The geometry of the cyclotomic trace

Speaker: 

Aaron Mazel-Gee

Institution: 

USC

Time: 

Monday, April 1, 2019 - 4:00pm to 5:00pm

Host: 

Location: 

RH 340P

K-theory is a means of probing geometric objects by studying their vector bundles, i.e. parametrized families of vector spaces.  Algebraic K-theory, the version applying to varieties and schemes, is a particularly deep and far-reaching invariant, but it is notoriously difficult to compute.  The primary means of computing it is through its "cyclotomic trace" map K→TC to another theory called topological cyclic homology.  However, despite the enormous computational success of these so-called "trace methods" in algebraic K-theory computations, the algebro-geometric nature of the cyclotomic trace has remained mysterious.

In this talk, I will describe a new construction of TC that affords a precise interpretation of the cyclotomic trace at the level of derived algebraic geometry.  By the end of the talk, you will be able to take home with you a very nice and down-to-earth fact about traces of matrices.  No prior knowledge of algebraic K-theory or derived algebraic geometry will be assumed.

This represents joint work with David Ayala and Nick Rozenblyum.

Joint Los Angeles Topology Seminar at Caltech

Institution: 

Joint Seminar

Time: 

Monday, November 5, 2018 - 4:00pm to 6:00pm

Location: 

Linde 310

Chris Gerig (Harvard): SW = Gr
Whenever the Seiberg-Witten (SW) invariants of a 4-manifold X are defined, there exist certain 2-forms on X which are symplectic away from some circles. When there are no circles, i.e. X is symplectic, Taubes' ``SW=Gr'' theorem asserts that the SW invariants are equal to well-defined counts of J-holomorphic curves (Taubes' Gromov invariants). In this talk I will describe an extension of Taubes' theorem to non-symplectic X: there are well-defined counts of J-holomorphic curves in the complement of these circles, which recover the SW invariants. This ``Gromov invariant'' interpretation was originally conjectured by Taubes in 1995.

Biji Wong (CIRGET Montreal): A Floer homology invariant for 3-orbifolds via bordered Floer theory
Using bordered Floer theory, we construct an invariant for 3-orbifolds with singular set a knot that generalizes the hat flavor of Heegaard Floer homology. We show that for a large class of 3-orbifolds the orbifold invariant behaves like HF-hat in that the orbifold invariant, together with a relative Z_2-grading, categorifies the order of H_1^orb. When the 3-orbifold arises as Dehn surgery on an integer-framed knot in S^3, we use the {-1,0,1}-valued knot invariant epsilon to determine the relationship between the orbifold invariant and HF-hat of the 3-manifold underlying the 3-orbifold.

Joint Los Angeles Topology Seminar at UCLA

Institution: 

Joint Seminar

Time: 

Monday, October 15, 2018 - 4:00pm to 6:00pm

Location: 

MS 6627

Lei Chen (Caltech): Section problems
In this talk, I will discuss a direction of study in topology: Section problems. There are many variations of the problem: Nielsen realization problems, sections of a surface bundle, sections of a bundle with special property (e.g. nowhere zero vector field). I will discuss some techniques including homology, Thurston-Nielsen classification and dynamics. Also I will share many open problems. Some of the results are joint work with Nick Salter.

Lisa Piccirillo (UT Austin): TBA

Combinatorics of orbit configuration spaces

Speaker: 

Christin Bibby

Institution: 

University of Michigan

Time: 

Monday, October 29, 2018 - 4:00pm to 5:00pm

Host: 

Location: 

RH 340P

From a group action on a space, define a variant of the configuration space by insisting that no two points inhabit the same orbit. When the action is almost free, this "orbit configuration space'' is the complement of an arrangement of subvarieties inside the cartesian product, and we use this structure to study its topology. We give an abstract combinatorial description of its poset of layers (connected components of intersections from the arrangement) which turns out to be of much independent interest as a generalization of partition and Dowling lattices. The close relationship to these classical posets is then exploited to give explicit cohomological calculations akin to those of (Totaro '96). Joint work with Nir Gadish.

Homotopy theory for Kan simplicial manifolds and a smooth analog of Sullivan's realization functor

Speaker: 

Chris Rogers

Institution: 

University of Nevado, Reno

Time: 

Monday, April 22, 2019 - 4:00pm to 5:00pm

Host: 

Location: 

RH 340P

Kan simplicial manifolds, also known as "Lie infinity-groupoids", are simplicial Banach manifolds which satisfy conditions similar to the horn lling conditions for Kan simplicial sets. Group-like Lie infinity-groupoids (a.k.a "Lie infinity-groups") have been used to construct geometric models for the higher stages of the Whitehead tower of the orthogonal group. With this goal in mind, Andre Henriques developed a smooth analog of Sullivan's realization functor from rational homotopy theory which produces a Lie infinity-group from certain commutative dg-algebras (i.e. L_infinity-algebras).

In this talk, I will present a homotopy theory for both these commutative dg-algebras and for Lie infinity-groups, and discuss some examples that demonstrate the compatibility between the two. Conceptually, this work can be interpreted either as a C^\infty-analog of classical results of Bouseld and Gugenheim in rational homotopy theory, or as a homotopy-theoretic analog of classical theorems from
Lie theory. This is based on joint work with A. Ozbek (UNR grad student) and C. Zhu (Gottingen).

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