Seminar Calendar
for Topology Seminar events the year of Friday, April 21, 2017.

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More information on this calendar program is available.
Questions regarding events or the calendar should be directed to Tori Corkery.
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Friday, January 20, 2017

Graduate Geometry/Topology Seminar
4:00 pm   in 241 Altgeld Hall,  Friday, January 20, 2017
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Submitted by dcarmod2.
Organizational Meeting
Abstract: This is the organizational meeting to get the schedule of talks down for the spring. If you think you might be interested in giving a talk at some point, please attend!

Friday, January 27, 2017

Graduate Geometry/Topology Seminar
4:00 pm   in 241 Altgeld Hall,  Friday, January 27, 2017
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Submitted by dcarmod2.
Daniel Carmody (UIUC Math)
Galois Categories and the Topological Galois Correspondence
Abstract: Classical Galois theory for fields gives a correspondence between closed subgroups of the Galois group of a Galois extension and intermediate subfields. The theory of covering spaces in topology gives a correspondence between connected coverings of nice spaces and subgroups of the fundamental group. The purpose of this talk is to explain the relationship between (and generalization) of these two theorems.

Tuesday, January 31, 2017

Topology Seminar
11:00 am   in 345 Altgeld Hall,  Tuesday, January 31, 2017
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Submitted by rezk.
Charles Rezk (Illinois)
Complex analytic elliptic cohomology and Looijenga line bundles
Abstract: I'll explain how, by taking the cohomology of suitable spaces and messing around a bit, you can get things like: the moduli stack of (analytic) curves, the universal curve, and Looijenga line bundles over these. This seems to have some relevance for the construction of complex analytic elliptic cohomology.

Friday, February 3, 2017

Graduate Geometry/Topology Seminar
4:00 pm   in 241 Altgeld Hall,  Friday, February 3, 2017
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Submitted by dcarmod2.
Marissa Loving (UIUC Math)
Train Tracks on Surfaces
Abstract: Our mantra throughout the talk will be simple, "Train tracks approximate simple closed curves." Our goal will be to explore some examples of train tracks, draw some meaningful pictures, and develop an analogy between train tracks and another well known method of approximation. No great knowledge of anything is required for this talk as long as one is willing to squint their eyes at the blackboard a bit at times.

Friday, February 10, 2017

Graduate Geometry/Topology Seminar
4:00 pm   in 241 Altgeld Hall,  Friday, February 10, 2017
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Submitted by dcarmod2.
Georgios Kydonakis (UIUC Math)
Opers and non-abelian Hodge theory
Abstract: We will describe two different families of flat $G$-connections over a compact Riemann surface for a complex, simple, simply connected Lie group $G$. The first is the family of $G$-opers, which for $G=\text{SL(2}\text{,}\mathbb{C}\text{)}$ can be thought of as global versions of the locally defined second order Schrödinger operators. The second comes from a particular subfamily of solutions to the so-called $G$-Hitchin equations. The physicist Davide Gaiotto conjectured that for $G=\text{SL(}n\text{,}\mathbb{C}\text{)}$ the second family in a scaling limit converges to a limiting connection which has the structure of an oper. We will describe a proof of this conjecture. This is joint work with Olivia Dumitrescu, Laura Fredrickson, Rafe Mazzeo, Motohico Mulase and Andrew Neitzke.

Friday, February 17, 2017

Graduate Geometry/Topology Seminar
4:00 pm   in 241 Altgeld Hall,  Friday, February 17, 2017
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Submitted by dcarmod2.
Melinda Lanius (UIUC Math)
Hyperbolic taxi cabs and conic kitty cats: a mathematical activity and coloring book
Abstract: In this extremely interactive talk, we will develop intuition for various metrics that I have encountered in my own research. We’ll work our way through understanding more familiar spaces such as the real plane as well as hyperbolic plane and disk, to less familiar objects: such as a surface with a Euclidean, cylindrical, or hyperbolic-funnel end. Some markers and colored pencils will be provided, but please feel free to bring your own fun office supplies.

Friday, February 24, 2017

Graduate Geometry/Topology Seminar
4:00 pm   in 241 Altgeld Hall,  Friday, February 24, 2017
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Submitted by dcarmod2.
Bill Karr (UIUC Math)
Geometry of convex hypersurfaces
Abstract: A convex hypersurface in Euclidean space or Minkowski space is the boundary of an open convex set. Smooth convex hypersurfaces have non-negative sectional curvature and indicate properties of more general Riemannian manifolds with non-negative curvature. I will discuss some properties of convex hypersurfaces. Finally, I will describe a problem that arises from Lorentzian geometry involving convex hypersurfaces and geodesic connectedness and discuss a possible solution to this problem.

Tuesday, March 7, 2017

Topology Seminar
11:00 am   in 345 Altgeld Hall,  Tuesday, March 7, 2017
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Submitted by rezk.
Guillaume Brunerie (IAS)
Invariant homotopy theory in homotopy type theory
Abstract: This talk will be about homotopy type theory and in particular the branch of it known as invariant homotopy theory, or synthetic homotopy theory.

The main idea is that homotopy type theory is a formal language which can be used to talk about "spaces-up-to-homotopy-equivalence". The basic objects can be thought of as spaces, but the language has the property that all the structures, properties, constructions and proofs that we can express are invariant under homotopy equivalence.

One advantage is that every construction or proof done in this setting is expected to be automatically valid in every infinity-topos, not just in the infinity-topos of spaces, while still looking elementary. In this sense, we can see homotopy type theory as an internal language for infinity-topoi. Moreover, such proofs are also amenable to computer formalization, as homotopy type theory is strongly related to computer proof assistants.

I will present the basic concepts and show what a few proofs and constructions look like in invariant homotopy theory. In particular, we will see the universal cover of the circle, the Hopf fibration, cohomology, and the Steenrod operations.


Graduate Student Analysis Seminar
4:00 pm   in 131 English Building,  Tuesday, March 7, 2017
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Submitted by compaan2.
Derek Jung   [email] (UIUC Math)
A variant of Gromov's H\"older equivalence problem for small step Carnot groups
Abstract: This is the second part of a talk I gave last semester in the Graduate Geometry/Topology Seminar. A Carnot group is a Lie group that may be identified with its Lie algebra via the exponential map. This allows one to view a Carnot group as both a sub-Riemannian manifold and a geodesic metric space. It is then natural to ask the following general question: When are two Carnot groups equivalent? In this spirit, Gromov studied the problem of considering for which $k$ and $\alpha$ there exists a locally $\alpha$-H\"older homeomorphism $f:\mathbb{R}^k\to G$. Very little is known about this problem, even for the Heisenberg groups. By tweaking the class of H\"older maps, I will discuss a variant of Gromov's problem for Carnot groups of step at most three. This talk is based on a recently submitted paper. Some knowledge of differential geometry and Lie groups will be helpful.

Friday, March 10, 2017

Graduate Geometry Topology Seminar
4:00 pm   in 241 Altgeld Hall,  Friday, March 10, 2017
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Submitted by penciak2.
Stefan Klajbor Goderich (UIUC Math)
Stability of relative equilibria and isomorphic vector fields
Abstract: We present applications of the notion of isomorphic vector fields to the study of nonlinear stability of relative equilibria. Isomorphic vector fields were introduced by Hepworth in his study of vector fields on differentiable stacks. Here we argue in favor of the usefulness of replacing an invariant vector field on a manifold by an isomorphic one to study nonlinear stability of relative equilibria. In particular, we use this idea to obtain a criterion for nonlinear stability. As an application, we sketch how to use this to obtain Montaldi and Rodrı́guez-Olmos’s criterion for stability of Hamiltonian relative equilibria.

Friday, March 17, 2017

Graduate Geometry/Topology Seminar
4:00 pm   in 241 Altgeld Hall,  Friday, March 17, 2017
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Submitted by dcarmod2.
Matthew Romney (UIUC Math)
A 50-minute peek into the quasi-world
Abstract: Quasiconformal geometry is the dominant research area which evolved from complex analysis in the 20th century and remains active today. This talk will give a friendly overview to the subject, from its roots in the classical Riemann mapping theorem and Liouville theorem on conformal mappings, to some of its compelling applications in other fields, including complex dynamics and geometric group theory.

Tuesday, March 28, 2017

Topology Seminar
11:00 am   in 345 Altgeld Hall,  Tuesday, March 28, 2017
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Submitted by cmalkiew.
Cary Malkiewich (UIUC)
Periodic orbits and topological restriction homology
Abstract: This talk is about an emerging connection between algebraic $K$-theory and free loop spaces on the one hand, and periodic orbits of continuous dynamical systems on the other. The centerpiece is a construction in equivariant stable homotopy theory called the "$n$th power trace," which relies on the equivariant norm construction of Hill, Hopkins, and Ravenel. This trace is a refinement of the Lefschetz zeta function of a map $f$, which detects not just fixed points but also periodic orbits of $f$. The applications so far include the resolution of a conjecture of Klein and Williams, and a new approach for the computation of transfer maps in algebraic $K$-theory. These projects are joint work with John Lind and Kate Ponto.

Friday, March 31, 2017

Graduate Geometry/Topology Seminar
4:00 pm   in 241 Altgeld Hall,  Friday, March 31, 2017
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Submitted by dcarmod2.
Sarah Mousley (UIUC Math)
Exotic limit sets of geodesics in Teichmuller space
Abstract: In 1975, Masur proved that the Teichmuller space of a surface of genus at least 2 is not Gromov hyperbolic. Since then, many have explored to what extent Teichmuller space has features of negative curvature. In a Gromov hyperbolic space, a geodesic ray converges to a unique point in the hierarchically hyperbolic space (HHS) boundary. We will present our result that a geodesic ray in Teichmuller space does not necessarily converge to a unique point in the HHS boundary of Teichmuller space. In fact, the limit set of a ray can be almost anything allowed by topology. The goal of this talk is not to prove the result, but rather to give necessary background to understand the statement. In particular, we will not assume knowledge of Teichmuller theory or HHS structures.

Tuesday, April 4, 2017

Topology Seminar
11:00 am   in 345 Altgeld Hall,  Tuesday, April 4, 2017
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Submitted by rezk.
Dan Ramras (IUPUI)
Coassembly for representation spaces
Abstract: I'll describe a homotopy-theoretical framework for studying the relationships between (families of) finite-dimensional unitary representations, vector bundles, and flat connections. Applications to surfaces, 3-manifolds, and groups with Kazhdan's property (T) will be discussed.

Friday, April 7, 2017

Graduate Geometry/Topology Seminar
4:00 pm   in 241 Altgeld Hall,  Friday, April 7, 2017
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Submitted by dcarmod2.
William Balderrama (UIUC Math)
Stable Phenomena in Algebraic Topology
Abstract: A phenomenon might be called stable if it happens the same way in every dimension. For example, if $C_\bullet$ is a chain complex, then $H_\ast C_\bullet = H_{\ast+1}C_{\bullet+1}$: ``taking homology'' is done the same in every dimension. In some cases, a construction might not be stable, but can be stabilized. For example, if $M$ is a smooth closed manifold, choice of distinct embeddings $i,j\colon M\rightarrow \mathbb{R}^n$ give rise to possibly nonisomorphic choices of normal bundles $N_iM$ and $N_jM$. However, we can stabilize this by adding trivial bundles: $N_iM\oplus k \simeq N_jM \oplus k$ for sufficiently large $k$, leading to the notion of the stable normal bundle. In this talk, I will introduce this notion of stability, and propose spectra, the main objects in stable homotopy theory, as a good way for dealing with it.

Tuesday, April 11, 2017

Topology Seminar
11:00 am   in 345 Altgeld Hall,  Tuesday, April 11, 2017
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Submitted by cmalkiew.
Kate Ponto (U Kentucky)
Traces for periodic point invariants
Abstract: Up to homotopy, the Lefschetz number and its refinement to the Reidemeister trace capture the essential information about fixed points of an endomorphism. These invariants can be applied to iterates of an endomorphism to describe periodic points, but in this case they provide far less complete information. I will describe an approach to refining these invariants through refinements of the associated symmetric monoidal and bicategorical traces. This gives richer invariants that also apply to endomorphisms of spaces with more structure (such as bundles).

Friday, April 14, 2017

Graduate Geometry/Topology Seminar
4:00 pm   in 241 Altgeld Hall,  Friday, April 14, 2017
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Submitted by dcarmod2.
Chris Gartland (UIUC Math)
Net and Filter Convergence Spaces
Abstract: A net or filter convergence space is a set together with a collection of data that axiomatizes the notion of convergence to an element of that set. In this sense, convergence spaces generalize topological spaces. More specifically, we will define the (equivalent) categories of net and filter convergence spaces and show that they contain the category of topological spaces (Top) as a full subcategory. We'll highlight some of the advantages these categories have over Top, especially in relation to Tychonoff's theorem. This talk is based off a series of blog posts by Jean Goubault-Larrecq, http://projects.lsv.ens-cachan.fr/topology/?page_id=785.

Tuesday, April 18, 2017

Topology Seminar
11:00 am   in 345 Altgeld Hall,  Tuesday, April 18, 2017
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Submitted by cmalkiew.
Nima Rasekh (UIUC)
Representable Cartesian Fibrations
Abstract: The goal of this talk is to introduce a notion of a representable Cartesian fibration. Getting there will take us on a journey to many different places in higher category land. We will start by discussing right fibrations, which model presheaves, and then move on to generalize it to Cartesian fibrations. Finally we will have our last stop in the realm of complete Segal objects, which will enable us to define and discuss representable Cartesian fibrations.

Friday, April 21, 2017

Graduate Geometry/Topology Seminar
4:00 pm   in 241 Altgeld Hall,  Friday, April 21, 2017
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Submitted by dcarmod2.
Joshua Wen (UIUC Math)
It’s hard being positive: symmetric functions and Hilbert schemes
Abstract: Macdonald polynomials are a remarkable basis of $q,t$-deformed symmetric functions that have a tendency to show up various places in mathematics. One difficult problem in the theory was the Macdonald positivity conjecture, which roughly states that when the Macdonald polynomials are expanded in terms of the Schur function basis, the corresponding coefficients lie in $\mathbb{N}[q,t]$. This conjecture was proved by Haiman by studying the geometry of the Hilbert scheme of points on the plane. I’ll give some motivations and origins to Macdonald theory and the positivity conjecture and highlight how various structures in symmetric function theory are manifested in the algebraic geometry and topology of the Hilbert scheme. Also, if you like equivariant localization computations, then you’re in luck!

Tuesday, April 25, 2017

Topology Seminar
11:00 am   in 345 Altgeld Hall,  Tuesday, April 25, 2017
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Submitted by rezk.
Carmen Rovi (Indiana)
The signature modulo 8 of a fiber bundle
Abstract: In this talk we shall be concerned with the residues modulo 4 and modulo 8 of the signature of a 4k-dimensional geometric Poincare complex. I will explain the relation between the signature modulo 8 and two other invariants: the Brown-Kervaire invariant and the Arf invariant. In my thesis I applied the relation between these invariants to the study of the signature modulo 8 of a fiber bundle, showing in particular that the non-multiplicativity of the signature modulo 8 is detected by an Arf invariant. In 1973 Werner Meyer used group cohomology to show that a surface bundle has signature divisible by 4. I will discuss current work with David Benson, Caterina Campagnolo and Andrew Ranicki where we are using group cohomology and representation theory of finite groups to detect non-trivial signatures modulo 8 of surface bundles.​

Tuesday, May 2, 2017

Topology Seminar
11:00 am   in 345 Altgeld Hall,  Tuesday, May 2, 2017
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Submitted by rezk.
Nathan Perlmutter (Stanford)
Parametrized Morse Theory, Cobordism Categories, and Positive Scalar Curvature
Abstract: In this talk I will construct a cobordism category consisting of manifolds equipped with a choice of Morse function, whose critical points occupy a prescribed range of degrees. I will identify the homotopy the of this cobordism category with the infinite loopspace of a certain Thom spectrum. Using the parametrized version of the Gromov-Lawson construction, I will then show how to use this cobordism category to probe the space of positive scalar curvature metrics on a closed, spin manifold of dimension > 4. Our main result detects many non-trivial homotopy groups in this space of positive scalar curvature metrics.