Alberto Verjovsky (Instituto de Matemáticas-UNAM, Mexico)

Compact 3-dimensional geometric solenoidal manifolds

Abstract: We present several results about solenoidal manifolds motivated by results by Dennis Sullivan in [1] with commentaries developed in [2] and on a join project with Dennis Sullivan [3]. Solenoidal manifolds of dimension n are topological spaces which are locally homeomorphic to the product of a Cantor set with an open subset of $R^n$. Geometric 3-dimensional solenoidal manifolds are the analog of geometric 3-manifolds in the sense of Thurston. We will give some results related to 3-dimensional geometric solenoidal manifolds.

[1] D. Sullivan, Solenoidal manifolds, J. Singul. 9 (2014), 203–205.
[2] A. Verjovsky, Commentaries on the paper “Solenoidal manifolds” by Dennis Sullivan. J. Singul. 9 (2014), 245–251.
[3] D. Sullivan, A. Verjovsky, Compact 3-dimensional geometric solenoidal manifolds. In preparation.

Luis Caffarelli (University of Texas at Austin, USA)
Diverse models on segregation
Abstract: In this lecture we will discuss give an overview of the analytical properties of several models describing segregation patterns under different circumstances depending on the interaction process, adjacent processes, like particle annihilation, segregation at a distance, and the interaction of species that diffuse continuously as insects with those dispersed by wind like some seeds.





Mike Hopkins (University of Harvard, USA)

Homotopy theory and algebraic vector bundles

Abstract:  This talk will describe joint work with Aravind Asok and Jean Fasel using the methods of homotopy theory to construct new examples of algebraic vector bundles.   I will describe a natural conjecture which, if true, implies that over the complex numbers the classification of algebraic vector bundles over smooth affine varieties admitting an algebraic cell decomposition coincides with the classification of topological complex vector bundles.

José Antonio de la Peña (Instituto de Matemáticas-UNAM, Mexico)
Degenerations of algebras and modules

Abstract:Let $A$ be a finite dimensional $k$-algebra.
We introduce some concepts on the geometry of representations and its applications.
(1) the representation type of the algebra (finite, tame or wild) can be read in the dimension of the module variety $mod_A(d)$ and the orbits $G(d) X$, where $G(d)$ if the group determining the iso-classes in $mod_A(d)$, $d \in \N$.
(2) Given two algebras $A$ and $B$, we say that $B$ is a degeneration of $A$ if there is an algebraic family $(A_z)_z \in Z$ such that $A_z$ is isomorphic to $A$ in a open and dense subset of $Z$ and $A_{z_0}$ is isomorphic to $B$ in some $z_0 \in Z$. If $B$ is a degeneration of $A$ then $\dim_k H^n(B) \ge \dim_k H^n(A)$, where $H^n$ denotes the Hochschild cohomology.
(3) Geiss degeneration theorem: a degeneration of a wild algebra is wild.
(4) Denote $q_A$ the (quadratic) Tits form of $A$, then:
(a) if $A$ is representation finite then $q_A$ is weakly positive (ie. $q_A(u) >0$ for $0 \ne u$ any vector with $0\le $ entries).
(b) if $A$ is tame the $q_A$ is weakly non-negative (ie. $q_A(u) \ge 0$ for any vector $u$ with $0\le $ entries).
(5) Bruestle-JAP-Skowronski theorem: let $A$ be a strongly simply connected algebra then $A$ is tame if and only if $q_A$ is wnn.
(6) We show that $A$ is strongly simply connected if and only if the trace of the Coxeter matrix satisfies $tr(\Phi_A)=-1$.

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