investigate how turbulent stripes form from invariant solutions of the Navier-Stokes equations. The origin of these turbulent stripes can be identified by applying concepts from the dynamical system theory. In doing so, we identify the exact coherent structures underlying stripes and their bifurcations and how they give rise to the turbulent attractor in phase space. We first report a family of localized nonlinear traveling wave solutions of the Navier-Stokes equations in channel flow. These solutions show structural similarities with turbulent stripes in experiments like obliqueness, quasi-streamwise streaks and vortices, etc. A parametric study of these traveling wave solution is performed, with parameters like Reynolds number, stripe tilt angle and domain size, including the stability of the solutions. These solutions emerge through saddle-node bifurcations and form a phase space skeleton for the turbulent stripes observed in the experiments. The lower branches of these TW solutions at different tilt angles undergo Hopf bifurcation and new solutions branches of relative periodic orbits emerge. These RPO solutions do not belong to the same family and therefore the routes to chaos for different angles are different. In shear flows, turbulence at onset is transient in nature. Consequently,turbulence can not be tracked to lower Reynolds numbers, where the dynamics may simplify. Before this happens, turbulence becomes short-lived and laminarizes. In the last part of the thesis, we show that using numerical simulations we can continue turbulent stripes in channel flow past the 'relaminarization barrier' all the way to their origin. Here, turbulent stripe dynamics simplifies and the fluctuations are no longer stochastic and the stripe settles down to a relative periodic orbit. This relative periodic orbit originates from the aforementioned traveling wave solutions. Starting from the relative periodic orbit, a small increase in speed i.e. Reynolds number gives rise to chaos and the attractor dimension sharply increases in contrast to the classical transition scenario where the instabilities affect the flow globally and give rise to much more gradual route to turbulence. AU - Paranjape, Chaitanya S ID - 6957 KW - Instabilities KW - Turbulence KW - Nonlinear dynamics TI - Onset of turbulence in plane Poiseuille flow ER - TY - JOUR AB - We consider large random matrices with a general slowly decaying correlation among its entries. We prove universality of the local eigenvalue statistics and optimal local laws for the resolvent away from the spectral edges, generalizing the recent result of Ajanki et al. [‘Stability of the matrix Dyson equation and random matrices with correlations’, Probab. Theory Related Fields 173(1–2) (2019), 293–373] to allow slow correlation decay and arbitrary expectation. The main novel tool is a systematic diagrammatic control of a multivariate cumulant expansion. AU - Erdös, László AU - Krüger, Torben H AU - Schröder, Dominik J ID - 6182 JF - Forum of Mathematics, Sigma TI - Random matrices with slow correlation decay VL - 7 ER - TY - JOUR AB - We prove that the local eigenvalue statistics of real symmetric Wigner-type matrices near the cusp points of the eigenvalue density are universal. Together with the companion paper [arXiv:1809.03971], which proves the same result for the complex Hermitian symmetry class, this completes the last remaining case of the Wigner-Dyson-Mehta universality conjecture after bulk and edge universalities have been established in the last years. We extend the recent Dyson Brownian motion analysis at the edge [arXiv:1712.03881] to the cusp regime using the optimal local law from [arXiv:1809.03971] and the accurate local shape analysis of the density from [arXiv:1506.05095, arXiv:1804.07752]. We also present a PDE-based method to improve the estimate on eigenvalue rigidity via the maximum principle of the heat flow related to the Dyson Brownian motion. AU - Cipolloni, Giorgio AU - Erdös, László AU - Krüger, Torben H AU - Schröder, Dominik J ID - 6186 IS - 4 JF - Pure and Applied Analysis SN - 2578-5893 TI - Cusp universality for random matrices, II: The real symmetric case VL - 1 ER - TY - JOUR AB - Across diverse biological systems—ranging from neural networks to intracellular signaling and genetic regulatory networks—the information about changes in the environment is frequently encoded in the full temporal dynamics of the network nodes. A pressing data-analysis challenge has thus been to efficiently estimate the amount of information that these dynamics convey from experimental data. Here we develop and evaluate decoding-based estimation methods to lower bound the mutual information about a finite set of inputs, encoded in single-cell high-dimensional time series data. For biological reaction networks governed by the chemical Master equation, we derive model-based information approximations and analytical upper bounds, against which we benchmark our proposed model-free decoding estimators. In contrast to the frequently-used k-nearest-neighbor estimator, decoding-based estimators robustly extract a large fraction of the available information from high-dimensional trajectories with a realistic number of data samples. We apply these estimators to previously published data on Erk and Ca2+ signaling in mammalian cells and to yeast stress-response, and find that substantial amount of information about environmental state can be encoded by non-trivial response statistics even in stationary signals. We argue that these single-cell, decoding-based information estimates, rather than the commonly-used tests for significant differences between selected population response statistics, provide a proper and unbiased measure for the performance of biological signaling networks. AU - Cepeda Humerez, Sarah A AU - Ruess, Jakob AU - Tkačik, Gašper ID - 6900 IS - 9 JF - PLoS computational biology TI - Estimating information in time-varying signals VL - 15 ER - TY - JOUR AB - Clathrin-mediated endocytosis (CME) is a highly conserved and essential cellular process in eukaryotic cells, but its dynamic and vital nature makes it challenging to study using classical genetics tools. In contrast, although small molecules can acutely and reversibly perturb CME, the few chemical CME inhibitors that have been applied to plants are either ineffective or show undesirable side effects. Here, we identify the previously described endosidin9 (ES9) as an inhibitor of clathrin heavy chain (CHC) function in both Arabidopsis and human cells through affinity-based target isolation, in vitro binding studies and X-ray crystallography. Moreover, we present a chemically improved ES9 analog, ES9-17, which lacks the undesirable side effects of ES9 while retaining the ability to target CHC. ES9 and ES9-17 have expanded the chemical toolbox used to probe CHC function, and present chemical scaffolds for further design of more specific and potent CHC inhibitors across different systems. AU - Dejonghe, Wim AU - Sharma, Isha AU - Denoo, Bram AU - De Munck, Steven AU - Lu, Qing AU - Mishev, Kiril AU - Bulut, Haydar AU - Mylle, Evelien AU - De Rycke, Riet AU - Vasileva, Mina K AU - Savatin, Daniel V. AU - Nerinckx, Wim AU - Staes, An AU - Drozdzecki, Andrzej AU - Audenaert, Dominique AU - Yperman, Klaas AU - Madder, Annemieke AU - Friml, Jiří AU - Van Damme, Daniël AU - Gevaert, Kris AU - Haucke, Volker AU - Savvides, Savvas N. AU - Winne, Johan AU - Russinova, Eugenia ID - 6377 IS - 6 JF - Nature Chemical Biology SN - 15524450 TI - Disruption of endocytosis through chemical inhibition of clathrin heavy chain function VL - 15 ER - TY - THES AB - Tissue morphogenesis in developmental or physiological processes is regulated by molecular and mechanical signals. While the molecular signaling cascades are increasingly well described, the mechanical signals affecting tissue shape changes have only recently been studied in greater detail. To gain more insight into the mechanochemical and biophysical basis of an epithelial spreading process (epiboly) in early zebrafish development, we studied cell-cell junction formation and actomyosin network dynamics at the boundary between surface layer epithelial cells (EVL) and the yolk syncytial layer (YSL). During zebrafish epiboly, the cell mass sitting on top of the yolk cell spreads to engulf the yolk cell by the end of gastrulation. It has been previously shown that an actomyosin ring residing within the YSL pulls on the EVL tissue through a cable-constriction and a flow-friction motor, thereby dragging the tissue vegetal wards. Pulling forces are likely transmitted from the YSL actomyosin ring to EVL cells; however, the nature and formation of the junctional structure mediating this process has not been well described so far. Therefore, our main aim was to determine the nature, dynamics and potential function of the EVL-YSL junction during this epithelial tissue spreading. Specifically, we show that the EVL-YSL junction is a mechanosensitive structure, predominantly made of tight junction (TJ) proteins. The process of TJ mechanosensation depends on the retrograde flow of non-junctional, phase-separated Zonula Occludens-1 (ZO-1) protein clusters towards the EVL-YSL boundary. Interestingly, we could demonstrate that ZO-1 is present in a non-junctional pool on the surface of the yolk cell, and ZO-1 undergoes a phase separation process that likely renders the protein responsive to flows. These flows are directed towards the junction and mediate proper tension-dependent recruitment of ZO-1. Upon reaching the EVL-YSL junction ZO-1 gets incorporated into the junctional pool mediated through its direct actin-binding domain. When the non-junctional pool and/or ZO-1 direct actin binding is absent, TJs fail in their proper mechanosensitive responses resulting in slower tissue spreading. We could further demonstrate that depletion of ZO proteins within the YSL results in diminished actomyosin ring formation. This suggests that a mechanochemical feedback loop is at work during zebrafish epiboly: ZO proteins help in proper actomyosin ring formation and actomyosin contractility and flows positively influence ZO-1 junctional recruitment. Finally, such a mesoscale polarization process mediated through the flow of phase-separated protein clusters might have implications for other processes such as immunological synapse formation, C. elegans zygote polarization and wound healing. AU - Schwayer, Cornelia ID - 7186 SN - 2663-337X TI - Mechanosensation of tight junctions depends on ZO-1 phase separation and flow ER - TY - THES AB - The first part of the thesis considers the computational aspects of the homotopy groups πd(X) of a topological space X. It is well known that there is no algorithm to decide whether the fundamental group π1(X) of a given finite simplicial complex X is trivial. On the other hand, there are several algorithms that, given a finite simplicial complex X that is simply connected (i.e., with π1(X) trivial), compute the higher homotopy group πd(X) for any given d ≥ 2. However, these algorithms come with a caveat: They compute the isomorphism type of πd(X), d ≥ 2 as an abstract finitely generated abelian group given by generators and relations, but they work with very implicit representations of the elements of πd(X). We present an algorithm that, given a simply connected space X, computes πd(X) and represents its elements as simplicial maps from suitable triangulations of the d-sphere Sd to X. For fixed d, the algorithm runs in time exponential in size(X), the number of simplices of X. Moreover, we prove that this is optimal: For every fixed d ≥ 2, we construct a family of simply connected spaces X such that for any simplicial map representing a generator of πd(X), the size of the triangulation of S d on which the map is defined, is exponential in size(X). In the second part of the thesis, we prove that the following question is algorithmically undecidable for d < ⌊3(k+1)/2⌋, k ≥ 5 and (k, d) ̸= (5, 7), which covers essentially everything outside the meta-stable range: Given a finite simplicial complex K of dimension k, decide whether there exists a piecewise-linear (i.e., linear on an arbitrarily fine subdivision of K) embedding f : K ↪→ Rd of K into a d-dimensional Euclidean space. AU - Zhechev, Stephan Y ID - 6681 SN - 2663-337X TI - Algorithmic aspects of homotopy theory and embeddability ER - TY - GEN AB - Suppose that $n\neq p^k$ and $n\neq 2p^k$ for all $k$ and all primes $p$. We prove that for any Hausdorff compactum $X$ with a free action of the symmetric group $\mathfrak S_n$ there exists an $\mathfrak S_n$-equivariant map $X \to {\mathbb R}^n$ whose image avoids the diagonal $\{(x,x\dots,x)\in {\mathbb R}^n|x\in {\mathbb R}\}$. Previously, the special cases of this statement for certain $X$ were usually proved using the equivartiant obstruction theory. Such calculations are difficult and may become infeasible past the first (primary) obstruction. We take a different approach which allows us to prove the vanishing of all obstructions simultaneously. The essential step in the proof is classifying the possible degrees of $\mathfrak S_n$-equivariant maps from the boundary $\partial\Delta^{n-1}$ of $(n-1)$-simplex to itself. Existence of equivariant maps between spaces is important for many questions arising from discrete mathematics and geometry, such as Kneser's conjecture, the Square Peg conjecture, the Splitting Necklace problem, and the Topological Tverberg conjecture, etc. We demonstrate the utility of our result applying it to one such question, a specific instance of envy-free division problem. AU - Avvakumov, Sergey AU - Kudrya, Sergey ID - 8182 T2 - arXiv TI - Vanishing of all equivariant obstructions and the mapping degree ER -