Tuesday, December 11th, 2018

SCGP Weekly Talk: Bruno Nachtergaele

Time: 1:00 PM - 2:00 PM

Location: 102

Description:

__Title__: Robustness of Topological Order by way of Stability of Superselection Sectors

__Abstract__: Kitaevâ€™s quantum double models provide a rich class of examples of two-dimensional lattice systems with topological order in the ground states and a spectrum described by anyonic elementary excitations. The infinite volume ground states of the abelian quantum double models come in a number of equivalence classes called superselection sectors. We prove that the superselection structure is stable under uniformly small perturbations of the quantum double Hamiltonians.

Wednesday, December 12th, 2018

**YITP Event:**Joint SBU/BNL Pheno seminar @ SBU: Tomer Volanski

Time: 2:30 PM - 3:30 PM

Location:

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**Math Event:**Algebraic Geometry: Bhargav Bhatt - Etale cohomology of affinoid spaces

Time: 4:00 PM - 5:30 PM

Location: Physics P-117

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__Title__: Etale cohomology of affinoid spaces

__Speaker__: Bhargav Bhatt [University of Michigan]

__Abstract__: This talk has two distinct but related parts. First, I will discuss a new Grothendieck topology (the arc topology) on the category of schemes and its usefulness in addressing some foundational questions in etale cohomology (including excision as well as new proofs of the Fujiwara-Gabber theorem and some results of Huber). Secondly, I will explain how to prove the analog of the Artin vanishing theorem in rigid analytic geometry. (Joint work with Akhil Mathew.) DIFFERENT ROOM: PHYSICS P-117

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**Math Event:**Math Colloquium: Manuel Rivera - A new point in topology

Time: 2:30 PM - 3:30 PM

Location: Math Tower P-131

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__Title__: A new point in topology

__Speaker__: Manuel Rivera [University of Miami]

__Abstract__: One knows in algebraic topology the homotopical properties of geometric spaces can be recast into the language of infinite dimensional topological groups determined by function spaces of closed loops in the geometric spaces. For example the zeroth homology of the function space of based loops can be naturally identified with the group algebra of the fundamental group of the geometric space. This group algebra has a compatible coproduct determined by saying the group elements are group-like for the coproduct. Conversely the group like elements in this coalgebra form a group equivalent to the given one. The new point in topology says this bialgebra determining the fundamental group and higher dimensional aspects can, in complete generality, be determined directly from algebraic structure on geometric chains in the geometric space itself. The algebraic construction that does this produces a free differential algebra from a differential coalgebra and was introduced sixty years ago for simply connected spaces. Remarkably it is understood only now to work fine for all geometric spaces if one adds something to it. The new idea beyond technique is to combine the algebraic construction from the past with the infinite homotopical symmetry of chain approximations to the diagonal mapping of the geometric space which is itself perfectly symmetrical. This makes the chain co-algebra on the geometric space cocommutative in a derived sense so that the construction from the past becomes enriched to a bi-algebra in a derived sense. One now sees the fundamental group in the zeroth homology of the enhanced algebraic construction and furthermore one sees higher dimensional homotopical information about the geometric space in the homotopy type of the enhanced algebraic construction.

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Thursday, December 13th, 2018

**Math Event:**Topology and Symplectic Geometry: Lev Tovstopyat-Nelip - The transverse invariant and braid dynamics

Time: 1:00 PM - 2:15 PM

Location: Math Tower 5-127

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__Title__: The transverse invariant and braid dynamics

__Speaker__: Lev Tovstopyat-Nelip [Boston College]

__Abstract__: Let K be a link braided about an open book (B,p) supporting a contact manifold (Y,x). K and B are naturally transverse links. We prove that the hat version of the transverse link invariant defined by Baldwin, Vela-Vick and Vertesi is non-zero for the union of K with B. As an application, we prove that the invariant of a transverse braid having fractional Dehn twist coefficient greater than one is non-zero. We discuss geometric consequences and future directions.

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**Math Event:**Math Colloquium: Bhargav Bhatt - Interpolating p-adic cohomology theories

Time: 4:00 PM - 5:00 PM

Location: Math Tower P-131

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__Title__: Interpolating p-adic cohomology theories

__Speaker__: Bhargav Bhatt [University of Michigan]

__Abstract__: Integration of differential forms against cycles on a complex manifold helps relate de Rham cohomology to singular cohomology, which forms the beginning of Hodge theory. The analogous story for p-adic manifolds, which is the subject of p-adic Hodge theory, is richer due to a wider variety of available cohomology theories (de Rham, etale, crystalline, and more) and torsion phenomena. In this talk, I will give a bird's eye view of this picture, guided by the recently discovered notion of prismatic cohomology that provides some cohesion to the story. (Based on joint work with Morrow and Scholze as well as work in progress with Scholze.)

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