The geometry and combinatorics of Springer fibers

This survey paper describes Springer fibers, which are used in one of the earliest examples of a geometric representation. We will compare and contrast them with Schubert varieties, another family of subvarieties of the flag variety that play an important role in representation theory and combinatorics, but whose geometry is in many respects simpler. The end of the paper describes a way that Springer fibers and Schubert varieties are related, as well as open questions.Comment: 18 page

Asymptotic mapping class groups of Cantor manifolds and their finiteness properties

We prove that the infinite family of asymptotic mapping class groups of surfaces of defined by Funar--Kapoudjian and Aramayona--Funar are of type $F_\infty$, thus answering questions of Funar-Kapoudjian-Sergiescu and Aramayona-Vlamis. As it turns out, this result is a specific instance of a much more general theorem which allows to deduce that asymptotic mapping class groups of Cantor manifolds, also introduced in this paper, are of type $F_\infty$, provide the underlying manifolds satisfy some general hypotheses. As important examples, we will obtain $F_\infty$ asymptotical mapping class groups that contain, respectively, the mapping class group of every compact surface with non-empty boundary, the automorphism group of every free group of finite rank, or infinite families of arithmetic groups. In addition, for certain types of manifolds, the homology of our asymptotic mapping class groups coincides with the stable homology of the relevant mapping class groups, as studied by Harer and Hatcher--Wahl.Comment: With an appendix by Oscar Randal-Williams. (v3: Rewritten introduction to include more motivation.) 63 pages, 7 figure

Total Positivity and Network Parametrizations: From Type A to Type C.

The Grassmannian Gr(k,n) of k-planes in n-space has a stratification by positroid varieties, which arises in the study of total nonnegativity. The positroid stratification has a rich combinatorial theory, introduced by Postnikov. In the first part of this thesis, we investigate the relationship between two families of coordinate charts, or parametrizations, of positroid varieties. One family comes from Postnikov's theory of planar networks, while the other is defined in terms of reduced words in the symmetric group. We show that these two families of parametrizations are essentially the same. In the second part of this thesis, we extend positroid combinatorics to the Lagrangian Grassmannian, a subvariety of Gr(n,2n) whose points correspond to maximal isotropic subspaces with respect to a symplectic form. Applying our results about parametrizations of positroid varieties, we construct network parametrizations for the analogs of positroid varieties in the Lagrangian Grassmannian using planar networks which satisfy a symmetry condition.PhDMathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/133290/1/rkarpman_1.pd

Critical varieties in the Grassmannian

We introduce a family of spaces called critical varieties. Each critical variety is a subset of one of the positroid varieties in the Grassmannian. The combinatorics of positroid varieties is captured by the dimer model on a planar bipartite graph $G$, and the critical variety is obtained by restricting to Kenyon's critical dimer model associated to a family of isoradial embeddings of $G$. This model is invariant under square/spider moves on $G$, and we give an explicit boundary measurement formula for critical varieties which does not depend on the choice of $G$. This extends our recent results for the critical Ising model, and simultaneously also includes the case of critical electrical networks. We systematically develop the basic properties of critical varieties. In particular, we study their real and totally positive parts, the combinatorics of the associated strand diagrams, and introduce a shift map motivated by the connection to zonotopal tilings and scattering amplitudes.Comment: 54 pages, 24 figures; v2: bibliography updated, various exposition improvement

Modelos bioinformáticos y estudio de receptores de proteínas mediante el uso de redes complejas para el desarrollo y diseño de fármacos eficaces en patologías del sistema nervioso central

La búsqueda y desarrollo de fármacos eficaces para el tratamiento de enfermedades neurodegenerativas ha generado grandes expectativas, debido a la relevancia que tienen sobre la economía de los sistemas sanitarios y la tremenda carga y desgaste que sufren familia y cuidadores. Por ello, la industria farmacéutica se ha volcado sobre estas patologías en las últimas tres décadas, pero las dificultades de realizar ensayos sobre el SN provoca que los gastos y tiempos de investigación se disparen, limitando de forma considerable la rentabilidad de los procesos tradicionales en el desarrollo de nuevos medicamentos. Es en este apartado donde realiza sus aportaciones el diseño de fármacos, dedicando una parte del mismo al desarrollo de modelos matemáticos que permitan predecir propiedades de interés para una gran variedad de sistemas químicos incluyendo moléculas de bajo peso molecular, polímeros, biopolímeros, sistemas heterogéneos, formulaciones farmacéuticas, conglomerados de moléculas e iones, materiales, nano-estructuras y otros. En dicho sentido, los estudios QSAR (Quantitative Structure-Activity-Relationships) son usados cada vez mas como herramientas para el descubrimiento molecular. Estos modelos QSAR pueden ser diseñados para que predigan la probabilidad de que un fármaco sea efectivo contra una enfermedad degenerativa determinada ya sea la enfermedad de Parkinson, Alzheimer o cualquier otra, actuando sobre una diana molecular específica. En esta memoria presentamos de manera conjunta la revisión de modelos previos y trabajos específicos novedosos, en los que se han introducido nuevos índices numéricos utilizados para describir tanto la estructura molecular de fármacos como la estructura macromolecular de sus dianas o receptores (proteínas y/o ADN/ARN). Con estos ITs hemos sido capaces de desarrollar nuevos modelos multiQSAR de gran interés por su doble función en la predicción de fármacos y sus dianas moleculares. Estos trabajos permitirán la introducción de nuevos conceptos teóricos y la evolución hacia modelos con posibles aplicaciones en la búsqueda de nuevos fármacos neuroprotectores útiles en el tratamiento de las enfermedades de Parkinson y Alzheimer y/o nuevas dianas moleculares para estos fármacos. Este tipo de investigación abarca un área general-básica en la que interactúan la Bioinformática y la Quimioinformática