Department of

Mathematics


Seminar Calendar
for Graduate Geometry events the next 12 months of Tuesday, August 1, 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, 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.

Friday, December 1, 2017

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

Fireworks, balloons, and singularities of curves

Alyssa Loving (UIUC)

Abstract: The aim of this talk is to explore a rather explosive topic, namely what fireworks, balloons, and singularities happen to have in common. All three can be "blown up". This talk will focus on blowing up singularities of curves in the plane, which is a valuable method of resolving singularities such as self intersections or cusps. I will touch on the generalization of this, which involves blowing up schemes. We will not get to blowing up balloons; that will have to be saved for a future talk.

Friday, December 8, 2017

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

Odd and grotesque continued fractions as geodesic flows

Claire Merriman (UIUC)

Abstract: Continued fractions are frequently studied in number theory, but they can also be described by geodesics on modular surfaces. This talk will look at continued fractions of the form $a_1\pm\frac{1}{a_2\pm\frac{1}{a_3\pm\ddots}}$, where the $a_i$ are odd, and how to use geodesic flows to represent the digits.

Friday, January 19, 2018

4:00 pm in 241 Altgeld Hall,Friday, January 19, 2018

Organizational Meeting

Abstract: We'll get the schedule of talks for the semester worked out. There will be cookies.

Friday, January 26, 2018

4:00 pm in 241 Altgeld Hall,Friday, January 26, 2018

Ice cream geometry: a mathematical activity and coloring book

Melinda Lanius (UIUC)

Abstract: Come explore metrics I use in my dissertation research. In a souped-up color-by-numbers, we'll develop a general notion of circle and ball. In geodesic connect-the-dots, we'll see what happens when straight lines curve. In metric mazes, we'll come to appreciate the wonky ways of nonhomogeneous spaces. After exploring the geometries of the real and hyperbolic plane, sphere, cone, and cylinder, we'll conclude by building less familiar objects: surfaces with a Euclidean, cylindrical, conical, or hyperbolic-funnel end. Please bring your fun office supplies!

Friday, February 2, 2018

4:00 pm in 241 Altgeld Hall,Friday, February 2, 2018

Integrable systems in algebraic geometry

Matej Penciak (UIUC)

Abstract: In this talk I'll introduce the definition of an algebraic integrable system. The definition axiomatizes what it means to have an integrable system in algebraic geometry. After connecting the definition to the more common notion in differential geometry, I'll give a few examples of my favorite integrable systems. Possible examples are the Hitchin system on the space of Higgs bundles, the Calogero-Moser system, the Toda lattice hierarchy, and, if time permits, I'll try to give a hint of how all these systems are related via the gauge theory in physics.

Friday, February 9, 2018

4:00 pm in 241 Altgeld Hall,Friday, February 9, 2018

What are spectra?

Tsutomu Okano (UIUC)

Abstract: This is an introductory talk on stable homotopy theory. I will begin by discussing stable phenomena in homotopy theory that led to the definition of spectra. Spectra represent generalized cohomology theories, such as various kinds of K-theories and cobordism theories. This implies that results about cohomology theories can be proven in the category of spectra, using analogies from topological spaces. Unfortunately, the earlier category of spectra was defective in formal properties and it was not until the 1990's that suitable categories of spectra were defined. This formality leads to the study of other sorts of homotopy theories, such equivariant and motivic homotopy theories.

Friday, February 16, 2018

4:00 pm in 241 Altgeld Hall,Friday, February 16, 2018

Canceled

(UIUC)

Friday, February 23, 2018

4:00 pm in 241 Altgeld Hall,Friday, February 23, 2018

Unoriented Bordism and the Algebraic Geometry of Homotopy Theory

Brian Shin (UIUC)

Abstract: The classification of manifolds up to bordism is an interesting geometric problem. In this talk, I will discuss how this problem connects to homotopy theory. In particular, I will demonstrate that viewing the homotopy theorist's toolbox through the lens of algebraic geometry leads naturally to the solution for the classification of manifolds up to bordism.

Friday, March 9, 2018

4:00 pm in 241 Altgeld Hall,Friday, March 9, 2018

Cancelled

Friday, March 16, 2018

4:00 pm in 241 Altgeld Hall,Friday, March 16, 2018

Operations in complex K theory

William Balderrama (UIUC)

Abstract: In nice cases, the correct algebraic setup can exert a large amount of control over a geometric situation. I will illustrate this with the example of complex K theory.

Friday, March 30, 2018

4:00 pm in 241 Altgeld Hall,Friday, March 30, 2018

Optimizing mesh quality

Sarah Mousley (UIUC)

Abstract: I will talk about a project in computational geometry I worked on during a summer internship at Sandia National Lab. Our work builds on that of Mullen et al., who introduced a new energy function for meshes (triangulations) and an algorithm for finding low energy meshes. The energy is a measure of the mesh’s quality for usage in Discrete Exterior Calculus (DEC), a method for numerically solving PDEs. In DEC, the PDE domain is triangulated and this mesh is used to obtain discrete approximations of the continuous operators in the PDE. While the motivation for this work is to obtain better solutions to PDEs, do not be turned off. I didn't solve a single PDE all summer. This is a geometry talk.

Friday, April 13, 2018

4:00 pm in 241 Altgeld Hall,Friday, April 13, 2018

What is group cohomology?

Elizabeth Tatum (UIUC)

Abstract: Group cohomology has many uses in algebra, topology, and number theory. In this talk, I will introduce group cohomology, give some basic examples, and discuss some applications to topology.

Friday, April 20, 2018

4:00 pm in 241 Altgeld Hall,Friday, April 20, 2018

The Volume Conjecture

Xinghua Gao (UIUC)

Abstract: It is a fundamental goal of modern knot theory to “understand” the Jones polynomial. The volume conjecture, which was initially formulated by Kashaev and later generalized by Murakami^2, relates quantum invariants of knots to the hyperbolic geometry of knot complements. In this talk, I will briefly explain the volume conjecture. No background in knot theory/hyperbolic geometry/physics required.

Friday, April 27, 2018

4:00 pm in 241 Altgeld Hall,Friday, April 27, 2018

Cannon-Thurston Maps

Elizabeth Field (UIUC)

Abstract: When a map between two objects extends to a continuous map between their boundaries, this boundary map is called the Cannon-Thurston map. When the objects at play are a hyperbolic group and a hyperbolic subgroup, there are two main questions which arise from this definition. First, when does the Cannon-Thurston map exist? Further, if such a map exists, can the fibers of this map be described in some nice algebraic or geometric way? In this talk, we will introduce the background needed to understand these questions. We will then briefly explore some of what is known about their answers, as well as some open questions which remain.