A categorical approach to the stable center conjecture

The stable center conjecture asserts that the space of stable distributions in the Bernstein center of a reductive p-adic is closed under convolution. It is closely related to the notion of an L-packet and endoscopy theory. We describe a categorical approach to the depth zero part of the conjecture. As an illustration of our method, we show that the Bernstein projector to the depth zero spectrum is stable.Comment: 74 pages, a grant acknowledgement is change

On the depth r Bernstein projector

In this paper we prove an explicit formula for the Bernstein projector to representations of depth at most r. As a consequence, we show that the depth zero Bernstein projector is supported on topologically unipotent elements and it is equal to the restriction of the character of the Steinberg representation. As another application, we deduce that the depth r Bernstein projector is stable. Moreover, for integral depths our proof is purely local.Comment: 42 pages, a grant acknowledgement is change

The Tannakian Formalism and the Langlands Conjectures

Let H be a connected reductive group over an algebraically closed field of characteristic zero, and let G be an abstract group. In this note we show that every homomorphism from the Grothendieck semiring of H to that of G which maps irreducible representations to irreducibles, comes from a group homomorphism from G to H. We also connect this result with the Langlands conjectures.Comment: 15 page

Designing de novo retroaldolase catalysts

Evolutionary history of native proteins, shaping observed sequences by complex interplay between mutational drift, maintaining stability and developing functionality, often complicates rationalization of protein engineering experiments making it hard to learn even from large datasets available with advent of high throughput screening and deep-sequencing technologies. Use of de novo protein scaffolds for gain of function design projects should, arguably, allow better understanding of fundamental principles underlying implementation of this function in nature and application of these principles to new protein engineering problems. Computational design of enzymatic activity in the de novo built idealized protein scaffolds instead of natural proteins from PDB has a promising advantages of avoiding limitations associated with evolutionary history and virtually unlimited number of geometric variants that can be generated for given scaffold to accommodate catalytic machinery. I am going to present computational strategy used to design de novo proteins with enzymatic activity and experimental data collected using recently identified de novo designed beta-barrels catalyzing retro-aldolase reaction. This information helps to narrow down range of catalytic mechanisms compatible with the structural model, which in turn help to highlight features and interactions potentially important for catalysis