Abstract: We will discuss the current work in producing an explicit kernel that induces the derived equivalence which arises from the Grassmannian flop. Specifically we will see that the essential image of the functor associated to this kernel aligns with a(n) (exceptional) collection first studied by Kappronav. Further we will explore some interesting geometric properties which the kernel of this functor enjoys and their curious ties to Geometric Invariant Theory. Future directions for a more general technique in producing these interesting kernels, will also be discussed. The pre-talk will discuss the previous work of Ballard et. al. in a similar kernel and it's geometric connections to the homology of projective spaces.

Abstract: The inner workings of the brain have intrigued generations of scientist and the general public alike. In recent years geometric methods have been successfully applied to describe how the brain reacts when introduced to outside stimuli. In fact, this application has seen so much success that the Nobel prize has been awarded for this work. In this talk we will see how, when given simple combinatorial data from the brain, these methods can be used to (partially) reconstruct the stimuli space. In other words, we will see how the use of geometry can read minds. In the remainder of the talk, we will discuss the most recent attempts to mathematically model this data using techniques from Algebraic Geometry.

Abstract: We develop a generalization of the $Q$-construction of the first author, Diemer, and the third author for Grassmann flops over an arbitrary field of characteristic zero. This generalization provides a canonical idempotent kernel on the derived category of the associated global quotient stack. This idempotent kernel, after restriction, induces a derived equivalence over any twisted form of a Grassmann flop. Furthermore its image, after restriction to the geometric invariant theory semistable locus, ``opens'' a canonical ``window'' in the derived category of the quotient stack. We check this window coincides with the set of representations used by Kapranov to form a full exceptional collection on Grassmannians. Even in the well-studied special case of standard Atiyah flops, the arguments yield a new proof of the derived equivalence.

Abstract: In this article we will build a universal imbedding of a regular Hom- Lie triple system into a Lie algebra and show that the category of regular Hom-Lie triple systems is equivalent to a full subcategory of pairs of $\mathbb{Z}_2$- graded Lie algebras and Lie algebra automorphism, then finally give some characterizations of this subcategory.

Abstract: We Show that every finite $\mathbb{Z}$-grading on the algebra of compact operators is induced by a decomposition of the underlying Complex Hilbert space. We do this by first showing that every $\mathbb{Z}$-grading of the compoact operators defines a Peirce decomposition of a certain ideal. Then show this decomposition induces the $\mathbb{Z}$-grading on the compact operators. While finally as a Corollary we will show that every finite $\mathbb{Z}$-grading on the bounded operators is also induced by a decomposition of the underlying Hilbert Space