This seminar is held on Tuesdays at 3pm in 210 Deady.

Summer Quarter, 2016

• July 7, Mark Walsh, (Wichita State)
  Positive scalar curvature and manifolds with boundary
  Abstract: A central theme in modern geometry is the relationship between curvature and topology. With regard to the scalar curvature, the problem of finding manifolds whose topology permits a Riemannian metric of positive scalar curvature is largely understood. More recently, attention has shifted to the problem of understanding the topology of the space of such metrics on a given manifold (and its corresponding moduli spaces). In this talk, I will review this problem and look at some recent developments with a particular emphasis on the case when the underlying manifold has a boundary.
• July 26, Sarah Rasmussen, (Cambridge)
  L-space Classification, Graph Manifolds, and Singularities
  Abstract: A closed 3-manifold is called an L-space if its reduced Heegaard Floer homology vanishes. Building on a joint result of Hanselman, J. Rasmussen, Watson, and myself, Némethi recently proved that a normal complex surface singularity is rational if and only if its link is an L-space. In this talk, I’ll introduce a new classification of graph manifold L-spaces, and discuss some applications for cabled knots and for complex singularities.
• September 6, David Wraith, (National U. of Ireland, Maynooth)
  Special Time: 4:00pm in 210 Deady Hall
  Positive Ricci curvature on highly connected manifolds
  Abstract: This talk concerns the existence of positive Ricci curvature metrics on closed (2n-2)-connected (4n-1)-manifolds. The focus will be largely topological: we will describe new constructions of these objects to which existing curvature results can be applied. The constructions are based on the technique of plumbing disc bundles. This is joint work with Diarmuid Crowley.
• September 23, Jessica Purcell, (Monash University)
  Special Time: 10:30am in 210 Deady Hall
  Limits of knots
  Abstract: There are different ways to define the convergence of knots. For example, the diagram graphs of a sequence of knots might converge to another graph, using ideas from graph theory, or the geometric structures on the knot complements might converge to a metric space, using ideas from geometry. In this talk, we will discuss both notions of convergence of knots, some consequences, and open questions.

Spring Quarter, 2016

• April 6-7, Cameron Gordon, (University of Texas at Austin)
  Part of the Niven Lecture series
• April 12, Felix Wierstra, (Stockholm University)
  Koszul spaces in rational homotopy theory
  Abstract: In this talk I will explain how the theory of Koszul duality can be used to compute the rational homotopy and cohomology groups of a special class of spaces called Koszul spaces. I will also explain how to compute the rational cohomology of n-fold loop spaces and the homotopy groups of n-fold suspensions of Koszul spaces. To make the talk more accessible for a larger audience I will start with a brief introduction to the theory of Koszul duality and rational homotopy theory.
• April 19, Lenhard Ng, (Duke University)
  Knot contact homology and string topology
  Abstract: Knot contact homology, a knot invariant with origins in symplectic geometry and holomorphic curves, has a surprising relation to a more classical knot invariant, the fundamental group of the knot complement. I’ll discuss how one can use string topology to make this relation a bit less surprising (in joint work with Kai Cieliebak, Tobias Ekholm, and Janko Latschev), and apply this to show that knot contact homology characterizes various types of knots. This talk is a spiritual follow-up to the colloquium from the day before, but I’ll try to make them functionally independent.
• April 26, Christine Escher, (OSU)
  Non-negative curvature and torus actions
  Abstract: The classification of Riemannian manifolds with positive and non-negative sectional curvature is a long-standing problem in Riemannian geometry. In this talk I will summarize recent joint work with Catherine Searle on the classification of closed, simply-connected, non-negatively curved Riemannian manifolds admitting an isometric, effective, maximal torus action. This classification has many applications, in particular the Maximal Symmetry Rank conjecture holds for this class of manifolds.
• May 3, Anibal Medina, (Stanford)
  E-infinity comodules and topological manifolds
  Abstract: The first story begins with a question of Steenrod. He asked if the product in the cohomology of a triangulated space, which is associative and graded commutative, can be induced from a cochain level product satisfying the same two properties. He answered it in the negative after identifying homological obstructions among a collection of chain maps he constructed. Using later language, his construction could be said to endow the simplicial chains with an E-infinity coalgebra structure. The second story also begins with a question: when is a space homotopy equivalent to a topological manifold? For dimensions greater than 4, an answer was provided by the work of Browder, Novikov, Sullivan and Wall in surgery theory, which in a later development was algebraically expressed by Ranicki as a single chain level invariant: the total surgery obstruction. After presenting the necessary parts of these stories, the goal of this talk will be to express the total surgery obstruction associated to a triangulated space in terms of comodules over the E-infinity coalgebra structure build by Steenrod on its chains.
• May 10, Laura Starkston, (Stanford)
  Doubling symplectic hypersurfaces and the topology of the complement
  Abstract: We study codimension 2 submanifolds of a smooth even dimensional manifold with certain cohomological constraints. We take inspiration from Donaldson’s construction of symplectic hypersurfaces (real codimension 2) of any symplectic manifold whose complements are Weinstein. Because Donaldson’s hypersurfaces are arbitrarily complicated in a way that depends on an auxiliary almost complex structure, we search for a more reasonable topological construction with similar properties. Specifically, we hope to find a topological decomposition where the pieces naturally support symplectic structures that can be glued together to a global symplectic structure. We take a first small step in this direction by understanding the notion of increasing the degree of the hypersurface at the topological level by doubling. We determine how the topology of the doubled hypersurface relates to the original, and how the topology of the complements are related.
• May 17, Aaron Mazel-Gee, (Berkeley)
  Resolution revolution: model categories, ∞-categories, and beyond

  Abstract: The notion of a resolution has played a central role in mathematics for over half a century. For example, a module over a ring admits a projective resolution, and in fact this is formally analogous to taking a CW-complex replacement of an arbitrary topological space.

  The reason that resolutions are a powerful tool is that they allow us to obtain “derived” versions of our ordinary operations, which often have better formal properties. Notable examples include: global functions (or more generally, global sections), whose derived version gives rise to “cohomology”; homomorphisms between abelian groups, whose derived functor gives rise to “Ext”; tensor product of abelian groups, whose derived version gives rise to “Tor”; and intersection — of manifolds, varieties, or even schemes — whose derived version agrees with the ordinary version if the intersection happens to be transverse.

  This talk is meant to be accessible to mathematicians of all stripes. In it, I will explain two approaches to accessing “derived” phenomena: the classical theory of model categories, and the more recent theory of ∞-categories. Each of these approaches each has its own merits and shortcomings, but miraculously it turns out that these almost perfectly balance each other out. As a result, together they form a robust toolkit that is useful in an extremely broad array of mathematical contexts.

  If time permits, I may also: (i) vaguely indicate the what & why of the theory of “model ∞-categories” that I explore in my thesis, or (ii) introduce “the ∞-category of manifolds” and explain how it can be used in manifold topology to do away with all sorts of pesky and distracting point-set issues (such as requiring certain maps to be fibrations, or the issue of certain maps / lifts of maps / diagrams only being well-defined up to homotopy).

• May 24, Faramarz Vafaee, (CalTech)
  ‘Zero’ surgery on knots in L-spaces
  Abstract: L-spaces are rational homology three-spheres with the same Heegaard Floer homology as lens spaces. Let K be a knot in an L-space Y with a Dehn surgery to a surface bundle over the circle. We show that K is rationally fibered, that is, the knot exterior admits a fibration over the circle. In particular, we show that every knot K in an L-space Y with a Dehn surgery to S1 x S2 is rationally fibered. Moreover, such a knot is Floer simple, that is, the rank of the knot Floer homology is equal to the order of the first singular homology of Y. By combining the latter two facts, we deduce that the induced contact structure on the ambient manifold Y is tight. This project is joint with Yi Ni.
• May 31, TriThang Tran, (UO)

Winter Quarter, 2016

• January 12, Dan Dugger (UO)
  The Grothendieck-Witt category of a field
  Abstract: The Grothendieck-Witt group of a field is the natural invariant that shows up when one wants to classify quadratic forms. It has been deeply studied by algebraists for decades. It seems to have not been noticed that these groups form the morphism sets in a certain category. I will describe this category and then explain how it shows up in motivic homotopy theory. I will also explain analogies with the Burnside category of a finite group, and the corresponding connection to equivariant homotopy theory. Regretfully, the talk will be more about category theory than either algebra or topology.
• January 19, Ian Zemke (UCLA)
  Basepoints in Heegaard Floer homology
  Abstract: We will sketch the construction of the Heegaard Floer homology invariants associated to a 3-manifold, and discuss the role of basepoints in the construction. We will discuss the effect of adding, removing, or moving basepoints, and talk about how these fit into a “graph TQFT”. We will focus on hat Heegaard Floer homology, the simplest version of the invariant.
• January 26, Robert Lipshitz (UO)
  Introduction to (Heegaard) Floer homology
  Abstract: Roughly, Lagrangian intersection Floer homology associates to a pair of Lagrangian submanifolds L, L’ a graded vector space HF(L,L’), whose Euler characteristic is the intersection number of L and L’. Lagrangian intersection Floer homology can be used to define invariants of knots, 3-manifolds, and 4-manifolds, in several ways; one is called Heegaard Floer homology. Some of these invariants now have elegant combinatorial definitions. In this talk, we will discuss the basic definitions of Lagrangian intersection Floer homology and illustrate how the play out in a combinatorial definition of the Heegaard Floer knot invariant. This may be the first talk in a series of two.
• February 2, Boris Botvinnik (UO)
  Stable Moduli Space of High-dimensional Handlebodies
• February 9, Nicholas Proudfoot (UO)
  Intersection cohomology of the symmetric reciprocal plane
  Abstract: I will give an introduction to intersection cohomology via the Weil conjectures, with a view toward understanding a particular fun class of examples.
• February 26, Robert Lipshitz (UO)
  Introduction to (Heegaard) Floer homology, Part 2
• March 1, Chad Giusti (University of Pennsylvania)
  Neural networks and the nerve theorem
  Abstract: In practice, many of the most successful models of abstract neural networks or the function of brain systems are constructed on a foundation of convex geometry, often without the knowledge of those using or constructing the models. Our initial attempts to develop theory about the coding properties such modeled systems has revealed deep connections to fundamental notions from algebraic geometry and topology. In this talk, I will describe some of these connections, give an outline of what we already know as a result of these connections, and describe a number of interesting open mathematical problems that we’ve found along the way. No knowledge of neuroscience is assumed.
• March 8, Nathan Perlmutter (Stanford)
  Cobordism categories and the moduli space of odd dimensional manifolds
  Abstract can be found here

Fall Quarter, 2015

• September 29, Organizational meeting.
• October 6, Alex Ellis (UO)
  Quantum gl(1|1) and tangle Floer homology
  Abstract: Quantum link polynomials can be computed by cutting up a link into elementary tangles (Reshetikhin-Turaev). For link homology theories which categorify these invariants, the story is more complicated. Knot Floer homology, which categorifies the Alexander polynomial, was first defined in a global way by counting pseudoholomorphic disks. The recent tangle Floer homology of Petkova-Vértesi gives a way to compute knot Floer homology using local pieces. We will sketch the idea of their construction and then discuss its relation with quantum gl(1|1).
• October 13, Robert Lipshitz (UO)
  Periodic knots and Hochschild homology
  Abstract: We will start by reviewing the definition of a 2-periodic knot in S^3, and two restrictions on periodic knots: Murasugi’s condition, in terms of the Alexander polynomial, and Edmonds’s condition, originally proved via minimal surfaces. We will then explain a common generalization, in terms of Heegaard Floer homology, due to Hendricks. Hendricks’s proof is, at heart, analytic. We will end with a sketch of how one can recover cases of Hendricks’s result, and similar results, using results on Hochschild homology and the formal structure of bordered Heegaard Floer homology. The last part of the talk is joint work with David Treumann.
• October 20, Dan Ramras (IUPUI)
  Stable representation theory and spaces of flat connections
  Abstract: Atiyah and Bott’s famous work on the Yang-Mills functional shows that the space of flat (unitary) connections on a trivial bundle over a Riemann surface is highly connected. For higher dimensional manifolds, the picture is very different. First, I’ll explain how certain homotopy classes in the gauge group of a flat bundle E can be used to construct non-trivial homotopy classes in the space of flat connections on E (this is joint work with Tom Baird). In particular examples, such as the Heisenberg manifold and the Hantzsche-Wendt manifold, it is possible to construct homotopy classes that do not come from the gauge group. To explain the latter examples, I’ll introduce methods from stable representation theory: deformation K-theory, the deformation representation ring, and the topological Atiyah-Segal map.
• October 26, Ben Elias (UO)
  Igusa diagrams and the K(pi, 1)-conjecturette (joint w/ Geordie Williamson)

  Abstract: The K(pi,1)-conjecture of Arnold-Brieskorn-Pham-Thom states that the hyperplane complement of a Coxeter hyperplane arrangement is a classifying space for the pure braid group. A combinatorial cell complex which is homotopy equivalent to the hyperplane complement was constructed by Salvetti; this is the preferred model for the supposed classifying space. The K(pi, 1)-conjecturette is just the statement that \pi_2 of the Salvetti complex is trivial.

  And why does one care?

  One usually constructs actions of a group using generators and relations. Generators and relations are a recipe for a 2D cell complex for which \pi_1 is the group. If one wants to construct a strict action of a group on a category, however, one needs a 3D cell complex for which \pi_1 is the group, and \pi_2 is trivial. Conjecturally, the Salvetti complex gives a recipe for constructing strict actions of the braid group on categories.

  How do you study \pi_2??

• November 6, Agnes Beaudry (U Chicago)
  Gluing data in chromatic homotopy theory
  Abstract: Understanding the stable homotopy groups of spheres is one of the great challenges of algebraic topology. They form a ring which, despite its simple definition, carries an amazing amount of structure. A famous theorem of Hopkins and Ravenel states that it is filtered by simpler rings called the chromatic layers. This point of view organizes the homotopy groups into periodic families and reveals patterns. There are many structural conjectures about the chromatic filtration. I will talk about one of these conjectures, the chromatic splitting conjecture, which concerns the gluing data between the different layers of the chromatic filtration.
• November 7-8, Cascade Topology Seminar
• November 10, Kristen Hendricks (UCLA)
  New versions of the Heegaard Floer correction term

  Abstract: Heegaard Floer homology is a suite of invariants of three- and four- manifolds, and knots and surfaces within them, introduced by P. Ozsvath and Z. Szabo (and, in the case of knots, J. Rasmussen) in the early 2000s. In this talk, we will focus on the Heegaard Floer correction term, a numerical invariant of a three-manifold with a spin-c structure which can be extracted from Heegaard Floer homology. This invariant has had a wide range of topological applications; after reviewing its basic structure, we will talk about several of these in detail. We will then briefly introduce a new variant of Heegaard Floer homology, called involutive Heegaard Floer homology, and discuss how to extract new versions of the correction term from this theory.

  Involutive Heegaard Floer homology is joint work with C. Manolescu.

• November 17, Dev Sinha (UO)
  Cohomology of symmetric and alternating groups

  Abstract: I’ll develop some topological ingredients used in recent calculations of the cohomology of alternating and symmetric groups. This includes configuration space models and cellular models of those, product and coproduct structures (which are defined for many other sequences of groups), as well as Nakaoka’s calculation of homology of symmetric groups, and a Gysin sequence relating cohomology symmetric and alternating groups.

  While much of this will be used in the algebra seminar which follows, these are all also of some independent interest.

• November 24, 2016-17 course discussion meeting
• December 2, Safia Chettih (UO)
  Topology of Configurations on Trees
  Abstract: Configuration spaces have a long developed history, with many remarkable structural results on the homology and cohomology of ordered and unordered configuration spaces. I will present a viewpoint of configurations on graphs with an eye towards geometrical understanding and the varied areas of mathematics where they may be applied. I will describe how recent results on unordered configurations may be extended to ordered configurations and give explicit presentations for homology and cohomology classes as well as pairings for ordered and unordered configurations of two and three points on trees, and talk about the geometric and combinatorial structures interrelating configurations on graphs.

Topology Seminar

Previous years: 2015 2013 2012 F11 S10 W10 F09 F08 S08 W08 F07 S07 W07 F06 F05-S06 F04-S05 F03-S04 F02-S03 F00-S01 F99-S00 F98-S99 F97-S98