Mathematics

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!

Localizing the E_2 page of the Adams spectral sequence

Eva Belmont (MIT)

Abstract: The Adams spectral sequence is one of the central tools for calculating the stable homotopy groups of spheres, one of the motivating problems in stable homotopy theory. In this talk, I will discuss an approach for computing the Adams E_2 page at p = 3 in an infinite region, by computing its localization by the non-nilpotent element b_{10}. This approach relies on computing an analogue of the Adams spectral sequence in Palmieri's stable category of comodules, which can be regarded as an algebraic analogue of stable homotopy theory. This computation fits in the framework of chromatic homotopy theory in the stable category of comodules.

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.

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.

Factorization homology and topological Hochschild cohomology of Thom spectra

Inbar Klang (Stanford)

Abstract: By a theorem of Lewis, the Thom spectrum of an n-fold loop map to BO is an E_n-ring spectrum. I will discuss a project studying the factorization homology and the E_n topological Hochschild cohomology of such Thom spectra, and talk about some applications, such as computations, and a duality between topological Hochschild homology and cohomology of certain Thom spectra. Time permitting, I will discuss connections to topological field theories. This talk will include an introduction to factorization homology via labeled configuration spaces.

Canceled

(UIUC)

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.

The generalized homology of $BU\langle 2k\rangle$

Phillip Jedlovec

Abstract: In their 2001 paper, ``Elliptic spectra, the Witten genus and the theorem of the cube,'' Ando, Hopkins, and Strickland use an algebro-geometric perspective to give a partial description of the generalized homology of the connective covers of BU. For any complex-orientable cohomology theory, E, they define homology elements $b_{i_1, ..., i_k}$ in $E_*BU\langle 2k\rangle$, prove the so called ``cocycle relations'' and ``symmetry relations'' on these elements, and show that when $E=H\mathbb{Q}$ or $k=1, 2,$ or $3$, these are in fact the defining relations for $E_*BU\langle 2k\rangle$. In this talk, I will sketch a new proof of these results which uses very little algebraic geometry, but instead uses facts about Hopf rings and the work of Ravenel and Wilson on the homology of the spaces in the $\Omega$-spectrum for Brown-Peterson cohomology. Time permitting, I will also discuss how this approach might be used to prove the Ando-Hopkins-Strickland theorem for $k>3$ and $E=H\mathbb{Z}_{(2)}$.

Cancelled

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.

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.

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.

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.

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.

Organizational Meeting

Jesse Huang (UIUC)

Abstract: We will draft a schedule of the seminar talks this semester. Please join us this afternoon and sign up if you have a topic in mind. As usual, cookies will be provided. All are welcome!

A generalization of pair of pants decompositions

Jesse Huang (UIUC)

Abstract: We will talk about higher dimensional pair of pants decompositions for smooth projective hypersurfaces.

Some Plethyistic Algebra

Charles Rezk (UIUC)

Abstract: This is a talk about an algebraic notion of a plethory. A plethory P determines a category of "P-rings", objects of which are commutative rings R equipped with a collection of functions $f_i : R \to R$ satisfying a list of axioms. Many interesting cohomology theories take values in a category of P-rings for some plethory P. The motivating example is K-theory, which takes values in "Lambda-rings", which is precisely the category of rings for the Lambda plethory. This talk will be expository, concentrating first on interesting examples of P-rings, then working backward to the definition of plethory. Then I'll talk about the "Witt ring" construction associated to any plethory, which includes and generalizes the classical construction of "Witt vectors".

3 invariants of manifolds you won’t believe are the same!

Hadrian Quan

Abstract: We’ll start by discussing everyone’s favorite invariant: the determinant of a linear map. After generalizing this to an invariant of a chain complex, we’ll talk about three different different ways to get a number from a representation of $\pi_1(M)$: topological, analytic, and dynamical. Number 3 might surprise you!

Unstable $v_1$-periodic Homotopy Groups through Goodwillie Calculus

Jens Kjær (Notre Dame Math)

Abstract: It is a classical result that the rational homotopy groups, $\pi_*(X)\otimes \mathbb{Q}$, as a Lie-algebra can be computed in terms of indecomposable elements of the rational cochains on $X$. The closest we can get to a similar statement for general homotopy groups is the Goodwillie spectral sequence, which computes the homotopy group of a space from its "spectral Lie algebra". Unfortunately both input and differentials are hard to get at. We therefore simplify the homotopy groups by taking the unstable $\nu_h$-periodic homotopy groups, $\nu_h^{-1}\pi_*(\ )$ (note $h=0$ recovers rational homotopy groups). For $h=1$ we are able to compute the $K$-theory based $\nu_1$-periodic Goodwillie spectral sequence in terms of derived indecomposables. This allows us to compute $\nu_1^{-1}\pi_*SU(d)$ in a very different way from the original computation by Davis.

The geometry of some low dimensional Lie groups

Ningchuan Zhang (UIUC)

Abstract: In this talk, I'll give explicit geometric descriptions of some low dimensional matrix groups. The goal is to show $\mathrm{SU}(2)\simeq S^3$ is a double cover of $\mathrm{SO}(3)$ and $\mathrm{SL}_2(\mathbb{C})$ is a double cover of the Lorentz group $\mathrm{SO}^+(1,3)$. Only basic knowledge of linear algebra and topology is assumed.

$A_n$ spaces in homotopy type theory

Egbert Rijke (UIUC)

Abstract: We will propose a definition of $A_n$ spacesin homotopy type theory.

Integrability of the Toda lattice

Matej Penciak (UIUC)

Abstract: I will introduce topics such as the Toda lattice, the Lax matrices, and integrability.

An introduction to Ratner's theorem

Venkata Sai Narayana Bavisetty

Abstract: This talk will be an introduction to ergodic theory. I will start out by explaining what ergodicity means and state Ratner's theorem. I will conclude by sketching the proof of Oppenheim conjecture(now a theorem).

Chromatic homotopy is algebraic when $p > n^2+n+1$

Piotr Pstrągowski (Northwestern Math)

Abstract: It is generally accepted that the structure of the $E(n)$-local homotopy theory, where $E(n)$ is the p-local Johnson-Wilson spectrum at height $n$, becomes increasingly algebraic when $p$ is large with respect to $n$. To give two examples of this phenomena, when the prime is large, the $E(n)$-based Adams spectral sequence collapses on the second page for degree reasons, giving an algebraic description of the homotopy of the $E(n)$-local sphere. Moreover, it is a result of Hovey and Sadofsky that the only invertible spectra in this range are the spheres, showing that the $E(n)$-local Hopkins' Picard group is isomorphic to the integers at large primes, in stark contrast to what happens when $p$ is small. In this talk, we show that when $p > n^2+n+1$, the homotopy category of $E(n)$-local spectra is equivalent to the homotopy category of differential $E(n)_*E(n)$-comodules, giving a precise sense in which chromatic homotopy is "algebraic" at large primes. This extends the work of Bousfield at $n = 1$ to all heights, and affirms a conjecture of Franke.

Supersymmetry and Morse theory

Lutian Zhao (UIUC)

Abstract: In 1982, Edward Witten discovered the topological invariant hidden inside the supersymmetric quantum field theory: the Morse complex can be constructed by the supersymmetric model. In this talk, I’ll try to explain the construction from the very beginning assuming no knowledge of supersymmetry as well as Morse theory. If time permitted, I’ll discuss some interpretation of index theorem by supersymmetry.

An Introduction to Persistent Homology

Dan Carmody (UIUC)

Abstract: In this talk, I'll start by introducing the Cech and Vietoris-Rips complexes, then compute some basic examples of persistent homology using the python library Gudhi (Maria et al., 2014). I'll introduce one of the standard metric structures on the space of persistence diagrams, then end by surveying some of the applications of persistent homology to crop science and human biology.

The Gross-Hopkins duals of higher real K-theory at prime 2

Guchuan Li (Northwestern Math)

Abstract: The Hopkins-Mahowald higher real K-theory spectra $E_n^G$ are generalizations of real K-theory; they are ring spectra which give some insight into higher chromatic levels while also being computable. This will be a talk based on joint work with Drew Heard and XiaoLin Danny Shi, in which we compute that higher real K-theory spectra with group $G=C_2$ at prime $2$ and height $n$ are Gross-Hopkins self duals with a shift $4+n$. This will allow us to detect exotic invertible $K(n)$-local spectra.

K3 surfaces and Hyperkahler manifolds

Sungwoo Nam (UIUC)

Abstract: In classification of complex surfaces, K3 surfaces take position similar to that of elliptic curves in smooth projective curves. With their higher-dimensional analogues, compact hyperkahler manifolds, they play an important role in string theory as well. In this talk, we will see their definition and basic properties, mostly about their cohomology. We’ll then discuss a theorem of Matsushita and Hwang, which shows rigidity of the structure of these manifolds.

Introduction to knot theory and the topology of knots

Chaeryn Lee (UIUC)

Abstract: This talk will introduce the very basic concepts and goals of knot theory. It will mainly focus on the topology of knots and how knot theory relates to 3-manifolds and surgery theory. Some topics to be covered will include Lens spaces, Heegaard splittings, Dehn surgery and knot exterior as a knot invariant.

The parametrized Tate construction

JD Quigley (Notre Dame Math)

Abstract: The Tate construction is a powerful tool in classical homotopy theory. I will begin by reviewing the Tate construction and surveying some of its applications. I will then describe an enhancement of the Tate construction to genuine equivariant homotopy theory called the ``parametrized Tate construction" and discuss some of its applications, including C_2-equivariant versions of Lin's Theorem and the Mahowald invariant, blueshift for Real Johnson-Wilson spectra (joint work with Guchuan Li and Vitaly Lorman), and trace methods for Real algebraic K-theory (work-in-progress with Jay Shah).