Department of

# Mathematics

Seminar Calendar
for Topology Seminar events the next 12 months of Saturday, August 1, 2015.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
      July 2015             August 2015           September 2015
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Tuesday, September 1, 2015

11:00 am in 243 Altgeld Hall,Tuesday, September 1, 2015

#### Chromatic splitting: the way things work at p=2

###### Agnes Beaudry (U Chicago)

Abstract: Recent computations show that the decomposition of $L_1L_{K(2)}S$ predicted by the chromatic splitting conjecture does not hold at $p=2$. I will explain how the situation differs and propose a different decomposition for $L_1L_{K(2)}S$ in this case.

Friday, September 4, 2015

4:00 pm in 243 Altgeld Hall,Friday, September 4, 2015

#### Classical Mechanics and Symplectic Geometry

###### Matej Penciak (UIUC Math)

Abstract: In this talk I will introduce the basics of the modern Lagrangian and Hamiltonian formulations of classical mechanics, and show the connection to symplectic and Poisson geometry. I will end with a discussion of how symmetries can be used to solve for the dynamics of classical systems and some formulations of Noether’s theorem on conserved quantities.

Tuesday, September 8, 2015

11:00 am in 243 Altgeld Hall,Tuesday, September 8, 2015

#### Group spectra and twisting structures

###### Marc Stephan (U Chicago)

Abstract: Group spectra are the group objects in Kan's category of semisimplicial spectra. They provide an algebraic, combinatorial model for the stable homotopy category. After introducing Kan's category of spectra, we will construct the analogues of Kan's loop group functor, of its right adjoint Wbar and corresponding classifying bundles such that the category of semisimplicial spectra becomes a twisted homotopical category in the sense of Farjoun and Hess. This will enable us to formulate and study an up-to-homotopy notion of normal subgroup spectra.

Friday, September 11, 2015

4:00 pm in 243 Altgeld Hall,Friday, September 11, 2015

#### Curvature bounds and bi-Lipschitz embeddings

###### Matthew Romney (UIUC Math)

Abstract: The first part of this talk will give an overview of uniformization and embedding problems in metric space geometry. In the second part, we consider these problems in the context of spaces of bounded curvature in the sense of Alexandrov. As the culmination, we present a new result giving sufficient conditions for a metric space to admit a bi-Lipschitz embedding in a finite-dimensional Euclidean space. The main requirement is that the space satisfy certain curvature bounds on the balls of a Whitney decomposition relative to a prescribed singular subset.

Tuesday, September 15, 2015

11:00 am in 243 Altgeld Hall,Tuesday, September 15, 2015

#### Computations in 2-local stable homotopy theory

###### Philip Egger (Northwestern)

Abstract: Computing the stable homotopy groups of spheres is a major focus of modern algebraic topology. One way of finding infinite families of nontrivial elements involves finite CW-complexes with non-nilpotent self-maps. I will show computationally (joint with Bhattacharya and Mahowald) that the spectra $A_1$ whose cohomology is $A(1)$ admits a self-map detected in Morava K-theory by $v_2^{32}$. Time permitting, I will describe Mahowald's proof of the $K(1)$-local telescope conjecture and introduce a finite complex $Z$ that I hope will shed light on the $K(2)$-local telescope conjecture.

Friday, September 18, 2015

4:00 pm in 243 Altgeld Hall,Friday, September 18, 2015

#### Pants, Primes and Projective Planes

###### Peter Nelson (UIUC Math)

Abstract: I'll talk about a basic but coarse notion of "equivalence" for manifolds, and then tease a connection between algebraic topology and number theory.

Monday, September 21, 2015

11:00 am in 443 Altgeld Hall,Monday, September 21, 2015

#### Cohomology theories associated to infinity-topoi, globally equivariant spectra, and elliptic cohomology

###### David Gepner (Purdue)

Abstract: In this talk I will explain how an infinity-topos equipped with a suitable ring or module object gives rise to a (co)homology theory, locally defined on the infinity-topos, and show how this recovers various versions of (co)homology. As an application, we will recover Schwede's category of global spectra as well as Lurie's construction of equivariant elliptic cohomology. Finally, in the presence of a ring structure, we will see how to find invertible and dualizable objects as well as maps which admit transfers with respect to these theories. This is joint work in progress with Thomas Nikolaus.

Tuesday, September 22, 2015

11:00 am in 243 Altgeld Hall,Tuesday, September 22, 2015

#### Real Johnson-Wilson Theories and Computations

###### Vitaly Lorman (Johns Hopkins)

Abstract: Complex cobordism and its relatives, the Johnson-Wilson theories, E(n), carry an action of C_2 by complex conjugation. Taking fixed points of the latter yields Real Johnson-Wilson theories, ER(n). These can be seen as generalizations of real K-theory and are similarly amenable to computations. We will outline their properties, describe a generalization of the \eta-fibration, and discuss recent computations of the ER(n)-cohomology of some well-known spaces, including CP^\infty.

Friday, September 25, 2015

4:00 pm in 243 Altgeld Hall,Friday, September 25, 2015

#### Poisson Manifolds as Families of Hamiltonian Systems

###### Joel Villatoro (UIUC Math)

Abstract: In this talk, we will introduce the subject of Poisson geometry from the point of view of classical mechanics where a Poisson manifold is a family of Hamiltonian phase spaces. This should be fairly expository and accessible. We will then give a brief idea of how to study the transverse geometry of this family.

Monday, September 28, 2015

11:00 am in 443 Altgeld Hall,Monday, September 28, 2015

#### A higher category theorist's take on the spin--statistics theorem

###### Theo Johnson-Freyd (Northwestern)

Abstract: This talk is about a pair of coincidences in homotopy theory which related quantum field theory with commutative algebra. The first coincidence is the fact that the etale homotopy type of Spec(R) matches the homotopy 1-type of BO(\infty), the classifying space of the stable orthogonal group. This coincidence, I will argue, is the reason for "unitary" phenomena in physics. The second coincidence is a categorification of this: I will describe a setting in which Spec(R) has an "etale" homotopy type that matches the homotopy 2-type of BO(\infty), and explain how this provides the "spin--statistics theorem" relating spinors to fermions.

Friday, October 2, 2015

4:00 pm in 243 Altgeld Hall,Friday, October 2, 2015

#### Goodwillie Hunting

###### Daniel Carmody (UIUC Math)

Abstract: In this talk I'll begin by introducing some of the basic machinery of homotopy theory including, but not limited to, homotopy pushouts/pullbacks and the category of $\Omega$-spectra. Once my hands are untied, I'll define 1-excisive functors and explain why we care about 1-excisive approximations to functors. I'll end by explaining the generalization to polynomial functors and briefly construct the Taylor tower. Time permitting, I'll answer the age-old question of whether or not Will Hunting gets the girl.

Friday, October 9, 2015

4:00 pm in 243 Altgeld Hall,Friday, October 9, 2015

#### Geometric properties of Higher Teichmüller Spaces

###### Georgios Kydonakis (UIUC Math)

Abstract: Higher Teichmüller Theory brings together different mathematical objects in describing the moduli space of fundamental group representations into a semisimple Lie group $G$. Realizing Teichmüller space as a subset of this moduli space in the case when $G=\text{PSL}\left( 2,\mathbb{R} \right)\,$, provides motivation to identify and study connected components of the representations variety, which share essential topological and geometric properties with the classical Teichmüller space. We will introduce particular examples of these components and point their special geometric significance.

Tuesday, October 13, 2015

11:00 am in 243 Altgeld Hall,Tuesday, October 13, 2015

#### Intermediate Hopf-Galois Extensions and the Nilpotence Theorem

###### Jon Beardsley (Johns Hopkins)

Abstract: We will define Hopf-Galois extensions of ring spectra and give some examples from chromatic homotopy theory. We also describe a method of producing intermediate extensions from normal-sub-Hopf-algebras, similarly to the way one produces intermediate Galois extensions from normal subgroups of the Galois group. We finally describe ongoing work to use this structure to streamline the proof of the Nilpotence Theorem of Devinatz, Hopkins and Smith.

Thursday, October 15, 2015

1:00 pm in 243 Altgeld Hall,Thursday, October 15, 2015

#### How to make predictions in topology (resp. arithmetic) using arithmetic (resp. topology)

###### Benson Farb (University of Chicago)

Abstract: Weil, Grothendieck, Deligne and others built an amazing bridge between topology and arithmetic. In this talk I will describe some recent attempts (some successful, some still conjectural) to add planks to this bridge. I hope to convince the audience that the question: "Why is $1/\zeta(n)$ the same as the 2-fold loop space of $CP^{n-1}$ ?" is not completely crazy. This is joint work with Jesse Wolfson.

Friday, October 16, 2015

4:00 pm in 243 Altgeld Hall,Friday, October 16, 2015

#### Some Constructions in Stable Homotopy Theory

###### Dileep Menon (UIUC Math)

Abstract: In this talk we’ll investigate some of the structure within the stable homotopy groups of spheres. We’ll start somewhere near the beginning. After introducing some basic notions, I’ll construct the Toda bracket which gives a way of creating new elements from old in the homotopy groups of spheres. We’ll use these explicit constructions to motivate some important global phenomena.

Tuesday, October 20, 2015

11:00 am in 243 Altgeld Hall,Tuesday, October 20, 2015

#### Weight structures and the algebraic K-theory of stable $\infty$-categories

###### Ernie Fontes (UT Austin)

Abstract: Algebraic K-theory is a spectral invariant of module categories with applications to number theory and manifold geometry. Recently, various people have used the technology of $\infty$-categories to establish universal characterizations for K-theory. Many of the basic structural results about K-theory have been elevated to apply in the $\infty$-categorical context. I will describe Waldhausen's sphere theorem, a new analogous result for the algebraic K-theory of stable $\infty$-categories, and some applications of the new theorem.

Friday, October 23, 2015

4:00 pm in 243 Altgeld Hall,Friday, October 23, 2015

#### Discrete Homotopy Theory and Finite Topological Spaces

###### Joseph Rennie (UIUC Math)

Abstract: I will use a simply phrased problem as a guide through the basics of Topology (General and Algebraic) for finite spaces, and how it connects to familiar spaces. I will then use a problem from Complexity Theory to motivate a discrete analogue to Homotopy Theory which coincides with the homotopy theory of finite spaces. If time permits, I will discuss some applications of algebraic Topology of finite spaces to the study of Sylow Subgroups of finite groups. Much of "Discrete Homotopy Theory" has some surprisingly nontrivial (haha) open problems, new ways of looking at old unsolved problems, and some unexpected results. My overall goal is to leave you with a sense of astonishment with an area which is too often neglected.

Tuesday, October 27, 2015

11:00 am in 243 Altgeld Hall,Tuesday, October 27, 2015

#### E_n cells and homological stability

###### Sander Kupers (Stanford)

Abstract: When studying objects with additional algebraic structure, e.g. algebras over an operad, it can be helpful to consider cell decompositions adapted to these algebraic structures. I will talk about joint work with Jeremy Miller on the relationship between E_n-cells and homological stability. Using this theory, we prove a local-to-global principle for homological stability, as well as give a new perspective on homological stability for various spaces including symmetric products and spaces of holomorphic maps.

Friday, October 30, 2015

4:00 pm in 243 Altgeld Hall,Friday, October 30, 2015

#### Homology of no k-equal manifolds

###### Nick Kosar (UIUC Math)

Abstract: No k-equal manifolds are generalizations of configuration spaces where we allow up to k-1 coordinates to be equal. Using a problem from computer science as motivation, I will talk about the homology of no k-equal manifolds of Euclidean space.

Tuesday, November 3, 2015

11:00 am in 243 Altgeld Hall,Tuesday, November 3, 2015

#### Framed Correspondences and the Milnor-Witt K-theory

###### Alexander Neshitov (Ottawa)

Abstract: The theory of framed motives developed by Garkusha and Panin based on ideas by Voevodsky, gives a tool to construct fibrant replacements of spectra in A^1-homotopy category. In the talk we will discuss how this construction gives an explicit identification of the motivic homotopy groups of the base field with its Milnor-Witt K-theory. In fact, this identification can be done similar to the theorem of Suslin-Voevodsky which identifies motivic cohomology of the base field with Milnor K-theory.

Friday, November 6, 2015

4:00 pm in 243 Altgeld Hall,Friday, November 6, 2015

#### Embeddings and analogies between right-angled Artin groups and mapping class groups

###### Elizabeth Field (UIUC Math)

Abstract: During this talk, we will discuss the relationships which arise between right-angled Artin groups and mapping class groups of surfaces. We will begin by exploring when a right-angled Artin group can be embedded into the mapping class group of a surface and conclude by discussing various analogies between these two types of groups. In particular, we will see how the acylindrical action of a right-angled Artin group on its extension graph leads us to a classification of the elements of a right-angled Artin group which is analogous to the Nielson-Thurston classification of the elements of a mapping class group. This talk will assume no prior knowledge of either mapping class groups or of right-angled Artin groups.

Tuesday, November 10, 2015

11:00 am in 243 Altgeld Hall,Tuesday, November 10, 2015

#### Motivic stable stems over finite fields

###### Glen Wilson (Rutgers)

Abstract: In the Morel-Voevodsky motivic stable homotopy category over the complex numbers, Marc Levine proved that the motivic stable stems $\pi_{n,0}$ are isomorphic to the topological stable stems $\pi_n^s$. What can we say about the motivic stable stems over fields of positive characteristic? In this talk, we will discuss calculations of the two-complete motivic stable stems over finite fields of odd characteristic using the motivic Adams spectral sequence. For n < 19, we find that after two-completion, $\pi_{n,0} = \pi_n^s + \pi_{n+1}^s$.

Friday, November 13, 2015

4:00 pm in 243 Altgeld Hall,Friday, November 13, 2015

#### Nielsen-Thurston Classification of Mapping Classes of Closed Surfaces

###### Sarah Mousley (UIUC Math)

Abstract: The mapping class group of a closed surface $S$, denoted by $Mod(S)$, is the collection of all orientation-preserving homeomorphisms of $S$ up to homotopy. We will begin by explicitly computing the mapping class group of the torus $T^2$ and see that there are three types of elements in $Mod(T^2)$: finite order, powers of Dehn twists, and Anosov. We will then look at closed surfaces of genus $g \geq 2$ and learn that a similar classification statement (known as the Nielsen-Thurston classification) holds. I hope to convince you that the Nielsen-Thurston classification is meaningful. After all, any small child can arbitrarily place objects into bins and call the result a classification. Were Nielsen and Thurston just a pair of toddlers? Come to my talk and find out.

Tuesday, November 17, 2015

11:00 am in 243 Altgeld Hall,Tuesday, November 17, 2015

#### K-theory computations for topological insulators

###### Aaron Royer (UT Austin)

Abstract: Over the past decade a new class materials called topological insulators, often with counter-intuitive electric properties, have been discovered. The mathematics involved in classifying such materials is twisted equivariant Real K-theory. I will briefly describe the setup and survey new K-theory computations coming from these considerations. This is joint work with Dan Freed.

Friday, November 20, 2015

4:00 pm in 243 Altgeld Hall,Friday, November 20, 2015

#### Fuchsian representations and geodesic flows with an eye towards Anosov structures.

###### Brian Collier (UIUC Math)

Abstract: Fuchsian representations are discrete faithful representations of the fundamental group of a closed surface into PSL(2,R) (the isometry group of the hyperbolic plane), and are in bijection with the Teichmüller space of hyperbolic structures on the surface. Using Higgs bundle techniques, Nigel Hitchin showed that there is a natural generalization of Fuchsian representations into the group PSL(n,R), however the geometry associated these representations remained mysterious until Labourie's work on Anosov representations. The aim of this talk is to motivate (especially to the speaker) why one should work to understand Anosov structures. After setting the scene, most of the time will be spent talking about Fuchsian representations and the geodesic flow on hyperbolic surfaces.

Tuesday, December 1, 2015

11:00 am in 243 Altgeld Hall,Tuesday, December 1, 2015

#### Uniqueness of smooth structures on spheres.

###### Zhouli Xu (U Chicago)

Abstract: In this talk, I will report recent progress that the 61-sphere has a unique smooth structure. Following results of Moise, Kervaire-Milnor, Browder and Hill-Hopkins-Ravenel, we show that the only odd dimensional spheres with a unique smooth structure are in dimension 1, 3, 5 and 61. Following recent work of Isaksen, we also show that in dimensions from 5 through 61, the only spheres with a unique smooth structure are in dimension 5, 6, 12, 56 and 61. Recent work of Behrens-Hill-Hopkins-Mahowald shows that the next sphere with a unique smooth structure, if exists, is in dimension at least 126. The computation of the stable homotopy groups of spheres at the prime 2 is essential to this result. I will review classical techniques and explain our new technique. This work is joint with Guozhen Wang.

Friday, December 4, 2015

4:00 pm in 243 Altgeld Hall,Friday, December 4, 2015

#### To Be Announced

###### IGL Groups (UIUC Math)

Tuesday, December 8, 2015

11:00 am in 243 Altgeld Hall,Tuesday, December 8, 2015

#### Effective field theories and elliptic cohomology

###### Daniel Berwick-Evans (UIUC)

Abstract: I will describe a geometric model for elliptic cohomology at the Tate curve whose cocycles are a class of 2-dimensional effective field theories. A geometrically-motivated modularity condition singles out cocycles whose Chern characters take values in the complexification of topological modular forms (TMF). The Witten genus of a string manifold and the moonshine module furnish examples.

Tuesday, January 19, 2016

11:00 am in 345 Altgeld Hall,Tuesday, January 19, 2016

#### Tensor triangular geometry of the stable motivic homotopy category

###### Kyle Ormsby (Reed)

Abstract: In Balmer's framework of tensor triangular geometry, the prime thick tensor ideals in a tensor triangulated category $\mathcal{C}$ form a space which admits a continuous map to the Zariski spectrum $\mathrm{Spec}^h(\mathrm{End}^\bullet_u(1))$ of homogeneous prime ideals in the graded endomorphism ring of the unit object. (Here the grading is induced by an element $u$ of the Picard group of $\mathcal{C}$.) If $\mathcal{C}$ is the stable motivic homotopy category and $u$ is the punctured affine line, then this endomorphism ring is the Milnor-Witt K-theory ring $K^{MW}_*(F)$ of the base field $F$. I will describe work by my student, Riley Thornton, which completely determines $\mathrm{Spec}^h(K^{MW}_*(F))$ in terms of the orderings of $F$. I will then comment on work in progress which uses the structure of this spectrum to study the thick subcategories of the stable motivic homotopy category.

Friday, January 22, 2016

4:00 pm in 241 Altgeld Hall ,Friday, January 22, 2016

#### Organizational Meeting

Tuesday, January 26, 2016

11:00 am in 345 Altgeld Hall,Tuesday, January 26, 2016

#### Towards a minimal projective resolution of bu<0>

###### Kirsten Wickelgren (Georgia Tech)

Abstract: Arone and Lesh have constructed sequences of spectra interpolating between certain spectra and the Eilenberg-MacLane spectrum HZ, and in certain cases relate their sequences to Goodwille and Weiss towers. They furthermore have conjectures relating their filtration of bu, the Weiss tower for V \mapsto BU(V), and a bu-analogue of the Whitehead conjecture. This talk will present aspects of this work of Arone and Lesh, and then discuss joint work with Julie Bergner, Ruth Joachimi, Kathryn Lesh, and Vesna Stojanoska towards proving these conjectures.

Friday, January 29, 2016

4:00 pm in 241 Altgeld Hall,Friday, January 29, 2016

#### Exploding Trousers: A New Symplectic Cobordism

###### Melinda Lanius (UIUC Math)

Abstract: Symplectic geometry is the study of a manifold $M$ with a particular type of smooth map $\omega$. We can glue two symplectic manifolds at a contact boundary only if one side is a strong convex filling of the contact boundary and the other a concave filling. By expanding our notion of symplectic in a very naturally occurring way, we will demonstrate how to glue two convex boundaries together. This new notion is called scattering-symplectic geometry and involves allowing the map $\omega$ to be singular at the contact boundary.

Friday, February 5, 2016

4:00 pm in 241 Altgeld Hall,Friday, February 5, 2016

#### Caucusing for Curvature Bounds

###### Bill Karr (UIUC Math)

Abstract: Alexandrov geometry is the study of non-smooth analogs of Riemannian manifolds with curvature bounded from below or above. Alexandrov spaces often arise as "limits" of Riemannian manifolds with sectional curvature bounds or as orbifolds where the underlying space is a Riemannian manifold with a sectional curvature bound. We will define Alexandrov spaces, Gromov-Hausdorff limits, and then look at some results as well as some open questions about spaces with curvature bounds. The talk is meant to be a showcase of some interesting topics that might be studied in an introductory reading group on Alexandrov geometry.

Tuesday, February 9, 2016

11:00 am in 345 Altgeld Hall,Tuesday, February 9, 2016

#### The EHP sequence in A1 algebraic topology

###### Ben Williams (UBC)

Abstract: The classical EHP sequence is a partial answer to the question of how far the unit map of the loop-suspension adjunction fails to be a weak equivalence. It can be used to move information from stable to unstable homotopy theory. I will explain why there is an EHP sequence in A1 algebraic topology, and what implications this has for the unstable A1 homotopy groups of spheres.

Friday, February 12, 2016

4:00 pm in 241 Altgeld Hall,Friday, February 12, 2016

#### De Rham homology, and some foliations

###### Daan Michiels (UIUC Math)

Abstract: While de Rham cohomology is standard material in differential geometry, de Rham homology isn't. We will define de Rham homology using the theory of currents, and give some elementary properties. To illustrate how de Rham homology can be useful, we discuss foliation currents and the way they relate to tautness of a codimension-1 foliation. The only prerequisite is some familiarity with differential forms and de Rham cohomology.

Thursday, February 18, 2016

11:00 am in 345 Altgeld Hall,Thursday, February 18, 2016

#### T-duality and iterated algebraic K-theory

###### John Lind (Regensburg)

Abstract: T-duality arose in string theory as an equivalence between the physics of two different but suitable related spacetimes. By considering only the underlying topological quantities, T-duality can be distilled into a mathematical theorem which states that the twisted K-theories of certain pairs of circle bundles equipped with U(1)-gerbes are isomorphic via a Fourier-Mukai transform. In this talk, I will describe a generalization of T-duality to higher rank sphere bundles. I will construct twists of the iterated algebraic K-theory of connective complex K-theory by higher gerbes and describe a T-duality isomorphism between the twisted iterated K-theories of a pair of suitably related sphere bundles. (Joint with H. Sati and C. Westerland)

Friday, February 19, 2016

4:00 pm in 241 Altgeld Hall,Friday, February 19, 2016

#### Yoneda Lemma and the Fundamental Group

###### Nima Rasekh (UIUC Math)

Abstract: The goal of this talk is to show how category theory relates to the fundamental group. We will first gloss over the basics of those two concepts and then we will introduce the overarching idea which relates these two: Higher Category Theory. Finally, we will see how everything in this world is connected. No particular knowledge of category theory is assumed.

Tuesday, February 23, 2016

11:00 am in 345 Altgeld Hall,Tuesday, February 23, 2016

#### An obstruction theory for producing exotic Picard elements

###### Robert Legg (Northwestern)

Abstract: Given an E_*E-comodule, for some homology theory E, can we find a spectrum whose homology is this comodule? I’ll describe an approach to answering a variant of this question using an obstruction theory and how this approach then can be used to produce elements of the K(n)-local exotic Picard group, that is, K(n)-local spectra that are invertible but which are indistinguishable from the sphere on homology.

Friday, February 26, 2016

4:00 pm in 241 Altgeld Hall,Friday, February 26, 2016

#### $\alpha$-continued Fractions

###### Claire Merriman (UIUC Math)

Abstract: Continued fractions are one way to represent positive real numbers as a sequence of integers with connections to number theory, geometry, and dynamics. Nakada’s $\alpha$-continued fraction expansions are a family of continued fraction expansions corresponding to the regular continued fraction expansion when $\alpha=1$. I will review the regular continued fraction representation, as well as introducing the Gauss map and its invariant measure. Most of the talk will focus on Nakada’s work constructing an invariant measure for the $\alpha$-Gauss map and its natural extension when $1/2\leq \alpha\leq 1$.

Tuesday, March 1, 2016

11:00 am in 345 Altgeld Hall,Tuesday, March 1, 2016

#### Orienting tmf with level structure

###### Dylan Wilson (Northwestern)

Abstract: Following Ando-Hopkins-Rezk, we build various highly structured genera for Spin and String manifolds valued in modular forms with level structure. We will try to indicate which pieces of this argument are formal, and what must be done if we wish to give orientations by other ring spectra such as TAF or conjectural cohomology theories attached to the moduli of K3 surfaces.

Friday, March 4, 2016

4:00 pm in 241 Altgeld Hall,Friday, March 4, 2016

#### The Topological Hochschild Homology of $\mathbb{Z}$

###### Juan Villeta-Garcia (UIUC Math)

Abstract: In the 1980's Bökstedt introduced Topological Hochschild Homology (THH), as a variant of algebraic Hochschild Homology, where tensoring over a ground ring was replaced by smashing over the sphere spectrum. Even for discrete rings, like the integers $\mathbb{Z}$, this construction provided new invariants. Bökstedt shortly thereafter calculated the THH of $\mathbb{Z}$ and $\mathbb{Z}/p\mathbb{Z}$, by exploiting heavy topological methods. Algebraically, though, a crucial tool was a spectral sequence relating classical Hochschild Homology to THH. In this talk we will introduce the spectral sequence, and sketch out a proof. We will then hint at a method of Brun to generalize to higher quotients, $\mathbb{Z}/p^n\mathbb{Z}$.

Tuesday, March 8, 2016

11:00 am in 345 Altgeld Hall,Tuesday, March 8, 2016

#### Topological Hochschild homology of the connective cover of the K(1)-local sphere

###### Gabriel Angelini-Knoll (Wayne State)

Abstract: In the early 1980’s Waldhausen proposed a program for studying the algebraic K-theory of the E(n) Bousfield localizations of the sphere using his localization theorem. The E(0) localization of the sphere, which is equivalent to the rational Eiilengberg-Maclane spectrum is the only case where the algebraic K-theory is known. Due to the work of McCarthy and Dundas, relative algebraic K-theory and relative topological cyclic homology are homotopy equivalent after p completion for certain maps of ring spectra. Topological cyclic homology is built out of topological Hochschild homology using the S^1-equivariant and cyclotomic structure. We therefore compute mod (p,v_1) homotopy of THH of the connective cover of the K(1)-local sphere, which is the p-completion of E(1)-local sphere. We construct a filtration of the connective cover of the K(1)-local sphere and then use a May-type spectral sequence to compute THH of the connective K(1)-local sphere. The connective K(1)-local sphere is a chromatic height 1 spectrum, so we expect height 2 phenomena in its algebraic K-theory according to the chromatic red-shift conjecture of Ausoni and Rognes. Though this is not visible in THH, it can already be seen by examining the homotopy fixed points with respect to the circle action. Specifically, I will show how the element v_2 appears in the homotopy fixed points with respect to the circle action.

Friday, March 11, 2016

4:00 pm in 241 Altgeld Hall,Friday, March 11, 2016

#### de Rham Cohomology and the Hodge Laplacian

Abstract: I will motivate a sequence of topological invariants on a smooth manifold: the de Rham cohomology groups. Starting from ideas in vector calculus, these invariants can be tell us much information while only being defined in terms of integration. Although defined simply, these invariants are comprised of cosets of an infinite dimensional vector space, so one may hope for a simple geometric way of choosing a 'canonical' representative for a coset. On a compact manifold such a choice exists, and in fact is the solution of a certain linear PDE. Time permitting, we will sketch a proof of this fact. Nothing beyond vector calculus and linear algebra will be assumed.

Tuesday, March 15, 2016

11:00 am in 243 Altgeld Hall,Tuesday, March 15, 2016

#### Picard groups of structured ring spectra and stable module categories

###### Akhil Mathew (Harvard)

Abstract: Let $G$ be a finite $p$-group. The stable module category of $G$ is defined as the quotient of the category of $G$-representations over a field $k$ of characteristic $p$ by those morphisms which factor through a projective. It can also be modeled as the category of module spectra over the Tate construction $k^{tG}$. It is a classical theorem of Dade that the Picard group of the stable module category contains no "exotic" objects when $G$ is abelian. This translates into a statement about the $E_\infty$-ring spectrum $k^{tG}$. We will discuss a general approach to studying the Picard groups of structured ring spectra using descent theory and describe a new proof of Dade's theorem based on Rognes's theory of Galois extensions of ring spectra and Galois descent.

Friday, March 18, 2016

4:00 pm in 241 Altgeld Hall,Friday, March 18, 2016

#### Computing the Goodwillie-Taylor Tower for Atomic Discrete Modules

###### Amelia Tebbe (UIUC Math)

Abstract: An atomic discrete module is a functor from finite sets to chain complexes of $R$-modules that is completely determined by its value at a particular set. For a general discrete module, that is, a functor from finite sets to chain complexes of $R$-modules, one can use left Kan extensions to construct a filtration by atomic discrete modules. Robinson gave an explicit description of a bicomplex for computing the stable homology of a general discrete module, in which the rows are given by the stable homology of the associated atomic discrete modules. Goodwillie’s calculus of functors gives us a way to approximate functors that is analogous to the Taylor series for real functions. The stable homology of a functor is the homology of the Goodwillie derivative of the functor. This fact inspires us to generalize Robinson's bicomplex to one for computing the higher order polynomial approximations produced by Goodwillie calculus. To this end, we give an explicit bicomplex for atomic functors such that truncation by rows allows us to compute all of the Goodwillie polynomial approximations.

Tuesday, March 29, 2016

11:00 am in 345 Altgeld Hall,Tuesday, March 29, 2016

#### Waldhausen K-theory and topological coHochschild homology

###### Kathryn Hess (Ecole Polytechnique Federale de Lausanne)

Abstract: I will present joint work with Brooke Shipley, in which we have defined a model category structure on the category of $\Sigma^{\infty}X_+$-comodule spectra such that the K-theory of the associated Waldhausen category of homotopically finite objects is naturally weakly equivalent to the usual Waldhausen K-theory of $X$, $A(X)$. I will describe the relation of this comodule approach to $A(X)$ to the more familiar description in terms of $\Sigma^\infty \Omega X_+$-module spectra. I will also explain the construction and properties of the topological coHochschild homology of $X$, which is a potentially interesting approximation to $A(X)$.

Friday, April 1, 2016

4:00 pm in 241 Altgeld Hall,Friday, April 1, 2016

#### Pure Braid Groups and Mapping Class Groups

###### Marissa Loving (UIUC Math)

Abstract: In 2009, Leininger and Margalit proved that given any two elements of the pure braid group they either commute or generate a free group. Their proof exploited the connection between the pure braid group and the mapping class group of a punctured sphere as well as using results from 3 dimensional topology and Bass-Serre Theory. I will give a sketch of their proof and talk about some current research in the area, namely generalizing this result to pure surface braid groups.

Tuesday, April 5, 2016

11:00 am in 345 Altgeld Hall,Tuesday, April 5, 2016

#### Lie algebras and v_n-periodic spaces

###### Gijs Heuts (Copenhagen)

Abstract: We use the Goodwillie tower of the category of pointed spaces to relate the telescopic homotopy theory of spaces (in the sense of Bousfield) to the homotopy theories of Lie algebras and commutative coalgebras in T(n)-local spectra, in analogy with rational homotopy theory. As a consequence, one can determine the Goodwillie tower of the Bousfield-Kuhn functor in terms of topological Andre-Quillen homology, giving a different perspective on recent work of Behrens and Rezk.

Friday, April 8, 2016

4:00 pm in 241 Altgeld Hall,Friday, April 8, 2016

#### Experiments in knot theory

###### Malik Obeidin (UIUC Math)

Abstract: One of the big benefits of restricting to low dimensions in topology is the combinatorial nature of a lot of the objects involved. This gives us leeway to use computer experiment to study what a "typical" object might look like. We'll take a look at knot theory - in particular, the geometry of random links.

Tuesday, April 12, 2016

11:00 am in 345 Altgeld Hall,Tuesday, April 12, 2016

#### A new approach to etale homotopy theory

###### David Carchedi (George Mason)

Abstract: Etale homotopy theory, as originally introduced by Artin and Mazur in the late 60s, is a way of associating to a suitably nice scheme a pro-object in the homotopy category of spaces, and can be used as a tool to extract topological invariants of the scheme in question. It is a celebrated theorem of theirs that, after profinite completion, the etale homotopy type of an algebraic variety of finite type over the complex numbers agrees with the homotopy type of its underlying topological space equipped with the analytic topology. We will present work of ours which offers a refinement of this construction which produces a pro-object in the infinity-category of spaces (rather than its homotopy category) and applies to a much broader class of objects, including all algebraic stacks. We will also present a generalization of the previously mentioned theorem of Artin-Mazur, which holds in much greater generality than the original result.

Friday, April 15, 2016

4:00 pm in 241 Altgeld Hall,Friday, April 15, 2016

#### $SL(2,\mathbb{C})$ Character Variety and Left-orderability.

###### Xinghua Gao (UIUC Math)

Abstract: It is conjectured that for irreducible $\mathbb{Q}$ homolgy 3-sphere $Y$, $\widehat{HF}(Y)$ being non-minimal, $\pi_1(Y)$ being left-orderable and $Y$ admitting a co-orientable taut foliation are equivalent. In this talk, I will tell you what an $SL(2,\mathbb{C})$-character variety is and how it could possibly be applied to the study of this conjecture.

Tuesday, April 19, 2016

11:00 am in 345 Altgeld Hall,Tuesday, April 19, 2016

#### A^1-homotopical classification of principal G-bundles

###### Marc Hoyois (MIT)

Abstract: Let k be an infinite field and G an isotropic reductive k-group. If X is a smooth affine k-variety, then locally trivial G-torsors over X are classified by maps X → BG in the A^1-homotopy category. I will discuss the proof of this statement and some applications. This is joint work with Aravind Asok and Matthias Wendt.

Friday, April 22, 2016

4:00 pm in 241 Altgeld Hall,Friday, April 22, 2016

#### Noncommutative Topology

###### Chris Gartland (UIUC Math)

Abstract: We'll review the classical Gelfand-Naimark theory establishing a duality between the category of compact Hausdorff spaces and the category of commutative unital C*-algebras. A noncommutative space can then be rightly viewed as a noncommutative unital C*-algebra. We show how to naturally extend the cohomology functors $K^0$, $H^0$, $H^1$, and $H^2$ from the category of classical spaces to the category of noncommutative spaces.

Tuesday, April 26, 2016

11:00 am in 345 Altgeld Hall,Tuesday, April 26, 2016

#### Motivic vector bundles on CP^n

###### Aravind Asok (USC)

Abstract: It is a classical problem to determine which complex topological vector bundles on CP^n admit an algebraic structure. We will discuss obstructions" to equipping a bundle with an algebraic structure coming from A^1-homotopy theory (though I won't presuppose any knowledge of A^1-homotopy theory). The main goal of the talk is to analyze the question of whether some rank 2 bundles on CP^n constructed by Elmer Rees using homotopy theoretic techniques admit an algebraic structure. This talk is based on joint work with Jean Fasel and Mike Hopkins.

Friday, April 29, 2016

4:00 pm in 241 Altgeld Hall,Friday, April 29, 2016

#### IGL Presentations

Tuesday, May 3, 2016

11:00 am in 345 Altgeld Hall,Tuesday, May 3, 2016

#### The infinite prime in chromatic homotopy theory

###### Nathaniel Stapleton (Max Planck Institute)

Abstract: The behavior of the category of E_n-local spectra simplifies in various ways as $p \rightarrow \infty$. For a collection of categories indexed by the prime numbers we construct a category "at the infinite prime" that captures behavior of all but finitely many of the input categories. We then apply this construction to the E_n-local and K(n)-local situations and analyze the resulting categories. This represents joint work with Barthel and Schlank.