Seminar Calendar
for Topology Seminar events the year of Thursday, September 20, 2012.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
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Thursday, January 19, 2012

2:00 pm   in 241 Altgeld Hall,  Thursday, January 19, 2012
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Submitted by lukyane2.
 Organizational MeetingAbstract: Organizational meeting for the seminar.

Tuesday, January 24, 2012

Topology Seminar
11:00 am   in 243 Altgeld Hall,  Tuesday, January 24, 2012
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Submitted by franklan.
 Michael Mandell (Indiana University)Localization sequences in THHAbstract: (Joint work with Andrew Blumberg, preprint arXiv:1111.4003.) For a discrete valuation ring R with quotient field F and residue field k, you have a cofibration sequence of K-theory spectra K(k) → K(R) → K(F). The corresponding sequence in THH is not a cofibration sequence, but both the cofiber of the map THH(k)→ THH(R) and the fiber of the map THH(R) → THH(F) have an interpretation in terms of the THH of Waldhausen categories. Thinking in terms of Waldhausen categories, we therefore get two cofibration sequences for THH, THH(k) → THH(R) → THH(F|R) (first constructed by Hesselholt and Madsen) and THH( Spec(R) on Spec(k) ) → THH(R) → THH(F) (generalizing to THH a well-known exact sequence in Hochschild homology). The first arises by looking at enrichments by connective spectra and the second by looking at enrichments in non-connective spectra.

Thursday, January 26, 2012

2:00 pm   in 241 Altgeld Hall,  Thursday, January 26, 2012
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Submitted by lukyane2.
 Mee Seong Im (UIUC Math)The Hamiltonian reduction of a certain affine variety Abstract: I will discuss certain theories in symplectic geometry and in algebraic geometry which give us various ways to view the same complex manifold. More specifically, the Hamiltonian reduction of the cotangent bundle of a certain variety can be thought of as the symmetric product of the complex plane while the GIT quotient of the same cotangent bundle but which is twisted by a character of the general linear group can be thought of as a certain Hilbert scheme. These varieties are related by the Hilbert-Chow morphism in the sense that one is a desingularization of the other. I will end with an analogous construction in which a notion of noncommutativity appears in the algebro-geometric quotient. Lots of examples will be provided throughout my talk.

Tuesday, January 31, 2012

Topology Seminar
11:00 am   in 243 Altgeld Hall,  Tuesday, January 31, 2012
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Submitted by franklan.
 Claire Tomesch (University of Chicago)Categories of cohesion and ‘discretized’ model categoriesAbstract: The purpose of this talk is to describe a notion of category of cohesion -- a concept of Lawvere introduced to describe 'relative discreteness' -- and its role in defining and understanding a model structure on Simpson-Tamsamani style versions of weak n-categories. The main payoff of this approach is an iterable construction of a model structure which takes into account the special role of 'discrete' objects.

Thursday, February 2, 2012

2:00 pm   in 241 Altgeld Hall,  Thursday, February 2, 2012
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Submitted by lukyane2.
 Ser-Wei Fu (UIUC Math)Length Spectra and Deformation FamiliesAbstract: This is a practice talk for the preliminary exam. I will define and describe the spectral rigidity problem and talk about known results. The talk will be focused on clearly defining every object and giving examples to illustrate a new approach to the problem. To be specific, two main topics that will be discussed are flat metrics and train tracks on surfaces.

Tuesday, February 7, 2012

Topology Seminar
11:00 am   in 243 Altgeld Hall,  Tuesday, February 7, 2012
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Submitted by franklan.
 Nathaniel Stapleton (MIT)Transchromatic twisted character mapsAbstract: The ring of coefficients of the codomain of the transchromatic generalized character maps is constructed to be the universal extension of the K(t)-localization of Morava En over which the p-divisible group associated to En splits as a sum of a height t connected p-divisible group and a height n-t constant p-divisible group. We will describe a refinement of this story to the universal extension of En over which the p-divisible group is a non-trivial extension of a height t connected p-divisible group by a height t constant p-divisible group. This refinement is able to recover the transchromatic generalized character maps and also recovers the classical generalized character theory of Hopkins, Kuhn, and Ravenel when t=0.

Thursday, February 9, 2012

2:00 pm   in 241 Altgeld Hall,  Thursday, February 9, 2012
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Submitted by lukyane2.
 Michael DiPasquale (UIUC Math)Resolutions and GeometryAbstract: Given a variety X inside of some projective space, one of the primary ways in which algebraic geometers study X is through its homogeneous coordinate ring S_X. One way to unpack the information hiding mysteriously inside of S_X is to study the free resolution of S_X. From this resolution come many fantastic invariants of X, primarily the betti diagram of X from which one can compute the Hilbert function of X and the regularity of X. Many difficult open conjectures for curves relate to the configuration of the betti diagram of the curve. Our modest goal is to see how such a seemingly arcane algebraic object as a resolution can actually reflect the geometry of a variety, primarily by looking at the case where X is a bunch of points in projective space. Interested folks may find David Eisenbud's book Geometry of Syzygies a good read; that is where much of my material will come from.

Tuesday, February 14, 2012

Topology Seminar
11:00 am   in 243 Altgeld Hall,  Tuesday, February 14, 2012
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Submitted by franklan.
 Thomas Kragh (MIT)Stable homotopy types and orientations in Hamiltonian Floer theoryAbstract: I will start by outlining the basic ideas in Morse theory and Conley index theory. Then I will describe Hamiltonian Floer homology using infinite dimensional Morse homology. I will then describe the ideas of finite dimensional approximations, and discuss existence and uniqueness. For cotangent bundles these finite dimensional approximations exists canonically - but are not natural. I will explain this in more detail for a nearby Lagrangian, and describe how this lead to new insights into the coherent orientations in Floer homology. If time permits I will talk about some generalizations of the finite dimensional approximations and relations to complex periodic cobordism.

Thursday, February 16, 2012

2:00 pm   in 241 Altgeld Hall,  Thursday, February 16, 2012
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Submitted by lukyane2.
 Jimmy Shan (UIUC Math)Polytopes, zeroes of polynomial equations and mixed volumeAbstract: We will present a bound of number of zeroes of polynomial equations using mixed volumes of polytopes which are constructed from the exponent vectors of the polynomials; Newton polytope associated to one polynomial is a prototype.

Tuesday, February 21, 2012

Topology Seminar
11:00 am   in 243 Altgeld Hall,  Tuesday, February 21, 2012
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Submitted by franklan.
 Nathaniel Rounds (Indiana University)Compactifying string topologyAbstract: String topology studies the algebraic topology of the free loop space of a manifold. In this talk, we describe a compact space of graphs and show how this space gives algebraic operations on the singular chains of the free loop space. In particular, our chain level operations induce Cohen and Godin's "positive boundary TQFT" on the homology the free loop space. This project is joint work with Kate Poirier.

Thursday, February 23, 2012

2:00 pm   in 241 Altgeld Hall,  Thursday, February 23, 2012
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Submitted by lukyane2.
 Brian Collier (UIUC Math)Motivating Hamiltonian Floer TheoryAbstract: The focus of this talk will be to motivate the main ideas in Hamiltonian Floer theory. To start, we will review the notion of a Hamiltonian vector field and other important things in symplectic geometry. We will then talk about Morse Theory in a manner that will generalize most naturally to the infinite dimensional situation. Finally I will introduce Hamiltonian Floer Theory as the analog of the Morse theory of a certain action functional.

Tuesday, February 28, 2012

Topology Seminar
11:00 am   in 243 Altgeld Hall,  Tuesday, February 28, 2012
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Submitted by franklan.
 Paul Goerss (Northwestern University)Brown-Comenetz duality in the K(2)-local categoryAbstract: The principal result I will discuss is the identification of the homotopy type of the Brown-Comenetz dual of the K(2)-local sphere at p=3. Given the rather technical nature of this computation, I will probably spend more time on why this is an interesting question than on techniques of proof.

Thursday, March 1, 2012

2:00 pm   in 241 Altgeld Hall,  Thursday, March 1, 2012
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Submitted by lukyane2.
 Juan Villeta-Garcia (Department of Mathematics, University of Illinois at Urbana-Champaign)The Harder-Narasimhan stratification of quiver representationsAbstract: A quiver is a finite graph with orientations, and their representations are defined by assigning vector spaces to each vertex and linear maps to each arrow. The theory of quiver representations is incredibly broad, with applications to such areas as quantum physics, Lie theory and invariant theory. We will give a brief overview of the category of quiver representations, and use a construction of Reineke that mimics the Harder-Narasimhan filtration for vector bundles, to analyze this category.

Tuesday, March 6, 2012

Topology Seminar
11:00 am   in 243 Altgeld Hall,  Tuesday, March 6, 2012
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Submitted by franklan.
 Ayelet Lindenstrauss (Indiana University)K-theory of formal power seriesAbstract: (Joint with Randy McCarthy.) We study the algebraic K-theory of parametrized endomorphisms of a unital ring R with coefficients in a simplicial R-bimodule M, and compare it with the algebraic K-theory of the ring of formal power series in M over R. Waldhausen defined an equivalence from the suspension of the reduced Nil K-theory of R with coefficients in M to the reduced algebraic K-theory of the tensor algebra TR(M). Extending Waldhausen's map from nilpotent endomorphisms to all endomorphisms, our map has to land in the ring of formal power series rather than in the tensor algebra, and is no longer in general an equivalence (it is an equivalence when the bimodule M is connected). Nevertheless, the map shows a close connection between its source and its target: it induces an equivalence on the Goodwillie Taylor towers of the two (as functors of M, with R fixed), and allows us to give a formula for the suspension of the invariant W(R;M) (which can be thought of as Witt vectors with coefficients in M, and is what the Goodwillie Taylor tower of the source functor converges to) as the inverse limit, as n goes to infinity, of the reduced algebraic K-theory of TR(M)/ (Mn).

Thursday, March 8, 2012

2:00 pm   in 241 Altgeld Hall,  Thursday, March 8, 2012
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Submitted by lukyane2.
 Sarah Yeakel (Department of Mathematics, University of Illinois at Urbana-Champaign)SpectraAbstract: I'll define spectra with motivation from homotopy theory, explain why they form a nicer category than topological spaces, and talk about how they are useful tools in pretty much any setting with a (co)homology theory.

Tuesday, March 13, 2012

Topology Seminar
11:00 am   in 243 Altgeld Hall,  Tuesday, March 13, 2012
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Submitted by franklan.
 Gabriel Drummond-Cole (Northwestern University)Homotopically trivializing the circle in the framed little disksAbstract: An action of the space of framed little disks on a target space induces a circle action on the target space. Kontsevich suggested that an action of the framed little disks along with a trivialization of the circle action could be encapsulated as follows: this data should be the same as an action of the Deligne-Mumford-Knudsen compactified genus zero moduli space. I'll present a rigorous formulation of this statement in the category of topological operads. There will be many pictures.

Thursday, March 15, 2012

2:00 pm   in 241 Altgeld Hall,  Thursday, March 15, 2012
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Submitted by lukyane2.
 Peter Nelson (Department of Mathematics, University of Illinois at Urbana-Champaign)Formal groups in algebraic topologyAbstract: Formal groups are objects that lie between Lie groups and Lie algebras. I'll motivate their application to geometry and topology by discussing Chern classes of vector bundles. Then I'll talk about their role in algebraic topology. This theory provides a deep connection between topology and algebraic geometry and even number theory.

Tuesday, March 27, 2012

Topology Seminar
11:00 am   Tuesday, March 27, 2012
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Submitted by franklan.
 Seminar canceled today

Thursday, March 29, 2012

2:00 pm   in 241 Altgeld Hall,  Thursday, March 29, 2012
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Submitted by lukyane2.
 Santiago Camacho (UIUC Math)Some first order definable properties of Tropical GeometryAbstract: There are many different ways one can approach tropical geometry, approximation by amoebas generated by algebraic varieties in $\mathbb{C}^n$, the valuation map over algebraic varieties of general algebraically closed valued fields, or just simply the geometry of the min-plus tropical semiring of $\mathbb{R}$. On this talk we focus on these last two and give a sketch of a proof of the fundamental theorem of tropical geometry linking them together, using elementary model theoretic tools. No previous knowledge on logic will be a assumed.

Tuesday, April 3, 2012

Topology Seminar
11:00 am   Tuesday, April 3, 2012
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Submitted by franklan.
 No seminar todayAbstract: We resume next week.

Thursday, April 5, 2012

2:00 pm   in 241 Altgeld Hall,  Thursday, April 5, 2012
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Submitted by lukyane2.
 Grace Work (UIUC Math)Ideal Triangulations of Hyperbolic 3-ManifoldsAbstract: In this talk we will see how link complements can be decomposed into finite unions of ideal tetrahedra. From this decomposition we can compute the hyperbolic structure on these manifolds. The tetrahedra provide us with information we can use to compute arithmetic invariants of the manifolds. Our main example will be the complement of the figure-eight knot.

Tuesday, April 10, 2012

Topology Seminar
11:00 am   in 243 Altgeld Hall,  Tuesday, April 10, 2012
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Submitted by franklan.
 Qayum Khan (University of Notre Dame)Rigidity of pseudo-free group actions on contractible manifoldsAbstract: Joint with Frank Connolly (U Notre Dame) and Jim Davis (Indiana U). We discuss Quinn's equivariant generalization of the Borel Conjecture. This concerns cocompact proper actions of a discrete group $\Gamma$ on a Hadamard manifold $X$. We give a complete solution when the action of $\Gamma$ is pseudo-free and when $X$ more generally is a $\mathrm{CAT}(0)$ manifold. Here, pseudo-free means that the singular set is discrete. A rich class of examples is obtained from crystallographic groups $\Gamma$ made out of isometric spherical space form groups $G$. If $\Gamma$ has no elements of order two, then we obtain equivariant topological rigidity of the pair $(X, \Gamma)$. Hence, if $\Gamma$ is torsion-free, then we generalize a recent theorem of A. Bartels and W. Lück, which validates the classical Borel Conjecture for $\mathrm{CAT}(0)$ fundamental groups. Otherwise, if $\Gamma$ has elements of order two, we show how to parameterize all possible counter-examples, in terms of Cappell's $\mathrm{UNil}$ summands of the L-theory of infinite dihedral groups. In certain cases, these are detected along hypersurfaces in the orbifold $X / \Gamma$ by generalized Arf invariants.

Thursday, April 12, 2012

Special Topology Seminar
11:00 am   in 241 Altgeld Hall,  Thursday, April 12, 2012
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Submitted by franklan.
 Daniel Cohen (Louisiana State University)Topological complexity of configuration spacesAbstract: The topological complexity of a space is a homotopy type invariant motivated by the motion planning problem from robotics. We discuss this invariant in the context of configuration spaces of ordered points on orientable surfaces.

2:00 pm   in 241 Altgeld Hall,  Thursday, April 12, 2012
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Submitted by lukyane2.
 Kelly Funk (Department of Mathematics, University of Illinois at Urbana-Champaign)Rigidity Across DynamicsAbstract: We will discuss the notions of rigidity and uniform rigidity for dynamical systems. We will explore the structure of these sequences and attempt to characterize them.

Tuesday, April 17, 2012

Topology Seminar
11:00 am   in 243 Altgeld Hall,  Tuesday, April 17, 2012
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Submitted by franklan.
 Kevin Costello (Northwestern University)The elliptic genus from quantum field theoryAbstract: Witten proposed that the elliptic genus of a manifold should be the partition function of a certain sigma-model. I'll describe a rigorous version of this result, which also has an interpretation in derived algebraic geometry.

Thursday, April 19, 2012

2:00 pm   in 241 Altgeld Hall,  Thursday, April 19, 2012
 Del Edit Copy
Submitted by lukyane2.
 Francesco Bei (Sapienza Università di Roma)L^2 de Rham and Hodge theorems on stratified pseudomanifoldsAbstract: Even when studying smooth objects one often runs into singular spaces, but fortunately these often come with some extra structure: a stratification. After recalling what a stratification is and how they come up, I will show that there exist a class of Riemannian metrics whose L^2 de Rham and Hodge cohomology groups are isomorphic to certain topologically defined intersection cohomology groups with a perversity' that depends on the metric.

Tuesday, April 24, 2012

Topology Seminar
11:00 am   in 243 Altgeld Hall,  Tuesday, April 24, 2012
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Submitted by franklan.
 Anna Marie Bohmann (Northwestern University)Global equivariant K-theoryAbstract: Equivariant K-theory is one of the original examples of an equivariant homology theory, but it is surprisingly difficult to construct as an orthogonal spectrum, and thus as a global spectrum. I will highlight some of the difficulties that arise in building good equivariant K-theory spectra and discuss Joachim's construction of equivariant K-theory via $C^*$-algebras. Finally, I will explain why this construction yields a global version of K-theory.

Thursday, April 26, 2012

2:00 pm   in 241 Altgeld Hall,  Thursday, April 26, 2012
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Submitted by lukyane2.
 Caglar Uyanik (Department of Mathematics, University of Illinois at Urbana-Champaign)What is an Outer Space?Abstract: I will try to explain the construction and the topology of Outer space introduced by Culler and Vogtmann. In particular, I will sketch the proof of contractibility.

Tuesday, May 1, 2012

Topology Seminar
11:00 am   Tuesday, May 1, 2012
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Submitted by franklan.
 No seminar todayAbstract: We will resume in the fall.

Thursday, May 3, 2012

Special Graduate Geometry and Topology Seminar
12:00 pm   in 241 Altgeld Hall,  Thursday, May 3, 2012
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Submitted by lukyane2.
 Amir Nayyeri (UIUC Computer Science)How to Walk Your Dog in the MountainsAbstract: We describe a O(log n)-approximation algorithm for computing the homotopic Frechet distance between two polygonal curves that lie on the boundary of a surface. Prior to this work, algorithms where known only for curves on the Euclidean plane with polygonal obstacles. A key technical ingredient in our analysis is a $O(\log n)$-approximation algorithm for computing the minimum height of a homotopy between two curves. No algorithms were previously known for approximating this parameter. Surprisingly, it is not even known if computing either the homotopic Frechet distance, or the minimum height of a homotopy, is in NP. Joint work with Sariel Har-Peled, Mohammad Salavatipour and Anastasios Sidiropoulos

Thursday, August 30, 2012

2:00 pm   in 241 Altgeld Hall,  Thursday, August 30, 2012
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Submitted by collier3.
 Informational and Organizational meetingAbstract: It's time for the graduate geometry and topology seminar to start again. We will meet for a short organizational meeting. Everyone in strongly encouraged to attend, especially new first years. COOKIES will be provided.

Tuesday, September 4, 2012

Topology Seminar
11:00 am   in 243 Altgeld Hall,  Tuesday, September 4, 2012
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Submitted by franklan.
 Jim McClure (Purdue University)Verdier duality and Poincare dualityAbstract: There is a well-known argument that deduces Poincare duality from Verdier duality. In the lecture I will review the relevant sheaf-theoretic background and show that the isomorphism obtained in this way is the same as the classical isomorphism obtained from the cap product. As a byproduct I will observe that Verdier duality is not actually needed for the well-known argument mentioned above. Everything I will say has an analogue for intersection homology (in particular, Verdier duality is not needed for the proof of Poincare duality in that situation either); I'll say something about this at the end if time allows.

Thursday, September 6, 2012

2:00 pm   in 241,  Thursday, September 6, 2012
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Submitted by collier3.
 Anton Lukyanenko (UIUC Math)What is geometric group theory and who cares?Abstract: How do you tell if two groups are isomorphic? This is an extremely difficult task, but in certain cases attaching geometric notions to the groups makes it tractable and leads to new, intriguing geometries. The main example will come from the Heisenberg group, which with a (sub-)Riemannian metric becomes one of the 8 Thurston geometries.

Tuesday, September 11, 2012

Topology Seminar
11:00 am   in 243 Altgeld Hall,  Tuesday, September 11, 2012
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Submitted by franklan.
 Ben Ward (Purdue University)From pre-Lie to BVAbstract: We will investigate several extensions of the notion of an operad and consider algebraic structures encoded by and arising from them, including (pre-)Lie, Frobenius, Gerstenhaber and BV structures. Then, motivated by topological questions, we will study the example of $A_{\infty}$ Frobenius algebras and the associated BV (resp. homotopy BV) structure on the Hochschild cohomology (resp. cochains). This is partially joint work with Ralph Kaufmann and Javier Zuniga.

Thursday, September 13, 2012

2:00 pm   in 241 Altgeld Hall,  Thursday, September 13, 2012
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Submitted by collier3.
 Peter Nelson (UIUC Math)What is a (co)homology theory and why you should care.Abstract: The main goal of algebraic topology is to study spaces via various algebraic invariants. I'll give a brief introduction to the primary type of these invariants, namely, homology and cohomology theories. Examples and "geometric" applications will abound.

Tuesday, September 18, 2012

Topology Seminar
11:00 am   in 243 Altgeld Hall,  Tuesday, September 18, 2012
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Submitted by franklan.
 Sean Tilson (Wayne State University)Power operations and the Kunneth Spectral SequenceAbstract: Power operations have been constructed and successfully utilized in the Adams and Homological Homotopy Fixed Point Spectral Sequences by Bruner and Bruner-Rognes. It was thought that such results were not specific to the spectral sequence, but rather that they arose because highly structured ring spectra are involved. In this talk, we show that while the Kunneth Spectral Sequence enjoys some nice multiplicative properties, the obvious algebraic operations are zero (other than the square). Despite the negative results we are able to use old computations of Steinberger's with our current work to compute operations in the homotopy of some relative smash products.

Thursday, September 20, 2012

2:00 pm   in 241 Altgeld Hall,  Thursday, September 20, 2012
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Submitted by collier3.
 Mike DiPasquale (UIUC Math)Bezout, Cayley-Bacharach, and PascalAbstract: We introduce some basic constructions of algebraic geometry in the process of exploring the geometry of curves in the complex projective plane. In particular we will discuss Bezout's theorem and the Cayley-Bacharach theorem for plane cubics, pointing out the special case of Pascal's 'mystic hexagon.' The object is to communicate the power of algebraic machinery in proving some beautiful geometric theorems.

Tuesday, September 25, 2012

Topology Seminar
11:00 am   in 243 Altgeld Hall,  Tuesday, September 25, 2012
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Submitted by franklan.
 David Gepner (Universität Regensburg)Brauer groups of commutative ring spectraAbstract: The Picard and Brauer groups of a commutative ring spectrum R can be interpreted as the first two negative homotopy groups of a nonconnective version of the spectrum of units of R. We'll focus on two techniques for computing these negative homotopy groups: if R is connective, then any Azumaya R-algebra is etale locally trivial, and these groups reduce to the etale cohomology of the sheaf of units $GL_1$; if R is nonconnective, then this probably fails, but nevertheless R often admits interesting finite G-Galois extensions whose group cohomology computes the relative Picard and Brauer groups. This is joint work with B. Antieau and T. Lawson, respectively.

Thursday, September 27, 2012

2:00 pm   in 241 Altgeld Hall,  Thursday, September 27, 2012
 Del Edit Copy
Submitted by collier3.
 Bill Karr (UIUC Math)Geodesics on Surfaces of Revolution in Minkowski SpaceAbstract: I will introduce some basic definitions from Lorentzian geometry and a notion of angle in the tangent space to a Lorentzian manifold. Then, I'll explain my REGS project about geodesics on surfaces of revolution in Minkowski space using a spacetime version Clairaut's relation from classical differential geometry.

Tuesday, October 2, 2012

Topology Seminar
11:00 am   in 243 Altgeld Hall,  Tuesday, October 2, 2012
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Submitted by franklan.
 Inna Zakharevich (University of Chicago)Ring structures on scissors congruence spectraAbstract: Hilbert's third problem asks the following question: given two polyhedra, when is it possible to dissect them into a finite number of pairwise congruent polyhedra? The answer, given by the Dehn-Sydler theorem (1901,1965) is that it is possible whenever two invariants -- the volume and the Dehn invariant -- are equal. Generalizing this problem, we can say that two polytopes in a nice enough manifold (such as $R^n$, $S^n$, or $H^n$) are "scissors congruent" if they can be dissected into a finite number of pairwise congruent polytopes and ask for a classification of scissors congruence types. This question was studied by Dupont and Sah, who assigned groups of scissors congruence types on manifolds and analyzed many structures on these groups. In particular, it turns out that in the case of $E^n$ and $S^n$, the groups assemble into a graded ring. In this talk we give a different perspective on scissors congruence groups by showing that they arise as the 0-th K-group of a particular type of Waldhausen category, and use Dupont and Sah's observations to construct these ring structures directly on the K-theoretic level.

Thursday, October 4, 2012

2:00 pm   in 241 Altgeld Hall,  Thursday, October 4, 2012
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Submitted by collier3.
 Sarah Yeakel (UIUC Math)What is Calculus of Functors?Abstract: Functors between categories give a wealth of information, but can be extremely complicated to compute. We teach our calculus students to approximate difficult real valued functions with polynomials. This technique can be applied to categories we know and love to produce some awesome results. I'll talk about some basic examples of functors on topological spaces and how the theory can be used on manifolds and vector spaces as well.

Tuesday, October 9, 2012

Topology Seminar
11:00 am   in 243 Altgeld Hall,  Tuesday, October 9, 2012
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Submitted by franklan.
 Marc Hoyois (Northwestern University)From algebraic cobordism to motivic cohomologyAbstract: I will present a famous theorem due to Hopkins and Morel which characterizes motivic cohomology of schemes as the universal oriented cohomology theory with additive formal group law. The talk will serve as a gentle introduction to motivic homotopy theory, geared towards understanding the statement of the theorem and some of its stunning applications.

Thursday, October 11, 2012

2:00 pm   in Altgeld Hall,  Thursday, October 11, 2012
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Submitted by collier3.
 Mychael Sanchez (UIUC Math)Homotopy theory and infinity categoriesAbstract: I'll talk about several objects of interest in homotopy theory, some properties they share, and some deficiencies of ordinary category theory when studying them. I'll then discuss a class of mathematical objects called infinity categories that correct some these deficiencies and how we might come up with them.

Tuesday, October 16, 2012

Topology Seminar
11:00 am   in 243 Altgeld Hall,  Tuesday, October 16, 2012
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Submitted by franklan.
 Anthony Elmendorf (Purdue University Calumet)Generalized and equivariant multicategoriesAbstract: Leinster developed a general procedure for defining multicategories using monads satisfying a Cartesian property. However, his definition only captures the usual notion of multicategory in the case when the multicategory doesn't have a symmetric structure. By looking at additional structure available when we start with a general $\Sigma$-free operad in Cat, we can account for the symmetric structure, and thereby generalize Leinster's construction to this case. The construction also easily accounts for the sort of equivariant operads considered by Guillou, May, and Merling in their ongoing study of equivariant infinite loop space theory via equivariant permutative categories. There are several natural conjectures that we will discuss.

Thursday, October 18, 2012

2:00 pm   in 241 Altgeld Hall,  Thursday, October 18, 2012
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Submitted by collier3.
 Seth Wolbert (UIUC Math)Stacks in Differential GeometryAbstract: A stack over the category of smooth manifolds is a structure that can be used to generalize the deconstructive (i.e., via restriction) and reconstructive (i.e.,via gluing) properties seen in fiber bundles. This talk is designed to give a gentle introduction to these structures and some of their nice properties. Given time, we will also discuss the stack of transport functors and how parallel transport induces an equivalence of categories between this stack and the stack of principal G-bundles with connections.

Tuesday, October 23, 2012

Topology Seminar
11:00 am   in 243 Altgeld Hall,  Tuesday, October 23, 2012
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Submitted by franklan.
 Agnes Beaudry (Northwestern University)The K(2)-local Moore spectrum at the prime 2Abstract: We use methods of Goerss, Henn, Karamanov, Mahowald, and Rezk to study the homotopy $\pi_*L_{K(2)}V(0)$ at the prime 2. In particular, we study the $E_2$-page of an Adams-Novikov spectral sequence converging to $\pi_*L_{K(2)}V(0)$ via another spectral sequence called the short resolution spectral sequence. Its $E_1$-page is composed of cohomology groups $H^n(G_k, (E_2)_*V(0))$ where $(E_2)_*$ is Morava $E$-theory and the $G_k$'s are finite subgroups of the Morava stabilizer group $\mathbb{G}_2$. These finite groups come from automorphisms of elliptic curves and we use the geometry thus made available to us to simplify computations. We explain how to obtain the $E_2$-page of the short resolution spectral sequence for $(E_2)_*V(0)$ and compute the complete spectral sequence for $v_1^{-1}(E_2)_*V(0)$. This gives the $E_2$-page of the Adams-Novikov spectral sequence, namely $H^*(\mathbb{G}_2, v_1^{-1}(E_2)_*V(0))$. We expect the differentials to follow classical patterns and explain what this would imply for $v_1^{-1}\pi_*L_{K(2)}V(0)$.

Thursday, October 25, 2012

2:00 pm   in 241 Altgeld Hall,  Thursday, October 25, 2012
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Submitted by collier3.
 Nathan Rehfuss (UIUC MATH)The Unknot and Why You Should Care.Abstract: I plan to give a brief general overview of knot theory, followed by an exploration of the methods, challenges, and applications of the unknotting problem.

Tuesday, October 30, 2012

Topology Seminar
11:00 am   in 243 Altgeld Hall,  Tuesday, October 30, 2012
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Submitted by franklan.
 Matthew Thibault (University of Chicago)Finite simplicial complexes via pro-posetsAbstract: One has a pair of functors between finite topological spaces and finite simplicial complexes. Via this correspondence, McCord proves that finite topological spaces up to weak homotopy equivalence coincides with finite simplicial complexes up to homotopy equivalence. Since finite topological spaces coincide with finite posets, this allows one to convert problems in algebraic topology into problems in combinatorics. However, due to a dearth of maps in the category of finite spaces, one must enlarge this category in order to describe all homotopy classes of maps between (finite) simplicial complexes. In this talk, I will describe the homotopy category of finite simplicial complexes in terms of the category of pro-posets.

Wednesday, October 31, 2012

Special Topology Seminar
4:00 pm   in 145 Altgeld Hall,  Wednesday, October 31, 2012
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Submitted by franklan.
 Irakli Patchkoria (Universität Bonn)Rigidity in equivariant stable homotopy theoryAbstract: Let G be a finite abelian group or finite (non-abelian) 2-group. We show that the 2-local G-equivariant stable homotopy category, indexed on a complete G-universe, has a unique G-equivariant model in the sense of Quillen model categories. This means that the suspension functor, homotopy cofiber sequences and the stable Burnside category determine all "higher order structure" of the 2-local G-equivariant stable homotopy category such as for example equivariant homotopy types of function G-spaces. The theorem can be seen as an equivariant generalization of Schwede's rigidity theorem at the prime 2.

Thursday, November 1, 2012

2:00 pm   in 241 Altgeld Hall,  Thursday, November 1, 2012
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Submitted by collier3.
 Juan Villeta-Garcia (UIUC Math)Beginner Intersection Theory in Algebraic GeometryAbstract: Given two varieties V and W in \mathbb{P}^n, understanding their intersection V\cap W has been a subject of constant research for most of the 20th century. Many definitions of what an intersection product should be have been given, and subsequently refined. We will give a gentle introduction from the algebraic approach, but also incorporate such constructions as Chern classes and Chow Rings, if time permits. We will have lots of examples!

Tuesday, November 6, 2012

Topology Seminar
11:00 am   in 243 Altgeld Hall,  Tuesday, November 6, 2012
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Submitted by franklan.
 Jim Davis (Indiana University)Every finite group acts freely and homologically trivially on a product of spheresAbstract: I show that if K is a finite CW complex with finite fundamental group G and universal cover homotopy equivalent to $X=S^{n_1} \times \ldots \times S^{n_k}$, then for every $n \geq dim X$, G acts freely on $X \times S^n$, with the action on homology given by $g \otimes 1 \colon H_*(X) \otimes H_*(S^n) \to H_*(X) \otimes H_*(S^n)$. Recently Unlu and Yalcin constructed for any finite group G, a finite CW complex K with universal cover homotopy equivalent to a product of spheres, where G acts trivially on the homology of the universal cover of K. As a corollary we get the title of the talk.

Thursday, November 8, 2012

2:00 pm   in 241 Altgeld Hall,  Thursday, November 8, 2012
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Submitted by collier3.
 (UIUC Math)Abstract: Cancelled this week. Go to women's seminars instead.

Tuesday, November 13, 2012

Topology Seminar
11:00 am   in 243 Altgeld Hall,  Tuesday, November 13, 2012
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Submitted by franklan.
 Jonathan Campbell (Stanford University)Topological Hochschild homology and Koszul dualityAbstract: Topological Hochschild homology (THH) is an invariant of ring spectra related both to K-theory and topological field theories. In this talk I'll state and prove a theorem concerning the relationship between THH and Koszul duality. I'll introduce the necessary definitions, and in particular say what I mean by "Koszul duality". I will also introduce some $\infty$-categorical background that will be necessary for the proof. Finally, I'll discuss some related results that I believe to be true, and applications of the work above to topological field theories.

Thursday, November 15, 2012

2:00 pm   in 241 Altgeld Hall,  Thursday, November 15, 2012
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Submitted by collier3.
 Brian Collier (UIUC Math)Flat Bundles and Representations of the Fundamental Group.Abstract: Given a manifold $M$ and a vector space V, a representation $\rho:\pi_1(M)\rightarrow GL(V)$ gives rise to a flat vector bundle via associated bundles and the action of $\pi_1(M)$ on the universal cover. Conversely given a vector bundle with a flat structure we get a representation of $\pi_1(M)$. To appreciate this correspondence we will need to discuss some general bundle theory and flat structures this will be done through many examples. This talk should be accessible to all graduate students interested in geometry.

Tuesday, November 27, 2012

Topology Seminar
11:00 am   in 243 Altgeld Hall,  Tuesday, November 27, 2012
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Submitted by franklan.
 Lena Folwaczny (University of Illinois at Chicago)New constructions of virtual knot polynomialsAbstract: Virtual knots and links can be described topologically as embeddings of circles in thickened surfaces (of arbitrary genus) taken up to surface homeomorphisms and 1-handle stabilization. In this talk we give an alternate definition of a virtual knot polynomial, the Affine Index Polynomial, using virtual linking numbers. We call this new definition the Wriggle Polynomial. The Affine Index Polynomial is defined in terms of an integer labeling system of a virtual knot diagram that derives from an essentially unique structure of an affine flat biquandle for flat virtual diagrams, and equality of the definitions is not immediately obvious. Interesting applications of this polynomial to Vassiliev Invariants, Mutant Knots, and the Cosmetic Crossing Change Conjecture are discussed.

Thursday, November 29, 2012

2:00 pm   in 241 Altgeld Hall,  Thursday, November 29, 2012
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Submitted by collier3.
 Nerses Aramian (UIUC Math)Milnor's Exotic SphereAbstract: In 1956 Milnor constructed an example of a differentiable manifold that is homeomorphic to $S^7$, but is not diffeomorphic to it. The existence of such an object is quite remarkable, since it shows that not everything about a manifold is determined by its topological structure. In fact, later it was shown that for the case of spheres it is a rare phenomenon to have a unique differentiable structure. I am going to attempt to walk you through the construction of the exotic $S^7$. As the discussion progresses there will be a need to introduce several tools, such as Poincare Duality, Oriented Bordism, Pontrjagin Classes, Hirzebruch Signature Formula. I will attempt to make some of the discussion more homotopical'', so as to convince you that homotopy theory may provide a gateway to generalizations of Milnor's argument.

Tuesday, December 4, 2012

Topology Seminar
11:00 am   in 243 Altgeld Hall,  Tuesday, December 4, 2012
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Submitted by franklan.
 Stephan Stolz (University of Notre Dame)2-dimensional field theories and modular formsAbstract: Graeme Segal suggested two decades ago that the generalized cohomology theory now known as "Topological Modular Form Theory" of a manifold X should be related to families of 2-dimensional field theories parametrized by X. This is an analog of the well-known statement that homotopy classes of families of Fredholm operators parametrized by X can be identified with the K-theory of X. In this talk on joint work with Peter Teichner, I will present a conjectural picture of TMF(X) as concordance classes of families of supersymmetric 2-dimensional Euclidean field theories parametrized by X. Evidence for the conjecture comes from an analogous description of K(X) in terms of 1-dimensional field theories, and our result that the partition function of a supersymmetric 2-dimensional Euclidean field theory is a modular form. The latter is the main focus of the talk.

Thursday, December 6, 2012

2:00 pm   in 241 Altgeld Hall,  Thursday, December 6, 2012
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Submitted by collier3.
 Daniel Hockensmith (UIUC Math)An Introduction to JetsAbstract: What is a jet bundle and how does one use it? One expects multiple answers to the latter question, but it is somewhat surprising that there are also multiple answers to the former. We will highlight one approach to jet bundles and apply it to the study of PDE's. In the process, we will discover relationships between geometric objects (i.e. curvature of connections on a fiber bundle) and historically analytic objects (i.e. PDE's and their solutions). Our focus will be placed upon the first jet bundle as there is a plethora of readily visualized examples, but we will certainly talk about jet bundles of arbitrary finite order.

Tuesday, December 11, 2012

Topology Seminar
11:00 am   in 243 Altgeld Hall,  Tuesday, December 11, 2012
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Submitted by franklan.
 John Harper (Purdue University)TQ-homology completion of nilpotent structured ring spectraAbstract: An important theme in current work in homotopy theory is the investigation and exploitation of enriched algebraic structures on spectra that naturally arise, for instance, in algebraic topology, algebraic K-theory, and derived algebraic geometry. Such structured ring spectra or `geometric rings'' are most simply viewed as algebraic-topological generalizations of the notion of ring from algebra and algebraic geometry. This talk will describe recent progress, in joint work with M. Ching, on developing standard tools of the homotopy theory of spaces in this new algebraic-topological context of structured ring spectra, with a special emphasis on recovering algebraic and topological structures from associated homology objects.

Thursday, December 13, 2012