Department of

Mathematics


Seminar Calendar
for Topology Seminar events the year of Sunday, April 11, 2021.

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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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Tuesday, January 26, 2021

11:00 am in Via Zoom,Tuesday, January 26, 2021

Models of Lubin-Tate spectra via Real bordism theory

XiaoLin "Danny" Shi (University of Chicago)

Abstract: In this talk, we will present Real-oriented models of Lubin-Tate theories at p=2 and arbitrary heights. For these models, we give explicit formulas for the action of certain finite subgroups of the Morava stabilizer groups on the coefficient rings. This is an input necessary for future computations. The construction utilizes equivariant formal group laws associated with the norms of the Real bordism theory. As a consequence, we will describe how we can use these models to prove periodicity theorems for Lubin-Tate theories and set up an inductive approach to prove differentials in their slice spectral sequences. This talk is based on several joint projects with Agnès Beaudry, Jeremy Hahn, Mike Hill, Guchuan Li, Lennart Meier, Guozhen Wang, Zhouli Xu, and Mingcong Zeng.

Please contact vesna@illinois.edu for Zoom info.

Friday, January 29, 2021

4:00 pm in Zoom,Friday, January 29, 2021

Organizaitonal Meeting

Brannon (UIUC)

Abstract: We will be having our first organizational meeting. Please email basilio3(at)illinois(dot)edu for the Zoom information.

Tuesday, February 2, 2021

11:00 am in Zoom,Tuesday, February 2, 2021

Redshift in algebraic K-theory

Jeremy Hahn (MIT)

Abstract: I will describe work, joint with Dylan Wilson, about the redshifting properties of algebraic K-theory. I will focus on concrete examples, sketching proofs at the prime 2 that K(ko) has chromatic height 2 and K(tmf) has chromatic height 3.

For Zoom info, please email vesna@illinois.edu

Friday, February 12, 2021

4:00 pm in Zoom,Friday, February 12, 2021

Geometry of Knots

Brannon Basilio (UIUC)

Abstract: In this talk, we give an introduction to the geometry of knots. We first start with an example of how to decompose a knot complement into ideal tetrahedron and the conditions needed in order to obtain a hyperbolic structure on the tetrahedron. We then talk briefly of recent work in knot theory that uses this decomposition to obtain bounds on the hyperbolic volume of the knot complement. For Zoom information, please email basilio3 (at) illinois (dot) edu.

Tuesday, February 16, 2021

11:00 am in Zoom,Tuesday, February 16, 2021

Understanding accessible infinity-categories

Charles Rezk   [email] (UICU)

Abstract: Lurie introduced the very important notion of "accessible infinity-category", a generalization of the more classical notion of "accessible category". These are (infinity-)categories which are produced from two pieces of data: a small (infinity-)category and a "regular cardinal". The goal of this talk is to give an introduction to some of the ideas surrounding these, and to put them in a broader context.

Email vesna@illinois.edu for zoom info.

Tuesday, February 23, 2021

11:00 am in Zoom,Tuesday, February 23, 2021

The Borel C_2-equivariant K(1)-local sphere

William Balderrama (UIUC)

Abstract: I'll talk about the structure of the Borel C_2-equivariant K(1)-local sphere. This captures Im J-type phenomena in C_2-equivariant and R-motivic stable stems, and gives a concise approach to understanding the K(1)-localizations of stunted projective spaces.

For Zoom info, please contact vesna@illinois.edu

Friday, February 26, 2021

4:00 pm in Zoom,Friday, February 26, 2021

A Length and an Area Walk Into a Bar Complex

Cameron Rudd (UIUC)

Abstract: I will discuss some occurrences of lengths and area in geometry. Please contact basilio3 (at) illinois (dot) edu for Zoom information.

Friday, March 5, 2021

4:00 pm in Zoom,Friday, March 5, 2021

To (Conformal) Infinity and Beyond!

Hadrian Quan (UIUC)

Abstract: In this talk I'll describe a few geometric and analytic questions in the context of asymptotically hyperbolic spaces: manifolds which look like hyperbolic space at infinity. I'll do my best to gesture at which of these questions were inspired by conjectures of physicists (such as the AdS-CFT correspondence), while remaining firmly in the context of well-defined mathematical objects. At the end I'll discuss how the geometry of the hyperbolic space inside of Minkowski space can be used to prove theorems on such asymptotically hyperbolic spaces. Email basilio3 (at) illinois (dot) edu for Zoom details.

Tuesday, March 9, 2021

11:00 am in Zoom,Tuesday, March 9, 2021

Cyclotomic Galois extensions in the chromatic homotopy

Tomer Schlank (Hebrew University)

Abstract: The chromatic approach to stable homotopy theory is "divide and conquer". That is, questions about spectra are studied through various localizations that isolate pure height phenomena and then are put back together. For each height n, there are two main candidates for pure height localization. The first is the generally more accessible K(n)-localization and the second is the closely related T(n)-localization. It is an open problem whether the two families of localizations coincide. One of the main reasons that the K(n)-local category is more amenable to computations is the existence of well understood Galois extensions of the K(n)-local sphere. In the talk, I will present a generalization, based on ambidexterity, of the classical theory of cyclotomic extensions, suitable for producing non-trivial Galois extensions in the T(n)-local and K(n)-local context. This construction gives a new family of Galois extensions of the T(n)-local sphere and allows to lift the well known maximal abelian extension of the K(n)-local sphere to the T(n)-local world. I will then describe some applications, including the study of the T(n)-local Picard group, a chromatic version of the Kummer theory, and interaction with algebraic K-theory. This is a joint project with Shachar Carmeli and Lior Yanovski.

for Zoom info, please email vesna@illinois.edu

Friday, March 19, 2021

4:00 pm in Zoom,Friday, March 19, 2021

Free products of Abelian groups in Mod(S)

Chris Loa (UIUC)

Abstract: In 2002, Farb and Mosher introduced a notion of convex cocompactness for mapping class groups. The original notion of convex cocompactness comes from Kleinian groups, where it is a special case of geometric finiteness. In recent work Dowdall, Durham, Leininger, and Sisto have introduced a notion of “parabolic” geometric finiteness for mapping class groups. Examples include convex cocompact groups (as one would hope) and finitely generated Veech groups by work of Tang. In this talk we’ll construct a new family of examples of parabolically geometrically finite groups and show why they are undistorted in Mod(S). Please email basilio3 (at) illinois (dot) edu for Zoom information.

Friday, March 26, 2021

4:00 pm in Zoom,Friday, March 26, 2021

An introduction to CAT(0) cube complexes

Marissa Miller (UIUC)

Abstract: In this talk, I will introduce the notion of a CAT(k) metric spaces, which are spaces that have geometry comparable to complete simply connected surfaces of constant curvature k. We will specifically focus on CAT(0) spaces and will explore CAT(0) cube complexes in some detail, looking at various examples of these complexes and their relationships to questions in geometric group theory. Please email basilio3 (at) illinois (dot) edu for Zoom details.

Friday, April 2, 2021

4:00 pm in Zoom,Friday, April 2, 2021

Oriented Cohomology Theories

Tsutomu Okano (UIUC)

Abstract: I will discuss oriented cohomology theories in both topological and algebro-geometric settings. They naturally come equipped with useful tools such as Thom isomorphisms, Chern classes and Gysin maps. I will give a sketch of the Thom-Pontrjagin construction, from which it follows that complex cobordism is the universal (complex) oriented cohomology theory. For Zoom details, please email basilio3 (at) illinois (dot) edu.

Tuesday, April 6, 2021

11:00 am in Zoom,Tuesday, April 6, 2021

A multiplicative theory of motivic infinite loop series

Brian Shin (UIUC)

Abstract: From a spectrum $E$ one can extract its infinite loop space $\Omega^\infty E = X$. The space $X$ comes with a rich structure. For example, since $X$ is a loop space, we know $\pi_0 X$ comes with a group structure. Better yet, since $X$ is a double loop space, we know $\pi_0 X$ is in fact an abelian group. How much structure does this space $X$ possess? In 1974 Segal gave the following answer to this question: the structure of an infinite loop space is exactly the structure of a grouplike $E_\infty$ monoid. In fact, this identification respects multiplicative structures. In this talk, I'd like to discuss the analogue of this story in the setting of motivic homotopy theory. In particular we'll see that the motivic story, while similar to the classical one, has a couple interesting twists.

For Zoom info, please email vesna@illinois.edu

Friday, April 9, 2021

4:00 pm in Zoom,Friday, April 9, 2021

The Yamabe Problem

Xinran Yu (UIUC)

Abstract: In a two-dimensional case, the fact that every Riemann surface has a metric with constant Gaussian curvature leads to a successful classification of Riemann surfaces. Generalizing this property to higher dimensions could be an interesting problem to consider. Thus we seek a conformal metric on a compact Riemannian manifold with constant scalar curvature. The Yamabe problem was solved in the 1980s, due to Yamabe, Trudinger, Aubin, and Schoen. Their solution to the Yamabe problem uses the techniques of calculus of variation and elliptic regularity of the Laplacian. The proof introduces a conformal invariant, so-called the Yamabe invariant, which shifts the focus from an analysis point of view to understanding a geometric invariant. The solution is separated nicely into two cases, regarding the dimension and flatness of a given Riemannian manifold. For Zoom details, please email basilio3 (at) illinois (dot) edu.

Friday, April 16, 2021

4:00 pm in Zoom,Friday, April 16, 2021

Geometric Aspects of Syntactic Categories

Joseph Rennie (UIUC)

Abstract: In this conceptual talk, a natural progression of abstractions will take us from ordinary metric-geometry to a kind of geometry over syntactic categories of first-order theories. No model theory background is assumed. The focus of the talk will be solely on the geometric side, but the talk will end where geometry and logic become indistinguishable. For Zoom details, please email basilio3 (at) illinois (dot) edu.

Tuesday, April 20, 2021

11:00 am in Zoom,Tuesday, April 20, 2021

Exotic $K(h)$-local Picard groups when $2p-1=h^2$ and the Vanishing Conjecture

Ningchuan Zhang (University of Pennsylvania)

Abstract: The study of Picard groups in chromatic homotopy theory was initiated by Hopkins-Mahowald-Sadofsky. By analyzing the homotopy fixed point spectral sequence for the $K(h)$-local sphere, they showed that the exotic $K(h)$-local Picard group $\kappa_h$ is zero when $(p-1)\nmid h$ and $2p-1>h^2$. In this joint work in progress with Dominic Culver, we study $\kappa_h$ when $2p-1=h^2$ and show that its vanishing is implied by a special case of Hopkins’ Chromatic Vanishing Conjecture. Goerss-Henn-Mahowald-Rezk defined an algebraic detection map for $\kappa_h$, which is injective in this case. We will use the Gross-Hopkins duality reduce the target of the detection map to some Greek letter element computations. The vanishing of $\kappa_h$ is then implied by some bounds on the divisibility of those Greek letter elements. At height $3$ and prime $5$, the Miller-Ravenel-Wilson computation implies that exotic elements in $\kappa_3$ are not detected by a type 2 complex. The full vanishing of $\kappa_3$ requires a bound on the $v_1$-divisibility of $\gamma$-family elements. Using the same duality argument, we can also reduce the mod-$p$ Homological Vanishing Conjecture to some bounds on the divisibility of Greek letter elements. By comparing the bounds in both cases, we conclude that the Vanishing Conjecture implies $\kappa_h=0$ when $2p-1=h^2$.

For Zoom info, please email vesna@illinois.edu

Friday, April 23, 2021

4:00 pm in Zoom,Friday, April 23, 2021

The Calabi conjecture on non-compact manifolds

Karthik Vasu (UIUC)

Abstract: The problem of finding Einstein metrics involves determining the metric tensor (which is solving for $n^2$ many functions) is in general hard. In the complex case where we have a K\"{a}hler structure this problem reduces to solving for one smooth function through a PDE. Aubin and Yau in their works around 1980 were able to solve this PDE in the zero and negative curvature case on compact manifolds. In this talk I will discuss the geometric setup of the compact case and results in the non-compact case that followed. I will also give a bit of background information to have a high level understanding of the problem. Please email basilio3 (at) illinois (dot) edu for Zoom info.

Tuesday, April 27, 2021

11:00 am in via Zoom,Tuesday, April 27, 2021

Equivariant quotients and localizations of norms of $BP_{\mathbb{R}}$

Agnès Beaudry (University of Colorado Boulder)

Abstract: Quotients, localizations and completions of $BP$ play a central role in chromatic homotopy theory. For example, the Johnson-Wilson spectra $E(h)$ obtained by a quotient and localization of $BP$ are key players in the chromatic story at height $h$. However, working only with $E(h)$, the equivariance inherent to the chromatic story coming from the Morava stablizer group is obscured. A first step is to instead consider $E_{\mathbb{R}}(h)$, the Real Johnson-Wilson $C_2$-spectrum. However, for many heights $h$ there are bigger subgroups of the Morava stabilizer group lurking and $E_{\mathbb{R}}(h)$ does not capture their action. Indeed, restricting to finite cyclic 2-groups and for $h=2^{n-1}m$, the stabilizer group contains a subgroup isomorphic to $C_{2^n}$. In this talk, I will explain how one can instead consider quotients of norms of $N_{C_2}^{C_{2^n}}BP_{\mathbb{R}}$ to construct height $h$, $C_{2^n}$-equivariant analogues of $E(h)$.

For Zoom info, please contact vesna@illinois.edu

Tuesday, May 4, 2021

11:00 am in Zoom,Tuesday, May 4, 2021

Splitting $BP\langle 2 \rangle \wedge BP \langle 2 \rangle$ at odd primes

Elizabeth Tatum (UIUC)

Abstract: In the 1980’s, Mahowald and Kane used Brown-Gitler spectra to construct splittings of $bo \wedge bo$ and $l \wedge l$. In this talk, we will construct an analogous splitting for the spectrum $BP\langle 2 \rangle \wedge BP \langle 2 \rangle$ at odd primes.

For Zoom info, please contact vesna@illinois.edu

Friday, May 7, 2021

4:00 pm in Zoom,Friday, May 7, 2021

3-manifolds and number theory

Robert Dicks (UIUC)

Abstract: Arithmetic hyperbolic 3-manifolds and algebraic number theory are intertwined in many ways. For this talk, the speaker will discuss the relation between these 3-manifolds and automorphic forms. Some topics will include the connection between the Langlands program and rational homology 3-spheres and work of Dunfield and Ramakrishnan relating the cohomology of an arithmetic hyperbolic 3-manifold M to the number of its finite covers which fiber over the circle. For Zoom details, please email basilio3 (at) illinois (dot) edu.