In this talk we will discuss the moduli spaces Simp^m_n of degree n+1 morphisms  \A^1_K\to \A^1_K  with "ramification length <m" over an algebraically closed field K. For each m, the moduli space Simp^m_n is a Zariski open subset of the space of degree n+1 polynomials over K up to Aut(\A^1_K). It is, in a way, orthogonal to the many papers about polynomials with prescribed zeroes- here we are prescribing, instead, the ramification data. We will also see why and how our results align, in spirit, with the long standing open problem of understanding the topology of the Hurwitz space.

The twisted rabbit problem is a celebrated problem in complex dynamics. Work of Thurston proves that up to equivalence, there are exactly three branched coverings of the sphere to itself satisfying certain conditions. When one of these branched coverings is modified by a mapping class, a map equivalent to one of the three coverings results. Which one?

After remaining open for 25 years, this problem was solved by Bartholdi-Nekyrashevych using iterated monodromy groups. In joint work with Belk, Lanier, and Margalit, we present an alternate solution using topology and geometric group theory that allows us to solve a more general problem.

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.

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

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).