Algebraic Topology is a library of techniques for, in an appropriate sense, measuring shapes.  It has long been understood, via the work of Weil, Quillen, Deligne, and many others that analogues of these techniques can give very interesting information about problems in other disciplines.  Deligne's solution of a family of conjectures of Weil in the 1970's is a prime example.  Over the last 10-15 years, a new family of such applications directed at the notion of "Big Data" has been constructed.  We will discuss these developments with examples.

Let X be a Noetherian space, let f be a continuous self-map on X, let Y be a closed subset of X, and let x be a point in X. We show that the set containing all positive integers n such that the n-th iterate of x under f lands in Y is a union of at most finitely many arithmetic progressions along with a set of Banach density 0. This is joint work with Jason Bell and Tom Tucker.

We show that if a strongly minimal theory has one recursive model then every model is recursive in 0^(3). In a special cases we can lower this bound to 0^(2). This answers a long-standing open question in recursive model theory. The analysis uses both model theoretic ideas as well as some recursion theoretic techniques. I will try to explain this interplay without assuming recursion theory knowledge beyond the existence of a Halting set, whose role I will briefly review.  (Work joint with Julia F. Knight)

Grobner bases for twisted commutative algebras
Speaker: Steven Sam
3:45 PM

Recent work of Draisma-Kuttler, Snowden, and Church-Ellenberg-Farb in commutative algebra, algebraic geometry, and homological algebra, seeks to understand stability results (say, of some invariant of a sequence of algebraic objects) as the finite generation of some other algebraic object.

Examples include the Delta-modules in the study of free resolutions of Segre embeddings and FI-modules in the study of cohomology of configuration spaces. This finite generation often reduces to establishing the Noetherian property: subobjects of finitely generated objects are again finitely generated. I will discuss the situation of modules over twisted commutative algebras, which realize some of these topics as special cases. I will introduce a Grobner basis theory and show how it proves these Noetherian results. The talk will be mostly combinatorial and I will suggest some open problems. This is based on joint work with Andrew Snowden.

Smoothing of limit linear series on metrized complexes of algebraic curves.
Speaker: Madhusudan Manjunath
5:00 PM

The theory of limit linear series on curves of compact type (reducible curves whose dual graph is a tree) was introduced by Eisenbud and Harris in

1986 and this theory has several applications to algebraic curves. This theory has recently been generalized  to objects called ``metrized complexes of curves" by Amini and Baker. A metrized complex of curves is essentially a metric graph with algebraic curves plugged into the vertices of this metric graph. Eisenbud and Harris showed that any limit $g^1_d$ on a curve of compact type can be smoothed. We study the question of smoothing a limit $g^1_d$ on a metrized complex.  We provide an effective characterization of a smoothable limit $g^1_d$ on a metrized complex and the talk will include examples demonstrating this characterization.  This is ongoing work with Matthew Baker and Luo Ye.

I’ll discuss homotopy theoretic constructions, ideas, and issues that come up in symplectic topology.  No prior knowledge of symplectic topology will be assumed.

The topic of this talk arises from two directions. On the one hand, Gödel's incompleteness theorem tell us that given any sufficiently strong, consistent, effectively axiomatizable theory T for first-order arithmetic, there is a statement that is true but not provable in T. On the other hand, over the past seventy years, a number of researchers studying witnessing functions for various combinatorial statements have realized the importance of fast-growing functions and the fact that their totality is often not provable over a given sufficiently strong, consistent, effectively axiomatizable theory T for first-order arithmetic (e.g. the Paris-Harrington and the Kirby-Paris theorems).

I will talk about the structure induced by giving the order (for a fixed T) of relative provability for totality of algorithms. That is, for algorithms describing functions f and g, we say f ≤ g if T along with the totality of g suffices to prove the totality of f. It turns out that this structure is rich, and encodes many facets of the nature of provability over sufficiently strong, consistent, effectively axiomatizable theories for first-order arithmetic. (Work joint with Mingzhong Cai, David Diamondstone, Steffen Lempp, and Joseph S. Miller.)

Adjunctions describe a “duality” between mathematical objects of different types. Monads encode algebraic structure. Descent refers to classical recognition or gluing problems in algebra, algebraic geometry, and homotopy theory that have been connected by the language of adjunctions, monads, and comonads. In this talk, we explain how adjunctions and monads should be thought of as diagrams (“resolutions”) and how algebras, coalgebras, and thus descent data are then described by particular “weighted” limits of these diagrams. Maps between the weights, which describe the shapes of each limit notion, recover the expected maps between these objects, allowing us to prove basic theorems completely independently of the mathematical context in which we are working.