Department of

# Mathematics

Seminar Calendar
for Topology Seminar events the year of Thursday, December 7, 2017.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
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Friday, January 20, 2017

4:00 pm in 241 Altgeld Hall,Friday, January 20, 2017

#### Organizational Meeting

Abstract: This is the organizational meeting to get the schedule of talks down for the spring. If you think you might be interested in giving a talk at some point, please attend!

Friday, January 27, 2017

4:00 pm in 241 Altgeld Hall,Friday, January 27, 2017

#### Galois Categories and the Topological Galois Correspondence

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

Abstract: Classical Galois theory for fields gives a correspondence between closed subgroups of the Galois group of a Galois extension and intermediate subfields. The theory of covering spaces in topology gives a correspondence between connected coverings of nice spaces and subgroups of the fundamental group. The purpose of this talk is to explain the relationship between (and generalization) of these two theorems.

Tuesday, January 31, 2017

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

#### Complex analytic elliptic cohomology and Looijenga line bundles

###### Charles Rezk (Illinois)

Abstract: I'll explain how, by taking the cohomology of suitable spaces and messing around a bit, you can get things like: the moduli stack of (analytic) curves, the universal curve, and Looijenga line bundles over these. This seems to have some relevance for the construction of complex analytic elliptic cohomology.

Friday, February 3, 2017

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

#### Train Tracks on Surfaces

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

Abstract: Our mantra throughout the talk will be simple, "Train tracks approximate simple closed curves." Our goal will be to explore some examples of train tracks, draw some meaningful pictures, and develop an analogy between train tracks and another well known method of approximation. No great knowledge of anything is required for this talk as long as one is willing to squint their eyes at the blackboard a bit at times.

Friday, February 10, 2017

4:00 pm in 241 Altgeld Hall,Friday, February 10, 2017

#### Opers and non-abelian Hodge theory

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

Abstract: We will describe two different families of flat $G$-connections over a compact Riemann surface for a complex, simple, simply connected Lie group $G$. The first is the family of $G$-opers, which for $G=\text{SL(2}\text{,}\mathbb{C}\text{)}$ can be thought of as global versions of the locally defined second order Schrödinger operators. The second comes from a particular subfamily of solutions to the so-called $G$-Hitchin equations. The physicist Davide Gaiotto conjectured that for $G=\text{SL(}n\text{,}\mathbb{C}\text{)}$ the second family in a scaling limit converges to a limiting connection which has the structure of an oper. We will describe a proof of this conjecture. This is joint work with Olivia Dumitrescu, Laura Fredrickson, Rafe Mazzeo, Motohico Mulase and Andrew Neitzke.

Friday, February 17, 2017

4:00 pm in 241 Altgeld Hall,Friday, February 17, 2017

#### Hyperbolic taxi cabs and conic kitty cats: a mathematical activity and coloring book

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

Abstract: In this extremely interactive talk, we will develop intuition for various metrics that I have encountered in my own research. We’ll work our way through understanding more familiar spaces such as the real plane as well as hyperbolic plane and disk, to less familiar objects: such as a surface with a Euclidean, cylindrical, or hyperbolic-funnel end. Some markers and colored pencils will be provided, but please feel free to bring your own fun office supplies.

Friday, February 24, 2017

4:00 pm in 241 Altgeld Hall,Friday, February 24, 2017

#### Geometry of convex hypersurfaces

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

Abstract: A convex hypersurface in Euclidean space or Minkowski space is the boundary of an open convex set. Smooth convex hypersurfaces have non-negative sectional curvature and indicate properties of more general Riemannian manifolds with non-negative curvature. I will discuss some properties of convex hypersurfaces. Finally, I will describe a problem that arises from Lorentzian geometry involving convex hypersurfaces and geodesic connectedness and discuss a possible solution to this problem.

Tuesday, March 7, 2017

11:00 am in 345 Altgeld Hall,Tuesday, March 7, 2017

#### Invariant homotopy theory in homotopy type theory

###### Guillaume Brunerie (IAS)

Abstract: This talk will be about homotopy type theory and in particular the branch of it known as invariant homotopy theory, or synthetic homotopy theory.

The main idea is that homotopy type theory is a formal language which can be used to talk about "spaces-up-to-homotopy-equivalence". The basic objects can be thought of as spaces, but the language has the property that all the structures, properties, constructions and proofs that we can express are invariant under homotopy equivalence.

One advantage is that every construction or proof done in this setting is expected to be automatically valid in every infinity-topos, not just in the infinity-topos of spaces, while still looking elementary. In this sense, we can see homotopy type theory as an internal language for infinity-topoi. Moreover, such proofs are also amenable to computer formalization, as homotopy type theory is strongly related to computer proof assistants.

I will present the basic concepts and show what a few proofs and constructions look like in invariant homotopy theory. In particular, we will see the universal cover of the circle, the Hopf fibration, cohomology, and the Steenrod operations.

4:00 pm in 131 English Building,Tuesday, March 7, 2017

#### A variant of Gromov's H\"older equivalence problem for small step Carnot groups

###### Derek Jung   [email] (UIUC Math)

Abstract: This is the second part of a talk I gave last semester in the Graduate Geometry/Topology Seminar. A Carnot group is a Lie group that may be identified with its Lie algebra via the exponential map. This allows one to view a Carnot group as both a sub-Riemannian manifold and a geodesic metric space. It is then natural to ask the following general question: When are two Carnot groups equivalent? In this spirit, Gromov studied the problem of considering for which $k$ and $\alpha$ there exists a locally $\alpha$-H\"older homeomorphism $f:\mathbb{R}^k\to G$. Very little is known about this problem, even for the Heisenberg groups. By tweaking the class of H\"older maps, I will discuss a variant of Gromov's problem for Carnot groups of step at most three. This talk is based on a recently submitted paper. Some knowledge of differential geometry and Lie groups will be helpful.

Friday, March 10, 2017

4:00 pm in 241 Altgeld Hall,Friday, March 10, 2017

#### Stability of relative equilibria and isomorphic vector fields

###### Stefan Klajbor Goderich (UIUC Math)

Abstract: We present applications of the notion of isomorphic vector fields to the study of nonlinear stability of relative equilibria. Isomorphic vector fields were introduced by Hepworth in his study of vector fields on differentiable stacks. Here we argue in favor of the usefulness of replacing an invariant vector field on a manifold by an isomorphic one to study nonlinear stability of relative equilibria. In particular, we use this idea to obtain a criterion for nonlinear stability. As an application, we sketch how to use this to obtain Montaldi and Rodrı́guez-Olmos’s criterion for stability of Hamiltonian relative equilibria.

Friday, March 17, 2017

4:00 pm in 241 Altgeld Hall,Friday, March 17, 2017

#### A 50-minute peek into the quasi-world

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

Abstract: Quasiconformal geometry is the dominant research area which evolved from complex analysis in the 20th century and remains active today. This talk will give a friendly overview to the subject, from its roots in the classical Riemann mapping theorem and Liouville theorem on conformal mappings, to some of its compelling applications in other fields, including complex dynamics and geometric group theory.

Tuesday, March 28, 2017

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

#### Periodic orbits and topological restriction homology

###### Cary Malkiewich (UIUC)

Abstract: This talk is about an emerging connection between algebraic $K$-theory and free loop spaces on the one hand, and periodic orbits of continuous dynamical systems on the other. The centerpiece is a construction in equivariant stable homotopy theory called the "$n$th power trace," which relies on the equivariant norm construction of Hill, Hopkins, and Ravenel. This trace is a refinement of the Lefschetz zeta function of a map $f$, which detects not just fixed points but also periodic orbits of $f$. The applications so far include the resolution of a conjecture of Klein and Williams, and a new approach for the computation of transfer maps in algebraic $K$-theory. These projects are joint work with John Lind and Kate Ponto.

Friday, March 31, 2017

4:00 pm in 241 Altgeld Hall,Friday, March 31, 2017

#### Exotic limit sets of geodesics in Teichmuller space

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

Abstract: In 1975, Masur proved that the Teichmuller space of a surface of genus at least 2 is not Gromov hyperbolic. Since then, many have explored to what extent Teichmuller space has features of negative curvature. In a Gromov hyperbolic space, a geodesic ray converges to a unique point in the hierarchically hyperbolic space (HHS) boundary. We will present our result that a geodesic ray in Teichmuller space does not necessarily converge to a unique point in the HHS boundary of Teichmuller space. In fact, the limit set of a ray can be almost anything allowed by topology. The goal of this talk is not to prove the result, but rather to give necessary background to understand the statement. In particular, we will not assume knowledge of Teichmuller theory or HHS structures.

Tuesday, April 4, 2017

11:00 am in 345 Altgeld Hall,Tuesday, April 4, 2017

#### Coassembly for representation spaces

###### Dan Ramras (IUPUI)

Abstract: I'll describe a homotopy-theoretical framework for studying the relationships between (families of) finite-dimensional unitary representations, vector bundles, and flat connections. Applications to surfaces, 3-manifolds, and groups with Kazhdan's property (T) will be discussed.

Friday, April 7, 2017

4:00 pm in 241 Altgeld Hall,Friday, April 7, 2017

#### Stable Phenomena in Algebraic Topology

###### William Balderrama (UIUC Math)

Abstract: A phenomenon might be called stable if it happens the same way in every dimension. For example, if $C_\bullet$ is a chain complex, then $H_\ast C_\bullet = H_{\ast+1}C_{\bullet+1}$: taking homology'' is done the same in every dimension. In some cases, a construction might not be stable, but can be stabilized. For example, if $M$ is a smooth closed manifold, choice of distinct embeddings $i,j\colon M\rightarrow \mathbb{R}^n$ give rise to possibly nonisomorphic choices of normal bundles $N_iM$ and $N_jM$. However, we can stabilize this by adding trivial bundles: $N_iM\oplus k \simeq N_jM \oplus k$ for sufficiently large $k$, leading to the notion of the stable normal bundle. In this talk, I will introduce this notion of stability, and propose spectra, the main objects in stable homotopy theory, as a good way for dealing with it.

Tuesday, April 11, 2017

11:00 am in 345 Altgeld Hall,Tuesday, April 11, 2017

#### Traces for periodic point invariants

###### Kate Ponto (U Kentucky)

Abstract: Up to homotopy, the Lefschetz number and its refinement to the Reidemeister trace capture the essential information about fixed points of an endomorphism. These invariants can be applied to iterates of an endomorphism to describe periodic points, but in this case they provide far less complete information. I will describe an approach to refining these invariants through refinements of the associated symmetric monoidal and bicategorical traces. This gives richer invariants that also apply to endomorphisms of spaces with more structure (such as bundles).

Friday, April 14, 2017

4:00 pm in 241 Altgeld Hall,Friday, April 14, 2017

#### Net and Filter Convergence Spaces

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

Abstract: A net or filter convergence space is a set together with a collection of data that axiomatizes the notion of convergence to an element of that set. In this sense, convergence spaces generalize topological spaces. More specifically, we will define the (equivalent) categories of net and filter convergence spaces and show that they contain the category of topological spaces (Top) as a full subcategory. We'll highlight some of the advantages these categories have over Top, especially in relation to Tychonoff's theorem. This talk is based off a series of blog posts by Jean Goubault-Larrecq, http://projects.lsv.ens-cachan.fr/topology/?page_id=785.

Tuesday, April 18, 2017

11:00 am in 345 Altgeld Hall,Tuesday, April 18, 2017

#### Representable Cartesian Fibrations

###### Nima Rasekh (UIUC)

Abstract: The goal of this talk is to introduce a notion of a representable Cartesian fibration. Getting there will take us on a journey to many different places in higher category land. We will start by discussing right fibrations, which model presheaves, and then move on to generalize it to Cartesian fibrations. Finally we will have our last stop in the realm of complete Segal objects, which will enable us to define and discuss representable Cartesian fibrations.

Friday, April 21, 2017

4:00 pm in 241 Altgeld Hall,Friday, April 21, 2017

#### It’s hard being positive: symmetric functions and Hilbert schemes

###### Joshua Wen (UIUC Math)

Abstract: Macdonald polynomials are a remarkable basis of $q,t$-deformed symmetric functions that have a tendency to show up various places in mathematics. One difficult problem in the theory was the Macdonald positivity conjecture, which roughly states that when the Macdonald polynomials are expanded in terms of the Schur function basis, the corresponding coefficients lie in $\mathbb{N}[q,t]$. This conjecture was proved by Haiman by studying the geometry of the Hilbert scheme of points on the plane. I’ll give some motivations and origins to Macdonald theory and the positivity conjecture and highlight how various structures in symmetric function theory are manifested in the algebraic geometry and topology of the Hilbert scheme. Also, if you like equivariant localization computations, then you’re in luck!

Tuesday, April 25, 2017

11:00 am in 345 Altgeld Hall,Tuesday, April 25, 2017

#### The signature modulo 8 of a fiber bundle

###### Carmen Rovi (Indiana)

Abstract: In this talk we shall be concerned with the residues modulo 4 and modulo 8 of the signature of a 4k-dimensional geometric Poincare complex. I will explain the relation between the signature modulo 8 and two other invariants: the Brown-Kervaire invariant and the Arf invariant. In my thesis I applied the relation between these invariants to the study of the signature modulo 8 of a fiber bundle, showing in particular that the non-multiplicativity of the signature modulo 8 is detected by an Arf invariant. In 1973 Werner Meyer used group cohomology to show that a surface bundle has signature divisible by 4. I will discuss current work with David Benson, Caterina Campagnolo and Andrew Ranicki where we are using group cohomology and representation theory of finite groups to detect non-trivial signatures modulo 8 of surface bundles.​

Tuesday, May 2, 2017

11:00 am in 345 Altgeld Hall,Tuesday, May 2, 2017

#### Parametrized Morse Theory, Cobordism Categories, and Positive Scalar Curvature

###### Nathan Perlmutter (Stanford)

Abstract: In this talk I will construct a cobordism category consisting of manifolds equipped with a choice of Morse function, whose critical points occupy a prescribed range of degrees. I will identify the homotopy the of this cobordism category with the infinite loopspace of a certain Thom spectrum. Using the parametrized version of the Gromov-Lawson construction, I will then show how to use this cobordism category to probe the space of positive scalar curvature metrics on a closed, spin manifold of dimension > 4. Our main result detects many non-trivial homotopy groups in this space of positive scalar curvature metrics.

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

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

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

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.