The study of Derived Categories, first introduced by Grothendieck and his student Verdier, has come a long a way in describing deep connections between algebraic geometry, commutative algebra, representation theory, number theory, symplectic geometry, and theoretical physics. Despite that a lot of work has been done with derived categories there are still many open questions and interesting conjectures. One example is from the mid 90's and early 2000's where Bondal, Orlov and (independently) Kawamata conjectured that if two smooth complex varieties are related by a flop then they should be derived equivalent. It has been shown by Reid and others that flops can be equivalently explained by wall crossings from variations of geometric invariant theory. Using this approach Ballard Diemer, Favero have suggested a Kernel as a candidate to realize this conjectured derived equivalence. In recent joint work with Ballard, Chidambaram, Favero, and McFaddin, we have shown that a generalization of this kernel realizes the derived equivalence of the Grassmann Flop, which was first introduced by Donovan and Segal. In this talk we will examine some of the curious algebraic and geometric properties of a brand new generalization which describes these previously studied kernels and may describe many more wall crossings.

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.

We will discuss the recent 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. This is joint work with Ballard, Chidambaram, Favero, and McFaddin.