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  AI In mathematics, systolic groups are groups that act geometrically (properly discontinuously and cocompactly) on a systolic complex . These groups are a primary focus in geometric group theory because they exhibit a form of "simplicial non-positive curvature," making them combinatorial analogs of CAT(0) or non-positively curved spaces. [ 1 , 2 , 3 , 4 , 5 ] Key Components Systolic Complex : A connected, simply connected simplicial complex where the "links" of all its simplices are 6-large . 6-large means the complex is "flag" (any set of vertices that are pairwise connected by edges spans a simplex) and has no embedded cycles of length less than 6 that are "full subcomplexes". Essentially, every simplicial loop of length 4 or 5 must have a diagonal. Geometric Action : For a group to be called systolic, it must act by simplicial automorphisms on one of these complexes such that the action is: Proper : Each compact subcomplex is moved away f...

Future of Math Symposium ~***~

~***~ AI on a Hierarchy of Dynamic Systems The hierarchy of dynamic systems generally refers to  structured levels of complexity and organization, ranging from simple, predictable behaviors to complex, chaotic, and emergent behaviors . Key frameworks include the Ergodic Hierarchy (characterizing randomness levels) and multi-scale modeling (micro/meso/macro scales).  [ 1 ,  2 ,  3 ] 1. The Ergodic Hierarchy (Randomness & Chaos) This hierarchy categorizes dynamical systems based on their level of "mixing" or unpredictability, commonly used in statistical mechanics and chaos theory: [ 1 ,  2 ] Ergodicity:  The lowest level, where the system’s trajectory passes arbitrarily close to any point in its phase space over time. Weak Mixing:  Systems that do not have distinct, invariant subspaces. Strong Mixing:  Systems that behave like a well-mixed fluid, losing memory of their initial conditions quickly. Kolmogorov (K-systems):  Systems with ...