Department of

Mathematics


Seminar Calendar
for Topology Seminar 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.

Tuesday, September 12, 2017

11:00 am in 345 Altgeld Hall,Tuesday, September 12, 2017

The cooperations algebra for the second truncated Brown-Peterson spectrum

Dominic Culver (UIUC math)

Abstract: In the 1980s Mahowald and Mahowald-Lellmann studied the Adams spectral sequence for the sphere based on the connective real K-theory spectrum. In particular, it was used to study the height 1 telescope conjecture at the prime 2 and to perform low dimensional calculations of the stable homotopy groups of the sphere. Motivated by this, Mahowald proposed using the connective spectrum of topological modular forms to study height 2 telescope phenomena and to perform calculations. This requires understanding the cooperations algebra for tmf, and work of Behrens, Ormsby, Stapleton, and Stojanoska give partial calculations of this cooperations algebra. In this talk, I will talk discuss the problem of computing the cooperations algebra for a variant of tmf, which is sometimes referred to as BP<2>.

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.

Tuesday, October 3, 2017

11:00 am in 345 Altgeld Hall,Tuesday, October 3, 2017

Homotopy cardinality and the l-adic continuity of Morava-Euler characteristic

Lior Yanovski (Hebrew University)

Abstract: (Joint with Tomer Schlank) A finite set has an interesting numerical invariant - its cardinality. There are two natural generalizations of "cardinality" to an (homotopy) invariant for (suitably finite) spaces. One is the classical Euler characteristic. The other is the Baez-Dolan "homotopy cardianlity". These two invariants, both natural from a certain perspective, seem to be very different from each other yet mysteriously connected. The question of the precise relation between them was popularized by John Baez as one of the "mysteries of counting". Inspired by this, we show that (p-locally) there is a unique common generalization of these two invariants satisfying some desirable properties. The construction of this invariant relies on a certain l-adic continuity property of the sequence of Morava-Euler characteristics of a given space, which seems to be an interesting "trans-chromatic" phenomenon by itself.

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.

Tuesday, October 17, 2017

11:00 am in 345 Altgeld Hall,Tuesday, October 17, 2017

Derived Azumaya algebras and twisted K-theory

Tasos Moulinos (UIC)

Abstract: Topological K-theory of dg-categories is a localizing invariant of dg-categories over C taking values in the infinity category of KU-modules. In this talk I describe a relative version of this construction; namely for X a quasi-compact, quasi-separated C-scheme I construct a functor valued in Shv_{Sp}(X(C)), the infinity category of sheaves of spectra on X(C). For inputs of the form Perf(X, A) where A is an Azumaya algebra over X, I characterize the values of this functor in terms of the twisted topological K-theory of X(C). From this I deduce a certain decomposition, for X a finite CW-complex equipped with a bundle of projective spaces P over X, of KU(P) in terms of the twisted topological K-theory of X; this is a topological analogue of a result of Quillen’s on the algebraic K-theory of Severi-Brauer schemes.

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.

Tuesday, October 24, 2017

11:00 am in 345 Altgeld Hall,Tuesday, October 24, 2017

Categories for $K$-theory and Devissage

Jonathan Campbell (Vanderbilt)

Abstract: What sorts of categories can K-theory be defined for? We know that exact categories and Waldhausen categories can be used as appropriate input. However, there are geometric categories where we would like to define K-theory where we are only allowed to ``cut and paste" rather than quotient --- examples of these include the category of varieties, and the category of polytopes. I'll define a more general context where one may talk about the algebraic K-theory of these categories, and outline a proof of a general version of Quillen's devissage. I'll outline applications to studying "derived motivic measures" and the scissors congruence problem. This is joint work with Inna Zakharevich.

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.

Tuesday, October 31, 2017

11:00 am in 345 Altgeld Hall,Tuesday, October 31, 2017

Infinite Loop Spaces in Algebraic Geometry

Elden Elmanto (Northwestern)

Abstract: A classical theorem in modern homotopy theory states that functors from finite pointed sets to spaces satisfying certain conditions model infinite loop spaces (Segal 1974). This theorem offers a recognition principle for infinite loop spaces. An analogous theorem for Morel-Voevodsky's motivic homotopy theory has been sought for since its inception. In joint work with Marc Hoyois, Adeel Khan, Vladimir Sosnilo and Maria Yakerson, we provide such a theorem. The category of finite pointed sets is replaced by a category where the objects are smooth schemes and the maps are spans whose "left legs" are finite syntomic maps equipped with a K​-theoretic trivialization of its contangent complex. I will explain what this means, how it is not so different from finite pointed sets and why it was a natural guess. In particular, I will explain some of the requisite algebraic geometry. Time permitting, I will also provide 1) an explicit model for the motivic sphere spectrum as a torsor over a Hilbert scheme and, 2) a model for all motivic Eilenberg-Maclane spaces as simplicial ind-smooth schemes.

Friday, November 3, 2017

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

Canceled

Thursday, November 9, 2017

11:00 am in 345 Altgeld Hall,Thursday, November 9, 2017

A synthetic theory of (∞,1)-categories in homotopy type theory

Emily Riehl (Johns Hopkins University)

Abstract: If homotopy type theory describes a "synthetic theory of ∞-groupoids" is there a similar "synthetic theory of ∞-categories"? In joint work with Mike Shulman, we propose foundations for such a theory motivated by the model of homotopy type theory in the category of Reedy fibrant simplicial spaces, which contains as a full subcategory the ∞-cosmos of Rezk spaces; this model of ∞-categories, first introduced by Rezk, satisfies the requirements of a framework for synthetic ∞-category theory in the sense of joint work with Verity. We introduce simplices and cofibrations into homotopy type theory to probe the internal categorical structure of types, and define Segal types, in which binary composites exist uniquely up to homotopy, and Rezk types, in which the categorical isomorphisms are additionally equivalent to the type-theoretic identities — a "local univalence" condition. In the model these correspond exactly to the Segal and Rezk spaces. We then demonstrate that these simple definitions suffice to develop the synthetic theory of ∞-categories, including functors, natural transformations, co- and contravariant type families with discrete fibers (∞-groupoids), a "dependent" Yoneda lemma that looks like "directed identity-elimination," and the theory of coherent adjunctions.

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.

Tuesday, November 14, 2017

11:00 am in 345 Altgeld Hall,Tuesday, November 14, 2017

Notes on the margins of E-theory

Paul VanKoughnett (Northwestern University )

Abstract: The deformation space of a height n formal group over a finite field has an exact interpretation into homotopy theory, in the form of height n Morava E-theory. The K(t)-localizations of E-theory, for t < n, force us to contend with the margins of the deformation space, where the formal group's height is allowed to change. We present a modular interpretation of these marginal spaces, and discuss applications to homotopy theory.

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.

Tuesday, November 28, 2017

11:00 am in 345 Altgeld Hall,Tuesday, November 28, 2017

Colimits, descent and equifibrant replacement

Egbert Rijke (CMU Philosophy)

Abstract: Homotopy type theory is an emerging field of mathematics, based on Martin-Löf's constructive theory of types. We think of types as spaces, and type families as fibrations. With the addition of the univalence axiom and higher inductive types doing homotopy theory in type theory (and in a proof assistant!) then becomes feasible. (Reflexive) coequalizers can be used to define a many homotopy colimits in type theory. The case of reflexive coequalizers is interesting because classically the topos of reflexive graphs is cohesive over the topos of sets. I will present analogous results in homotopy type theory.

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.

Tuesday, December 5, 2017

11:00 am in 345 Altgeld Hall,Tuesday, December 5, 2017

Algebraic K-theory, polynomial functors, and lambda-rings

Akhil Mathew (University of Chicago)

Abstract: The Grothendieck group K_0 of a commutative ring is well-known to be a lambda-ring, via taking exterior powers of modules. In joint work in progress with Barwick, Glasman, and Nikolaus, we study space-level refinements of this structure. Namely, we show that the K-theory space of a category is naturally functorial for polynomial functors, and describe a universal property of the extended K-theory functor. This leads to a natural spectral refinement of the notion of a lambda-ring.

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.

Tuesday, December 12, 2017

11:00 am in 345 Altgeld Hall,Tuesday, December 12, 2017

The Dold--Thom theorem via factorization homology

Lauren Bandklayder (Northwestern University)

Abstract: The Dold--Thom theorem is a classical result in algebraic topology giving isomorphisms between the homology groups of a space and the homotopy groups of its infinite symmetric product. The goal of this talk is to outline a new proof of this theorem, which is direct and geometric in nature. The heart of this proof is a hypercover argument which identifies the infinite symmetric product as an instance of factorization homology.

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!

Tuesday, January 30, 2018

11:00 am in 345 Altgeld Hall,Tuesday, January 30, 2018

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.

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.

Tuesday, February 13, 2018

11:00 am in 345 Altgeld Hall,Tuesday, February 13, 2018

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.

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.

Tuesday, March 6, 2018

11:00 am in Psychology Building 21,Tuesday, March 6, 2018

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 $E=H\mathbb{Z}_{(2)}$.

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.