Department of

Mathematics


Seminar Calendar
for Graduate Geometry/Topology Seminar events the year of Wednesday, September 13, 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

4:00 pm in 241 Altgeld Hall,Friday, January 20, 2017

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

4:00 pm in 241 Altgeld Hall,Friday, January 27, 2017

Galois Categories and the Topological Galois Correspondence

Daniel Carmody (UIUC Math)

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.

Friday, February 3, 2017

4:00 pm in 241 Altgeld Hall,Friday, February 3, 2017

Train Tracks on Surfaces

Marissa Loving (UIUC Math)

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

4:00 pm in 241 Altgeld Hall,Friday, February 10, 2017

Opers and non-abelian Hodge theory

Georgios Kydonakis (UIUC Math)

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

4:00 pm in 241 Altgeld Hall,Friday, February 17, 2017

Hyperbolic taxi cabs and conic kitty cats: a mathematical activity and coloring book

Melinda Lanius (UIUC Math)

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

4:00 pm in 241 Altgeld Hall,Friday, February 24, 2017

Geometry of convex hypersurfaces

Bill Karr (UIUC Math)

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

4:00 pm in 131 English Building,Tuesday, March 7, 2017

A variant of Gromov's H\"older equivalence problem for small step Carnot groups

Derek Jung   [email] (UIUC Math)

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 17, 2017

4:00 pm in 241 Altgeld Hall,Friday, March 17, 2017

A 50-minute peek into the quasi-world

Matthew Romney (UIUC Math)

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.

Friday, March 31, 2017

4:00 pm in 241 Altgeld Hall,Friday, March 31, 2017

Exotic limit sets of geodesics in Teichmuller space

Sarah Mousley (UIUC Math)

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.

Friday, April 7, 2017

4:00 pm in 241 Altgeld Hall,Friday, April 7, 2017

Stable Phenomena in Algebraic Topology

William Balderrama (UIUC Math)

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.

Friday, April 14, 2017

4:00 pm in 241 Altgeld Hall,Friday, April 14, 2017

Net and Filter Convergence Spaces

Chris Gartland (UIUC Math)

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.

Friday, April 21, 2017

4:00 pm in 241 Altgeld Hall,Friday, April 21, 2017

It’s hard being positive: symmetric functions and Hilbert schemes

Joshua Wen (UIUC Math)

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!

Friday, September 1, 2017

4:00 pm in 241 Altgeld Hall,Friday, September 1, 2017

Organizational Meeting

Abstract: This is the organizational meeting at which we schedule the talks for the semester. If you think you might be interested in giving a talk at some point, you're highly encouraged to come. As usual, there will be cookies.

Friday, September 8, 2017

4:00 pm in 241 Altgeld Hall,Friday, September 8, 2017

An Introduction to Classifying Spaces

Daniel Carmody (UIUC Math)

Abstract: In this introductory talk I'll begin by recalling the notion of a principal $G$-bundle, then I'll move to discussing how one constructs classifying spaces of such things. In the process, I'll introduce simplicial spaces and describe the relationship between simplicial spaces and topological spaces.

Friday, September 15, 2017

4:00 pm in 241 Altgeld Hall,Friday, September 15, 2017

From Groups to Metric Spaces: A brief introduction to coarse geometry

Marissa Loving (UIUC)

Abstract: In this talk we will explore the ways in which we can think of a group as a metric space and some of the notions of equivalence that can be developed to make this analogy precise. Our overarching goal will be to recognize some of the powerful tools coarse geometry makes available to us, from retrieving fine algebraic data from coarse geometric information to relating purely topological objects to purely algebraic ones. No special background will be needed as we will aim to define anything beyond what would typically be covered in Math 500.

Friday, September 22, 2017

4:00 pm in 241 Altgeld Hall,Friday, September 22, 2017

Spectral Gaps, Dynamic Maps, Groups convex and co-compact

Hadrian Quan (UIUC)

Abstract: What do limit sets of group actions, solutions of the linear wave equation, and zeta functions all have in common? They’ll all appear in this talk in surprising ways. By the end of the talk I hope to convince you that their relation is more than surface-deep. This will be an introductory talk, with lots of pictures and examples and little assumed beyond knowledge of the fundamental group of a surface.

Friday, September 29, 2017

4:00 pm in 241 Altgeld Hall,Friday, September 29, 2017

Framed cobordisms in algebraic topology

Pedro Mendes De Araujo (UIUC)

Abstract: The Thom-Pontryagin construction was the first machinery developed to compute homotopy groups of spheres, in terms of framed cobordism classes of manifolds embedded in Euclidean space. Although much less successful at that than the algebraic machinery developed later, it has the advantage of being highly geometric and intuitive. In this talk, which will hopefully be fun, full of pictures, and requiring little more than an acquaintance with manifolds (and a high tolerance to geometric hand waving), we'll look at how it can be used to compute pi_{n+1} S^n. On the way, we'll give a very explicit proof of the Freudenthal suspension theorem.

Friday, October 6, 2017

4:00 pm in 241 Altgeld Hall,Friday, October 6, 2017

Homological Mirror Symmetry for P^1

Jesse Huang (UIUC)

Abstract: Homological Mirror Symmetry, conjectured by Kontsevich in 1994, is a fairytale between invariants of $X$ and its mirror $X^{mir}$, exchanging algebraic and symplectic data between the pair. In this talk, I will try to make sense of this for $\mathbb P^1$, which is very well understood and usually spoken of as a popular example advertising one version of the conjecture.

Friday, October 13, 2017

4:00 pm in 241 Altgeld Hall,Friday, October 13, 2017

Ribbon Graphs in Geometry

Cameron Rudd (UIUC)

Abstract: Ribbon graphs are combinatorial structures that correspond naturally to Riemann surfaces. In this talk I will introduce ribbon graphs and describe their relationship with Riemann surfaces, emphasizing their geometric content. Depending on time, I will also discuss some striking applications of these objects in mathematics.

Friday, October 20, 2017

4:00 pm in 241 Altgeld Hall,Friday, October 20, 2017

H-principle for Symplectic structure

Venkata Sai Narayana Bavisetty (UIUC)

Abstract: H-principle is a tool which helps reduce problems involving analysis and geometry into problems involving just geometry. I will start out by motivating the basic idea of H-principle and then use the idea of Holonomic approximation to sketch a proof of H-principle for symplectic structure. This will be an introductory talk and even though the title sounds esoteric, I hope to convince you that this is a central theme in Geometry.

Friday, October 27, 2017

4:00 pm in 241 Altgeld Hall,Friday, October 27, 2017

Topology from critical points

Nachiketa Adhikari (UIUC)

Abstract: "Every mathematician has a secret weapon. Mine is Morse theory." - Raoul Bott Morse theory is a tool that allows one to study the topology of a manifold by looking at special functions on it. In this introductory talk, we'll first look at some of the fundamental ideas relating critical points of these functions and the homotopy type of the manifold. We will then try to understand how the gradient flows of such functions can yield topological invariants for it. No knowledge beyond the words in this abstract will be assumed.

Friday, November 3, 2017

4:00 pm in 241 Altgeld Hall,Friday, November 3, 2017

Canceled

Friday, November 10, 2017

4:00 pm in 241 Altgeld Hall,Friday, November 10, 2017

Manifolds of Mappings

Timothy Drake (UIUC)

Abstract: This talk will introduce the notion of manifolds locally modeled on arbitrary (possibly infinite-dimensional) topological vector spaces, focusing on Banach and Fréchet manifolds. We will discuss the manifold structure on the space of $C^k$ mappings between finite-dimensional manifolds and give applications of infinite-dimensional manifold theory to finite-dimensional geometry and topology.

Friday, November 17, 2017

4:00 pm in 241 Altgeld Hall,Friday, November 17, 2017

Riemannian Popcorn, the Heat Kernel in Geometric Analysis

Aubrey Laskowski (UIUC)

Abstract: The heat kernel is the fundamental solution to the heat equation $(\partial_t-\Delta)u=0$ on a specified domain. The heat equation has shown up in geometry in unexpected ways, such as in a proof of the Atiyah-Singer index theorem. Varadhan and Norris each provided asymptotic relationships between the heat kernel and the Riemannian distance function. We will be discussing these results among others at a high level, without in-depth proofs. If time allows, we will discuss the natural connections of the heat kernel to probability theory. Some background in analysis will be useful, but not necessary.