## Spring 2010

### Title: Galois Theory of Algorithms.

Abstract: Many different programs are the implementation of the same algorithm. This gives us a surjective map from the collection of programs to the collection of algorithms. Similarly, there are many different algorithms that perform the same function. This gives us a surjective map from the collection of algorithms to the collection of computable functions. Algorithms are intermediate between programs and functions:
Programs - -> Algorithms - - > Functions.
We investigate how rigid programs are by looking at the group of automorphism of programs that preserve functionality. The fundamental theorem of Galois theory says that the subgroup lattice of this group is isomorphic to the dual lattice of intermediate types of algorithms. Only basic category theory is needed to understand this talk.

### Title: Operads and Homotopy Theory for Algebraic Structures (Part I).

Abstract: I will begin with a hands-on definition of Operads in Sets, Topological Spaces, dg-Vect, and examples of operads. This will motivate a hands-on definition of Operads in an arbitrary symmetric-monoidal category C. In nice cases, one can construct a symmetric monoidal category of "collections in C," whose monoids are exactly Operads. I will explain the relationship between Collections(C) and End(C), the exact statement of which will make precise the idea that Operads are just "representable" monads.

### Title: Operads and Homotopy Theory for Algebraic Structures (Part II).

Abstract: For an operad O, I will define a category of O-algebra objects in C. For an O-algebra object A in C, we define a category of A-module objects in C. We will discuss some nice properties that these categories have, and some nice properties that the forgetful functor O-Alg(C) ----> C has. I will then discuss, without too much technical detail, how a "homotopy theory" on the underlying category C induces a homotopy theory on the category of O-algebra objects in C. I will try to illustrate that the functor O-Alg(C) ---> C is not particularly well behaved with respect to this homotopy theory. This motivates us to develop a package of theorems we wish were true about the functor but are not.

### Title: Operads and Homotopy Theory for Algebraic Structures (Part III).

Abstract: Finally, I will choose a particular symmetric monoidal category C with a homotopy theory: dg-Vect. I will describe a class of Operads {O_{\infty} } for which the package of theorems about O_{\infty}-Alg(C)----> C are true, and describe how one may pass from any operad O to an operad O_{\infty}, so that O_{\infty} is an operad for which our package of theorems are true, while O-Alg(C) ----> O_{\infty}-Alg(C) is an equivalence of homotopy theories.

### Title: Natural weak factorization systems in model structures

Abstract: Factorization systems abound in mathematics: orthogonal factorization systems (ofs) have nice categorical properties but are too strict to describe many homotopical situations, while weak factorization systems (wfs) do figure prominently, especially in the definition of a model structure on a category, but aren't as well behaved. Natural weak factorization systems (nwfs) "algebraicize" the notion of wfs and sit somewhere in between the two, and thanks to Richard Garner's small object argument, it's possible to construct a plethora of examples.

My work explores the consequences of incorporating nwfs into model categories: a natural model structure will consist of a pair of nwfs together with a comparison map between them such that the underlying classes of maps form a model structure in the ordinary sense. After giving the basic definitions and exploring some of the properties of nwfs, I'll devote the remainder of the talk to persuading my audience that this was a good idea. The most significant results build toward an algebraization of the notion of a Quillen adjunction, each of which include five adjunctions of nwfs. In addition to supplying the definition, I am able to show that this extra algebraic structure can be found for a large family of known examples of Quillen adjunctions.