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# Seminars (HHH)

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Event When Speaker Title Presentation Material
HHHW01 2nd July 2018
10:00 to 11:00
Emily Riehl The model-independent theory of (∞,1)-categories (1)
Co-author: Dominic Verity (Macquarie University)

In these talks we use the nickname "∞-category" to refer to either a quasi-category, a complete Segal space, a Segal category, or 1-complicial set (aka a naturally marked quasi-category) - these terms referring to Quillen equivalent models of (∞,1)-categories, these being weak infinite-dimensional categories with all morphisms above dimension 1 weakly invertible. Each of these models has accompanying notions of ∞-functor, and ∞-natural transformation and these assemble into a strict 2-category like that of (strict 1-)categories, functors, and natural transformations.

In the first talk, we'll use standard 2-categorical techniques to define adjunctions and equivalences between ∞-categories and limits and colimits inside an ∞-category and prove that these notions relate in the expected ways: eg that right adjoints preserve limits. All of this is done in the aforementioned 2-category of ∞-categories, ∞-functors, and ∞-natural transformations. In the 2-category of quasi-categories our definitions recover the standard ones of Joyal/Lurie though they are given here in a "synthetic" rather than their usual "analytic" form.

In the second talk, we'll justify the framework introduced in the first talk by giving an explicit construction of these 2-categories. This makes use of an axiomatization of the properties common to the Joyal, Rezk, Bergner/Pellissier, and Verity/Lurie model structures as something we call an ∞-cosmos.

In the third talk, we'll encode the universal properties of adjunction and of limits and colimits as equivalences of comma ∞-categories. We also introduce co/cartesian fibrations in both one-sided and two-sided variants, the latter of which are used to define "modules" between ∞-categories, of which comma ∞-categories are the prototypical example.

In the fourth talk, we'll prove that theory being developed isn’t just "model-agnostic” (in the sense of applying equally to the four models mentioned above) but invariant under change-of-model functors. As we explain, it follows that even the "analytically-proven" theorems that exploit the combinatorics of one particular model remain valid in the other biequivalent models.

HHHW01 2nd July 2018
11:30 to 12:30
Thomas Nikolaus Higher categories and algebraic K-theory (1)
HHHW01 2nd July 2018
13:30 to 14:30
Tobias Dyckerhoff Higher Segal spaces (1)
HHHW01 2nd July 2018
14:30 to 15:30
John Francis Factorization homology (1)
HHHW01 2nd July 2018
16:00 to 17:00
Sarah Yeakel Isovariant homotopy theory
An isovariant map between G-spaces is an equivariant map which preserves isotropy groups. Isovariant homotopy theory appears in situations where homotopy is applied to geometric problems, for example, in surgery theory. We will describe some new results in isovariant homotopy theory, including two model structures and an application to intersection theory. This is work in progress, based on conversations with Cary Malkiewich, Mona Merling, and Kate Ponto.
HHHW01 3rd July 2018
10:00 to 11:00
Emily Riehl The model-independent theory of (∞,1)-categories (2)
Co-author: Dominic Verity (Macquarie University)

In these talks we use the nickname "∞-category" to refer to either a quasi-category, a complete Segal space, a Segal category, or 1-complicial set (aka a naturally marked quasi-category) - these terms referring to Quillen equivalent models of (∞,1)-categories, these being weak infinite-dimensional categories with all morphisms above dimension 1 weakly invertible. Each of these models has accompanying notions of ∞-functor, and ∞-natural transformation and these assemble into a strict 2-category like that of (strict 1-)categories, functors, and natural transformations.

In the first talk, we'll use standard 2-categorical techniques to define adjunctions and equivalences between ∞-categories and limits and colimits inside an ∞-category and prove that these notions relate in the expected ways: eg that right adjoints preserve limits. All of this is done in the aforementioned 2-category of ∞-categories, ∞-functors, and ∞-natural transformations. In the 2-category of quasi-categories our definitions recover the standard ones of Joyal/Lurie though they are given here in a "synthetic" rather than their usual "analytic" form.

In the second talk, we'll justify the framework introduced in the first talk by giving an explicit construction of these 2-categories. This makes use of an axiomatization of the properties common to the Joyal, Rezk, Bergner/Pellissier, and Verity/Lurie model structures as something we call an ∞-cosmos.

In the third talk, we'll encode the universal properties of adjunction and of limits and colimits as equivalences of comma ∞-categories. We also introduce co/cartesian fibrations in both one-sided and two-sided variants, the latter of which are used to define "modules" between ∞-categories, of which comma ∞-categories are the prototypical example.

In the fourth talk, we'll prove that theory being developed isn’t just "model-agnostic” (in the sense of applying equally to the four models mentioned above) but invariant under change-of-model functors. As we explain, it follows that even the "analytically-proven" theorems that exploit the combinatorics of one particular model remain valid in the other biequivalent models.

HHHW01 3rd July 2018
11:30 to 12:30
Thomas Nikolaus Higher categories and algebraic K-theory (2)
HHHW01 3rd July 2018
13:30 to 14:30
Tobias Dyckerhoff Higher Segal spaces (2)
HHHW01 3rd July 2018
14:30 to 15:30
John Francis Factorization homology (2)
HHHW01 3rd July 2018
16:00 to 17:00
Andrew Blumberg Thirteen ways of looking at an equivariant stable category
Motivated both by new higher categorical foundations as well as the technology emerging from Hill-Hopkins-Ravenel, there has been a lot of recent work by various authors on formal characterizations of the equivariant stable category. In this talk, I will give an overview of this story, with focus on the perspective coming from the framework of N-infinity operads. In particular, I will describe incomplete stable categories of "O-spectra" associated to any N-infinity operad O as well as some indications of what we can say in the case when G is an infinite compact Lie group.
HHHW01 4th July 2018
10:00 to 11:00
Emily Riehl The model-independent theory of (∞,1)-categories (3)
Co-author: Dominic Verity (Macquarie University)

In these talks we use the nickname "∞-category" to refer to either a quasi-category, a complete Segal space, a Segal category, or 1-complicial set (aka a naturally marked quasi-category) - these terms referring to Quillen equivalent models of (∞,1)-categories, these being weak infinite-dimensional categories with all morphisms above dimension 1 weakly invertible. Each of these models has accompanying notions of ∞-functor, and ∞-natural transformation and these assemble into a strict 2-category like that of (strict 1-)categories, functors, and natural transformations.

In the first talk, we'll use standard 2-categorical techniques to define adjunctions and equivalences between ∞-categories and limits and colimits inside an ∞-category and prove that these notions relate in the expected ways: eg that right adjoints preserve limits. All of this is done in the aforementioned 2-category of ∞-categories, ∞-functors, and ∞-natural transformations. In the 2-category of quasi-categories our definitions recover the standard ones of Joyal/Lurie though they are given here in a "synthetic" rather than their usual "analytic" form.

In the second talk, we'll justify the framework introduced in the first talk by giving an explicit construction of these 2-categories. This makes use of an axiomatization of the properties common to the Joyal, Rezk, Bergner/Pellissier, and Verity/Lurie model structures as something we call an ∞-cosmos.

In the third talk, we'll encode the universal properties of adjunction and of limits and colimits as equivalences of comma ∞-categories. We also introduce co/cartesian fibrations in both one-sided and two-sided variants, the latter of which are used to define "modules" between ∞-categories, of which comma ∞-categories are the prototypical example.

In the fourth talk, we'll prove that theory being developed isn’t just "model-agnostic” (in the sense of applying equally to the four models mentioned above) but invariant under change-of-model functors. As we explain, it follows that even the "analytically-proven" theorems that exploit the combinatorics of one particular model remain valid in the other biequivalent models.

HHHW01 4th July 2018
11:30 to 12:30
Thomas Nikolaus Higher categories and algebraic K-theory (3)
HHHW01 5th July 2018
10:00 to 11:00
Emily Riehl The model-independent theory of (∞,1)-categories (4)
Co-author: Dominic Verity (Macquarie University)

In these talks we use the nickname "∞-category" to refer to either a quasi-category, a complete Segal space, a Segal category, or 1-complicial set (aka a naturally marked quasi-category) - these terms referring to Quillen equivalent models of (∞,1)-categories, these being weak infinite-dimensional categories with all morphisms above dimension 1 weakly invertible. Each of these models has accompanying notions of ∞-functor, and ∞-natural transformation and these assemble into a strict 2-category like that of (strict 1-)categories, functors, and natural transformations.

In the first talk, we'll use standard 2-categorical techniques to define adjunctions and equivalences between ∞-categories and limits and colimits inside an ∞-category and prove that these notions relate in the expected ways: eg that right adjoints preserve limits. All of this is done in the aforementioned 2-category of ∞-categories, ∞-functors, and ∞-natural transformations. In the 2-category of quasi-categories our definitions recover the standard ones of Joyal/Lurie though they are given here in a "synthetic" rather than their usual "analytic" form.

In the second talk, we'll justify the framework introduced in the first talk by giving an explicit construction of these 2-categories. This makes use of an axiomatization of the properties common to the Joyal, Rezk, Bergner/Pellissier, and Verity/Lurie model structures as something we call an ∞-cosmos.

In the third talk, we'll encode the universal properties of adjunction and of limits and colimits as equivalences of comma ∞-categories. We also introduce co/cartesian fibrations in both one-sided and two-sided variants, the latter of which are used to define "modules" between ∞-categories, of which comma ∞-categories are the prototypical example.

In the fourth talk, we'll prove that theory being developed isn’t just "model-agnostic” (in the sense of applying equally to the four models mentioned above) but invariant under change-of-model functors. As we explain, it follows that even the "analytically-proven" theorems that exploit the combinatorics of one particular model remain valid in the other biequivalent models.

HHHW01 5th July 2018
11:30 to 12:30
Thomas Nikolaus Higher categories and algebraic K-theory (4)
HHHW01 5th July 2018
13:30 to 14:30
Rune Haugseng Higher categories of higher categories
I will describe a construction of a higher category of enriched (infinity,n)-categories, and attempt to convince the audience that this is interesting.
HHHW01 5th July 2018
14:30 to 15:30
Christopher Schommer-Pries The Relative Tangle Hypothesis
I will describe recent progress on a non-local variant of the cobordism hypothesis for higher categories of bordisms embedded into finite Euclidean spaces.
HHHW01 5th July 2018
16:00 to 17:00
Kathryn Hess A Künneth theorem for configuration spaces of products
Bill Dwyer, Ben Knudsen, and I recently constructed a model for the configuration space of a product of parallelizable manifolds in terms of the derived Boardman-Vogt tensor product of right modules over the operads of little cubes of the appropriate dimensions. In this talk, after recalling our earlier work, I will introduce a spectral sequence Ben and I have developed for computing the homology of this model.
HHHW01 6th July 2018
10:00 to 11:00
Claudia Scheimbauer Dualizability in the higher Morita category
In this talk I will explain how one can use geometric arguments to obtain results on dualizablity in a factorization version’’ of the Morita category. We will relate our results to previous dualizability results (by Douglas-Schommer-Pries-Snyder and Brochier-Jordan-Snyder on Turaev-Viro and Reshetikin-Turaev theories). We will discuss applications of these dualiability results: one is to construct examples of low-dimensional field theories “relative” to their observables. An example will be given by Azumaya algebras, for example polynomial differential operators (Weyl algebra) in positive characteristic and its center. (This is joint work with Owen Gwilliam.)
HHHW01 6th July 2018
11:30 to 12:30
Markus Spitzweck Hermitian K-theory for Waldhausen infinity categories with genuine duality
In the talk we will introduce the concept of an infinity category with genuine duality, a refinement of the notion of a duality on an infinity category. We will then define the infinity category of Waldhausen infinity categories with genuine duality and study the hermitian/real K-theory functor on this category. In particular we will state and indicate a proof of the Additivity theorem in this context. This is joint work with Hadrian Heine and Paula Verdugo.
HHH 10th July 2018
14:00 to 15:00
Ben Knudsen Connectivity and growth in the homology of graph braid groups
HHH 10th July 2018
15:30 to 16:30
Tyler Lawson Stable power operations
HHH 12th July 2018
15:30 to 15:45
Paul Goerss Approaches to chromatic splitting
HHH 12th July 2018
15:45 to 16:00
John Greenlees Sheaves over an elliptic curve and U(1)-spectra'
HHH 12th July 2018
16:00 to 16:15
Lennart Meier Splittings of tmf_1(n)
HHH 12th July 2018
16:15 to 16:30
Stefan Schwede Categories and orbispaces
HHH 17th July 2018
14:00 to 15:00
Julie Bergner The Waldhausen S-construction as an equivalence of homotopy theories
The notion of unital 2-Segal space was defined independently by Dyckerhoff-Kapranov and Galvez-Carrillo-Kock-Tonks as a generalization of a category up to homotopy. The notion of unital 2-Segal space was defined independently by Dyckerhoff-Kapranov and Galvez-Carrillo-Kock-Tonks as a generalization of a category up to homotopy. A key example of both sets of authors is that the output of applying Waldhausen's S-construction to an exact category is a unital 2-Segal space. In joint work with Osorno, Ozornova, Rovelli, and Scheimbauer, we expand the input of this construction to augmented stable double Segal spaces and prove that it induces an equivalence on the level of homotopy theories. Furthermore, we prove that exact categories and their homotopical counterparts can be recovered as special cases of augmented stable double Segal spaces.

HHH 17th July 2018
15:30 to 16:30
Bjorn Ian Dundas The Geometric Diagonal
HHH 19th July 2018
15:30 to 15:45
Jesper Grodal Have you seen this homology class??
HHH 19th July 2018
15:45 to 16:00
Kathryn Lesh Labeled complexes and homological stability
HHH 19th July 2018
16:00 to 16:15
Michael Mandell 1.3 Thoughts on N∞ Ring Spectra
HHH 19th July 2018
16:15 to 16:30
Brooke Shipley Commuting homology and homotopy inverse limits
HHH 24th July 2018
14:00 to 15:00
Thomas Schick Geometric models of twisted K-homology
HHH 24th July 2018
15:30 to 16:30
Clark Barwick Exodromy
HHH 26th July 2018
15:30 to 15:45
Tobias Barthel HHH Gong Show - On beyond Chouinard
HHH 26th July 2018
15:45 to 16:00
Irina Bobkova HHH Gong Show - Small Picard groups
HHH 26th July 2018
16:00 to 16:15
Fabian Hebestreit HHH Gong Show - Algebraic cobordism categories
HHH 26th July 2018
16:15 to 16:30
Tomer Schlank HHH Gong Show - Non-commutative rational spectra and marked configuration spaces
HHH 31st July 2018
14:00 to 15:00
Clemens Berger Derived modular envelopes and moduli spaces of bordered Riemann surfaces
We study the derived modular envelope of several cyclic operads. More specifically, using a cyclic W-construction as cofibrant replacement we get a derived modular envelope (of the cyclic operad of planar structures) which can be interpreted as a ribbon graph model for the corresponding moduli space of bordered Riemann surfaces. (Joint work with Ralph Kaufmann).

HHH 31st July 2018
15:30 to 16:30
Gregory Arone Tree complexes and obstructions to embeddings.
Using the framework of the calculus of functors (a combination of manifold and orthogonal calculus) we define a sequence of obstructions for embedding a smooth manifold (or more generally a CW complex) M in R^d. The first in the sequence is essentially Haefliger’s obstruction. The second one was studied by Brian Munson. We interpret the n-th obstruction as a cohomology of configurations of n points on M with coefficients in the homology of a complex of trees with n leaves. The latter can be identified with the cyclic Lie_n representation. When M is a union of circles, we conjecture that our obstructions are closely related to Milnor invariants. When M is of dimension 2 and d=4, we speculate that our obstructions are related to ones constructed by Schneidermann and Teichner. This is very much work in progress.

HHH 2nd August 2018
15:30 to 15:45
Agnes Beaudry Linearize this!
HHH 2nd August 2018
15:45 to 16:00
Joshua Hunt Lifting endotrivial modules
HHH 2nd August 2018
16:00 to 16:15
Magdalena Kedziorek Galois extensions, a fairy tale
HHH 2nd August 2018
16:15 to 16:30
Vesna Stojanoska Dreamy Pics
HHH 7th August 2018
14:00 to 15:00
Oscar Randal-Williams Cellular E_k-algebras and homological stability
I will explain recent joint work with S Galatius and A. Kupers in which we use the notion of cellular E_k-algebras and their derived indecomposables to study homological stability for things such as mapping class groups, automorphism groups of free groups, or general linear groups. Using these tools also suggests new types of results in homological stability, such as a second-order form of stability. In this talk I will suppress the details specific to each case, and instead focus on the general calculational aspects of the homology of cellular E_k-algebras with few cells.
HHH 7th August 2018
15:30 to 16:30
Nicholas Kuhn A tutorial on constructions of finite complexes with specified cohomology (after Steve Mitchell and Jeff Smith)
Central to the study of modern homotopy theory is the Periodicity Theorem of Mike Hopkins and Jeff Smith, which says that any type n finite complex admits a v_n self map. Their theorem follows from the Devanitz-Hopkins-Smith Nilpotence Theorem once one has constructed at least one example of v_n self map of a type n complex. The construction of such an ur-example uses a construction due to Jeff Smith making use of the modular representation theory of the symmetric groups. This followed the first construction of a type n complex for all n by Steve Mitchell, which used the modular representation theory of the general linear groups over Z/p. The fine points of the Smith construction are not in the only published source: Ravenel's write-up in his book on the Nilpotence Theorems. I'll discuss some of this, and illustrate the ideas with a construction of a spectrum whose mod 2 cohomology is free on one generator as a module over A(3), the 1024 dimensional subalgebra of the Steenrod algebra generated by Sq^1, Sq^2, Sq^4, and Sq^8.
HHH 9th August 2018
15:30 to 15:45
Agnes Beaudry Linearize this!
HHH 9th August 2018
15:45 to 16:00
Hans-Werner Henn Unexpected and confusing Pics
HHH 9th August 2018
16:00 to 16:15
Gijs Heuts A Whitehead theorem for periodic homotopy groups?
HHH 9th August 2018
16:15 to 16:30
Dylan Wilson Koszul to keep cool
HHH 10th August 2018
14:00 to 15:00
Markus Land On the K-theory of pullbacks
In this talk I will report on joint work with Georg Tamme about excision results in K-theory and related invariants.   We show that, associated to any pullback square of E_1 ring spectra, there is an associated pullback diagram of K-theory spectra in which one of the corners is the K-theory of a new E_1 ring canonically associated to the original pullback diagram.   I will explain the main ingredients for the proof and then concentrate on simple consequences of this theorem. These include Suslin’s results on excision (for Tor-unital ideals) and the fact that (what we call) truncating invariants satisfy excision, nil invariance and cdh descent. If time permits I will also discuss how to deduce that in certain cases (bi)relative K-groups are torsion groups of bounded exponent, improving results of Geisser—Hesselholt.
HHHW02 13th August 2018
10:00 to 11:00
Daniel Dugger Surfaces with involutions
I will talk about the classification of surfaces with involution, together with remarks on the RO(Z/2)-graded Bredon cohomology of these objects.
HHHW02 13th August 2018
11:30 to 12:30
Jeremiah Heller A motivic Segal conjecture
HHHW02 13th August 2018
14:30 to 15:30
Agnes Beaudry Pic(E-Z/4) and Tools to Compute It
In this talk, we describe tools to compute the Picard group of the category of E-G-module spectra where E is Morava E-theory at n=p=2 and G is a finite cyclic subgroup of the Morava stabilizer of order 4. We discuss two tools: 1) A group homomorphism from RO(G) to Pic(E-G) and methods for computing it. 2) The Picard Spectral sequence and some of its equivariant properties. This work is joint with Irina Bobkova, Mike Hill and Vesna Stojanoska
HHHW02 13th August 2018
16:00 to 17:00
Clark Barwick Exodromy and endodromy
On the étale homotopy type of a scheme, there is a natural stratification. We describe both it and its dual, and we discuss some of the ramification information it captures. Joint work with Saul Glasman, Peter Haine, Tomer Schlank.
HHHW02 14th August 2018
09:00 to 10:00
This is report on part of a program to give refinements of numerical invariants arising in enumerative geometry to invariants living in the Grothendieck-Witt ring over the base-field. Here we define an invariant in the Grothendieck-Witt ring for counting'' rational curves. More precisely, for a del Pezzo surface S over a field k and a positive degree curve class $D$ (with respect to the anti-canonical class $-K_S$), we define a class in the Grothendiek-Witt ring of k, whose rank gives the number of rational curves in the class D containing a given collection of distinct closed points $\mathfrak{p}=\sum_ip_i$ of total degree $-D\cdot K_S-1$. This recovers Welschinger's invariants in case $k=\mathbb{R}$ by applying the signature map. The main result is that this quadratic invariant depends only on the $\mathbb{A}^1$-connected component containing $\mathfrak{p}$ in $Sym^{3d-1}(S)^0(k)$, where $Sym^{3d-1}(S)^0$ is the open subscheme of $Sym^{3d-1}(S)$ parametrizing geometrically reduced 0-cycles.
HHHW02 14th August 2018
11:30 to 12:30
Magdalena Kedziorek Algebraic models for rational equivariant commutative ring spectra
I will recall different levels of commutativity in G-equivariant stable homotopy theory and show how to understand them rationally when G is finite or SO(2) using algebraic models. This is joint work with D. Barnes and J.P.C. Greenlees.
HHHW02 14th August 2018
14:30 to 15:30
Marc Hoyois Motivic infinite loop spaces and Hilbert schemes
Co-authors: Elden Elmanto (Northwestern University), Adeel A. Khan (Universität Regensburg), Vladimir Sosnilo (St. Petersburg State University), Maria Yakerson (Universität Duisburg-Essen)We prove a recognition principle for motivic infinite loop spaces over a perfect field of characteristic not 2. This is achieved by developing a theory of framed motivic spaces, which is a motivic analogue of the theory of E-infinity-spaces. Our main result is that grouplike framed motivic spaces are equivalent to the full subcategory of motivic spectra generated under colimits by suspension spectra. As a consequence, we deduce some representability results for suspension spectra of smooth varieties and various motivic Thom spectra in terms of Hilbert schemes of points in affine spaces.Related Linkshttps://arxiv.org/abs/1711.05248 - preprint
HHHW02 14th August 2018
16:00 to 17:00
Vesna Stojanoska Galois extensions in motivic homotopy theory
There are two notions of homotopical Galois extensions in the motivic setting; I will discuss what we know about each, along with illustrative examples. This is joint work in progress with Beaudry, Heller, Hess, Kedziorek, and Merling.
HHHW02 15th August 2018
09:00 to 10:00
Doug Ravenel Model category structures for equivariant spectra
We will discuss methods for constructing a model structure  the category of orthogonal equivariant spectra for a finite group G.   The most convenient one is two large steps away from the most obvious one.
HHHW02 15th August 2018
10:00 to 11:00
Oliver Roendigs On very effective hermitian K-theory
We argue that the very effective cover of hermitian K-theory in the sense of motivic homotopy theory is a convenient algebro-geometric generalization of the connective real topological K-theory spectrum. This means the very effective cover acquires the correct Betti realization, its motivic cohomology has the desired structure as a module over the motivic Steenrod algebra, and that its motivic Adams and slice spectral sequences are amenable to calculations. The latter applies to provide the expected connectivity for its unit map from the motivic sphere spectrum. This is joint work with Alexey Ananyevskiy and Paul Arne Ostvaer.
HHHW02 15th August 2018
11:30 to 12:30
Teena Gerhardt Hochschild homology for Green functors
Hochschild homology of a ring has a topological analogue for ring spectra, topological Hochschild homology (THH), which plays an essential role in the trace method approach to algebraic K-theory. For a C_n-equivariant ring spectrum, one can define C_n-relative THH. This leads to the question: What is the algebraic analogue of C_n-relative THH? In this talk, I will define twisted Hochschild homology for Green functors, which allows us to describe this algebraic analogue. This also leads to a theory of Witt vectors for Green functors, as well as an algebraic analogue of TR-theory. This is joint work with Andrew Blumberg, Mike Hill, and Tyler Lawson.
HHHW02 16th August 2018
09:00 to 10:00
Mark Behrens C_2 equivariant homotopy groups from real motivic homotopy groups
The Betti realization of a real motivic spectrum is a genuine C_2 spectrum. It is well known (c.f. the work of Dugger-Isaksen) that the homotopy groups of the Betti realization of a complex motivic spectrum can be computed by "inverting tau". I will describe a similar theorem which describes the C_2-equivariant RO(G) graded homotopy groups of the Betti realization of a cellular real motivic spectrum in terms of its bigraded real motivic homotopy groups. This is joint work with Jay Shah.
HHHW02 16th August 2018
10:00 to 11:00
Tom Bachmann Motivic normed spectra and Tambara functors
Motivic normed spectra are a motivic analogue of the G-commutative ring spectra from genuine equivariant stable homotopy theory. The category of G-commutative ring spectra such that the underlying genuine equivariant spectrum is concentrated in degree zero (i.e. is a Mackey functor) is in fact equivalent to the category of G-Tambara functors. I will explain an analogous motivic result: the category of normed motivic spectra (over a field), such that the underlying motivic spectrum is an effective homotopy module, is equivalent to a category of homotopy invariant presheaves with generalized transfers and étale norms.
HHHW02 16th August 2018
11:30 to 12:30
Anna Marie Bohmann Graded Tambara functors
Let E be a G-spectrum for a finite group G. It's long been known that homotopy groups of E have the structure of "Mackey functors." If E is G commutative ring spectrum, then work of Strickland and of Brun shows that the zeroth homotopy groups of E form a "Tambara functor." This is more structure than just a Mackey functor with commutative multiplication and there is much recent work investigating nuances of this structure. I will discuss work with Vigleik Angeltveit that extends this result to include the higher homotopy groups of E. Specifically, if E has a commutative multiplication that enjoys lots of structure with respect to the G action, the homotopy groups of E form a graded Tambara functor. In particular, genuine commutative G ring spectra enjoy this property.
HHHW02 16th August 2018
14:30 to 15:30
Mona Merling Toward the equivariant stable parametrized h-cobordism theorem
Waldhausen's introduction of A-theory of spaces revolutionized the early study of pseudo-isotopy theory. Waldhausen proved that the A-theory of a manifold splits as its suspension spectrum and a factor Wh(M) whose first delooping is the space of stable h-cobordisms, and its second delooping is the space of stable pseudo-isotopies. I will describe a joint project with C. Malkiewich aimed at telling the equivariant story if one starts with a manifold M with group action by a finite group G.
HHHW02 16th August 2018
16:00 to 17:00
Fabien Morel Etale and motivic variations on Smith's theory on action of cyclic groups
In this talk, after recalling some facts on Smith's theory in classical topology, I will present some analogues in algebraic geometry using \'etale cohomology first, and I will also give analogues in \A^1-homotopy theory and motives of Smith's theory concerning the multiplicative group \G_m, which behaves like a  "cyclic of order 2".
HHHW02 17th August 2018
09:00 to 10:00
Aravind Asok On Suslin's Hurewicz homomorphism
I will discuss some recent progress on an old conjecture of Suslin about the image of a certain Hurewicz" map from Quillen's algebraic K-theory of a field F to the Milnor K-theory of F. This is based on joint work with J. Fasel and T.B. Williams.
HHHW02 17th August 2018
10:00 to 11:00
Carolyn Yarnall Klein four-slices of HF_2
The slice filtration is a filtration of equivariant spectra analogous to the Postnikov tower that was developed by Hill, Hopkins, and Ravenel in their solution to the Kervaire invariant-one problem. Since that time there have been several new developments, many dealing with cyclic groups. In this talk, we will focus our attention on a noncyclic group! After recalling a few essential definitions and previous results, we will investigate some computational tools that allow us to leverage homotopy results of Holler-Kriz to determine the slices of integer suspensions of HF_2 when our group is the Klein four group. We will end with a few slice tower and spectral sequence examples demonstrating the patterns that arise in the filtration. This is joint work with Bert Guillou.
HHHW02 17th August 2018
11:30 to 12:30
Kirsten Wickelgren Some results in A1-enumerative geometry
We will discuss several applications of A1-homotopy theory to enumerative geometry. This talk includes joint work with Jesse Kass and Padmavathi Srinivasan.
HHH 17th August 2018
13:15 to 14:15
Lifting G-stable endotrivial modules
HHHW02 17th August 2018
14:30 to 15:30
Thomas Nikolaus Cyclotomic spectra and Cartier modules
We start by reviewing the notion of cyclotomic spectra and basic examples such as THH. Then we introduce the classical algebraic notion of an (integral, p-typical) Cartier module and its higher generalization to spectra (topological Cartier modules). We present examples such as Witt vectors and K-theory of endomorphisms. The main result, which is joint work with Ben Antieau, is that there is a close connection between the two notions.
HHH 21st August 2018
14:00 to 15:00
Wolfgang Steimle An Additivity Theorem for cobordism categories, with applications to Hermitian K-theory
The goal of this talk is to explain that Genauer's computation of the cobordism category with boundaries is a precise analogue of Waldhausen's additivity theorem in algebraic K-theory, and to give a new, parallel proof of both results. The same proof technique also applies to cobordism categories of Poincaré chain complexes in the sense of Ranicki. Here we obtain that its classifying space is the infinite loop space of a non-connective spectrum which has similar properties as Schlichting's Grothendieck-Witt spectrum when 2 is invertible; but it turns out that these properties still hold even if 2 is not invertible. This talk is partially based on joint work with B. Calmès, E. Dotto, Y. Harpaz, F. Hebestreit, M. Land, K. Moi, D. Nardin and Th. Nikolaus.
HHH 21st August 2018
15:30 to 16:30
Fabian Hebestreit Parametrised homotopy theory via symmetric retractive spectra
I shall explain a new framework for parametrised homotopy theory, based on symmetric spectra objects in retractive spaces. In contrast to previous model categorical frameworks it encompasses a well-behaved symmetric monoidal structure, that models the Day-convolution present in the quasi-categorical set-up. As a small application I will give a simple proof, that K-theory as defined through stable homotopy theory agrees with the original geometric construction (even as highly structured objects), a fact that seems to be missing from the literature so far.
HHH 23rd August 2018
15:30 to 15:45
Steffen Sagave Commutative cochains
HHH 23rd August 2018
15:45 to 16:00
Marc Levine T for (SL) two
HHH 23rd August 2018
16:00 to 16:15
Tobias Lenz Homotopy (Pre-)Derivators of Cofibration Categories and Quasi-Categories
HHH 23rd August 2018
16:15 to 16:30
Inbar Klang Homology vs cohomology
HHH 28th August 2018
15:30 to 16:30
Lior Yanovski Ambidexterity in the T(n)-Local Stable Homotopy Theory
The monochromatic layers of the chromatic filtration on spectra, that is The K(n)-local (stable 00-)categories Sp_{K(n)} enjoy many remarkable properties. One example is the vanishing of the Tate construction due to  Hovey-Greenlees-Sadofsky.  The vanishing of Tate construction can be considered as a natural equivalence between the colimits and limits in Sp_{K(n)}  parametrized by finite groupoids. Hopkins and Lurie proved a generalization of this result where finite groupoids are replaced by arbitrary \pi-finite  00-groupoids. There is another possible sequence of  (stable 00-)categories who can be considered as "monochromatic layers", Those are the T(n)-local 00-categories Sp_{T(n)}. For the Sp_{T(n)} the vanishing of the Tate construction was proved by Kuhn. We shall prove that the analog of  Hopkins and Lurie's result in for Sp_{T(n)}.  Our proof will also give an alternative proof for the K(n)-local case. This is a joint work with Shachar Carmieli and Lior Yanovski
HHH 30th August 2018
15:30 to 15:45
David Barnes Equivariant sheaves aren't always Weyl
HHH 30th August 2018
15:45 to 16:00
Anna Marie Bohmann Multiplication, not multifarious
HHH 30th August 2018
16:00 to 16:15
Tyler Lawson Unit groups are confusing
HHH 30th August 2018
16:15 to 16:30
Markus Szymik Quo whatis quandles?
HHH 11th September 2018
14:00 to 15:00
Robert Bruner The mod 2 Adams spectral sequence for tmf_*
HHH 11th September 2018
15:30 to 16:30
Irakli Patchkoria Polynomial maps, Witt vectors and Real THH
A result of Hesselholt and Madsen asserts that the ring of components of the topological restriction homology (TR) is isomorphic to the ring of Witt vectors. Trying to prove a real analog of this result leads to a surprising algebraic fact: Witt vectors have an extra functoriality with respect to certain polynomial maps. In this talk we will discuss the extra functoriality and its applications to some problems in algebra and topology. This is joint work with E. Dotto and K. Moi.
HHH 13th September 2018
15:30 to 15:45
Andrew Baker BG_2 vs the Joker
HHH 13th September 2018
15:45 to 16:00
Michael Joachim On twisted spin-c bordism and twisted K-theory
HHH 13th September 2018
16:00 to 16:15
Viktoriya Ozornova Case of Identity
HHH 13th September 2018
16:15 to 16:30
Mingcong Zeng RO(G)-coefficient ring of constant Mackey functor via duality
HHH 18th September 2018
14:00 to 15:00
Christian Schlichtkrull Topological Hochschild homology via generalized Thom spectra
We realize certain quotient spectra of an even ring spectrum as generalized Thom spectra and we use this to calculate their topological Hochschild homology. The main point is to analyze how the topological Hochschild homology varies with different choices of multiplicative structure on the quotient spectra. We also compare the topological Hochschild homology to twisted homology of free loop spaces. This represents joint work with Samik Basu.
HHH 18th September 2018
15:30 to 16:30
Nathalie Wahl Slopes in homological stability, the classical way
In joint work with David Sprehn, we prove homological stability with "slope 1" for symplectic, orthogonal and unitary groups over fields other than F_2, following an argument of Quillen for general linear groups. Stability with slope 2 has long been known for all four types of groups over very general rings by a similar, though not equal, argument. I'll present these results and give an overview of the players in such homological stability arguments
HHH 20th September 2018
15:30 to 15:45
Lars Hesselholt K-theory of division algebras over local fields
HHH 20th September 2018
15:45 to 16:00
Viet-Cuong Pham Brown-Comenetz dual of some K(2)-local spectrum
HHH 20th September 2018
16:00 to 16:15
Constanze Roitzheim Twisted, not bitter
HHH 20th September 2018
16:15 to 16:30
Hiro Tanaka MorseTheory(On.Point) vs (MorseTheory.On)Point
HHHW03 24th September 2018
10:00 to 11:00
Agnes Beaudry K(n)-local homotopy from a Galois theory perspective -1
The goal of these two talks is to give an overview of a perspective on K(n)-local homotopy theory that lends itself to computations. I will introduce some of the key players of the workshop such as the Morava K and E-theories. The introduction will emphasize the Galois theory inherent to the situation and be designed to lead us to consider finite resolutions of the K(n)-local sphere. I will then explain how these perspectives have helped us gain a better understanding on problems in chromatic homotopy theory such as the study of invertible elements and chromatic reassembly.
HHHW03 24th September 2018
11:30 to 12:30
Niko Naumann The role of p-divisible groups -1
We will try to give an account of Lurie's treatment of p-divisible groups in spectral algebraic geometry, to the extend available by the date of the workshop.
HHHW03 24th September 2018
14:30 to 15:30
Christian Ausoni On the topological Hochschild homology of Johnson-Wilson spectra
Let E(n) denote the n-th Johnson-Wilson spectrum at an odd prime p. The spectrum E(1) coincides with the Adams summand of p-local topological K-theory. McClure and Staffeldt offered an intriguing computation of THH(E(1)), showing that it splits as a wedge sum of E(1) and a rationalized suspension of E(1).   In joint work with Birgit Richter, we study the Morava K-theories of THH(E(n)), with an aim at investigating if McClure-Staffeldt's splitting in lower chromatic pieces generalizes.  Under the assumption that E(2) is commutative, we show that THH(E(2)) splits as a wedge sum of E(2) and its lower chromatic localizations.
HHHW03 24th September 2018
16:00 to 17:00
Lennart Meier Topological modular forms with level structure
Topological modular forms with level structure are spectra associated with moduli of elliptic curves with extra structure. These come in a huge variety. We will report on progress on the conjecture that all of these spectra split additively into a few simple pieces. Another goal is to obtain an understanding of connective variants of TMF with level structure. We will present some constructions and a conjecture on their associated stacks. This work is partially joined with Viktoriya Ozornova.
HHHW03 25th September 2018
10:00 to 11:00
Agnes Beaudry Duality and invertibility using finite resolutions - 2
In this talk, I will explain applications of the finite resolutions constructed in the first talk to the study of invertibility and duality in the K(n)-local category.
HHHW03 25th September 2018
11:30 to 12:30
Niko Naumann The role of p-divisible groups - 2
We will try to give an account of Lurie's treatment of p-divisible groups in spectral algebraic geometry, to the extend available by the date of the workshop.
HHHW03 25th September 2018
14:30 to 15:30
Jeremy Hahn The spectrum of units of a height 2 theory
The space BSU admits two infinite loop space structures, one arising from addition of vector bundles and the other from tensor product. A surprising fact, due to Adams and Priddy, is that these two infinite loop spaces become equivalent after p-completion. I will explain how the Adams-Priddy theorem allows for an identification of sl_1(ku_p), the spectrum of units of p-complete complex K-theory. I will then describe work, joint with Andrew Senger, that attempts to similarly understand the spectrum of units of the 2-completion of tmf_1(3). Our computations seem suggestive of broader phenomena, and I will include discussion of several open questions.
HHHW03 25th September 2018
16:00 to 17:00
Tomer Schlank Modes, Ambidexterity and Chromatic homotopy.
Spectra can be considered as the universal presentable stable 00-category, similarly Set can be considered as the universal presentable 1-category and pointed spaces can be considered as the universal pointed presentable. More generally some properties of presentable 00-categories can be classified as equivalent to being a module over a universal symmetric monoidal 00-category. We call such universal symmetric monoidal 00-category "Modes". We describe certain facts about the general theory of modes and present how one can generate new ones from old ones.
HHHW03 26th September 2018
10:00 to 11:00
Ben Antieau Derived algebraic geometry I
In this series of lectures I will outline the major features of derived algebraic geometry in both the connective and nonconnective settings. The connective setting is what is closer to classical algebraic geometry and the study of geometric moduli spaces, for example moduli of objects in derived categories; the nonconnective setting is what is needed for applications to topological modular forms.
HHHW03 26th September 2018
11:30 to 12:30
Tobias Barthel, Nathaniel Stapleton Transchromatic homotopy theory 1
HHHW03 27th September 2018
10:00 to 11:00
Ben Antieau Derived algebraic geometry II
In this series of lectures I will outline the major features of derived algebraic geometry in both the connective and nonconnective settings. The connective setting is what is closer to classical algebraic geometry and the study of geometric moduli spaces, for example moduli of objects in derived categories; the nonconnective setting is what is needed for applications to topological modular forms.
HHHW03 27th September 2018
11:30 to 12:30
Tobias Barthel, Nathaniel Stapleton Transchromatic homotopy theory 2
HHHW03 27th September 2018
14:30 to 15:30
Gijs Heuts Lie algebras and v_n-periodic spaces
HHHW03 27th September 2018
16:00 to 17:00
Vesna Stojanoska Characteristic classes determine dualizing modules
I will address the question of determining the K(n)-local Spanier-Whitehead dual of the Lubin-Tate spectrum, equivariantly with respect to the action of the Morava stabilizer group. A dualizing module can be constructed abstractly, and we use characteristic classes to relate it to a certain representation sphere, at least when we restrict the action to a finite subgroup. As a consequence in specific examples, explicit calculations of characteristic classes also give explicit formulas for the Spanier-Whitehead duals of spectra like TMF and higher real K-theories. This is work in progress, joint with Agnes Beaudry, Paul Goerss, and Mike Hopkins.
HHHW03 28th September 2018
10:00 to 11:00
Zhouli Xu Motivic Ctau-modules and Stable Homotopy Groups of Spheres
I will discuss the equivalence of stable infinity categories, between the motivic Ctau-modules over the complex numbers and the derived category of BP_*BP-comodules. As a consequence, the motivic Adams spectral sequence for Ctau is isomorphic to the algebraic Novikov spectral sequence. This isomorphism of spectral sequences allows computations of classical stable stems at least to the 90-stem, with ongoing computations into even higher dimensions. I will also discuss the situation in the real motivic world, and some connections to the new Doomsday Conjecture, if time permits. This is joint work with Mark Behrens, Bogdan Gheorghe, Dan Isaksen and Guozhen Wang.
HHHW03 28th September 2018
11:30 to 12:30
Gereon Quick Examples of non-algebraic classes in the Brown-Peterson tower
It is a classical problem in algebraic geometry to decide whether a class in the singular cohomology of a smooth complex variety X is algebraic, that is if it can be realized as the fundamental class of an algebraic subvariety of X. One can ask a similar question for motivic spectra: Given a motivic spectrum E, which classes in the topological E-cohomology of X come from motivic classes. I would like to discuss this question and examples of non-algebraic classes for the tower of Brown-Peterson spectra.
HHHW03 28th September 2018
14:30 to 15:30
Mark Behrens The tmf-based Adams spectral sequence for Z
I will describe some specific and qualitative aspects of the tmf-based Adams spectral sequence for the Bhattacharya-Egger type 2 spectrum called "Z". This is joint work with Agnes Beaudry, Prasit Bhattacharya, Dominic Culver, and Zhouli Xu.
HHH 2nd October 2018
14:00 to 15:00
Scott Balchin Adelic models for Noetherian model categories (joint work with John Greenlees)
The algebraic models for rational G-equivariant cohomology theories (Barnes, Greenlees, Kedziorek, Shipley) are constructed by assembling data from each closed subgroup of G. We can compare this to the usual Hasse square, where we see abelian groups being constructed from data at each point of Spec(Z) in an adelic fashion.   This style of assembly can be described in a general fashion using the data contained in the Balmer spectrum of the corresponding tensor-triangulated categories. We show that given a Quillen model category whose homotopy category is a suitably well behaved tensor-triangulated category, that we can construct a Quillen equivalent model from localized p-complete data at each Balmer prime in an adelic fashion.
HHH 2nd October 2018
15:30 to 16:30
Jesper Grodal The Picard group of the stable module category
The stable module category of a finite group is currently maybe my favourite presentable symmetric monoidal stable infinity category. The goal of the talk is to harness its Picard group, using our old friend the orbit category. Part of this talk is a report on joint works with Barthel, Hunt, Carlson, Nakano, Mazza....
HHH 4th October 2018
15:30 to 15:45
George Raptis Transfer maps in A-theory
HHH 4th October 2018
15:45 to 16:00
Doug Ravenel The first English group theorist
HHH 4th October 2018
16:00 to 16:15
Birgit Richter Higher THH of Z/p^m with reduced coefficients
HHH 4th October 2018
16:15 to 16:30
Stefan Schwede Proper, equivariant, and stable!
HHH 8th October 2018
16:00 to 17:00
Lars Hesselholt Rothschild Lecture: Higher algebra and arithmetic - Monday 8th October 2018
This talk concerns a twenty-thousand-year old mistake: The natural numbers record only the result of counting and not the process of counting. As algebra is rooted in the natural numbers, the higher of Joyal and Lurie is rooted in a more basic notion of number which also records the process of counting. Long advocated by Waldhausen, the arithmetic of these more basic numbers should eliminate denominators. Notable manifestations of this vision include the Bökstedt-Hsiang-Madsen topological cyclic homology, which receives a denominator-free Chern character, and the related Bhatt-Morrow-Scholze integral p-adic Hodge theory, which makes it possible to exploit torsion cohomology classes in arithmetic geometry. Moreover, for schemes smooth and proper over a finite field, the analogue of de Rham cohomology in this setting naturally gives rise to a cohomological interpretation of the Hasse-Weil zeta function by regularized determinants, as envisioned by Deninger.
HHH 9th October 2018
14:00 to 15:00
Michael Ching Tangent ∞-categories and Goodwillie calculus
Goodwillie calculus is a set of tools in homotopy theory developed, to some extent, by analogy with ordinary differential calculus. The goal of this talk is to make that analogy precise by describing a common category-theoretic framework that includes both the calculus of smooth maps between manifolds, and Goodwillie calculus of functors, as examples.   This framework is based on the notion of "tangent category" introduced first by Rosicky and recently developed by Cockett and Cruttwell in connection with models of differential calculus in logic, with the category of smooth manifolds as the motivating example. In joint work with Kristine Bauer and Matthew Burke (both at Calgary) we generalize to tangent structures on an (∞,2)-category and show that the (∞,2)-category of presentable ∞-categories possesses such a structure. This allows us to make precise, for example, the intuition that the ∞-category of spectra plays the role of the real line in Goodwillie calculus. As an application we show that Goodwillie's definition of n-excisive functor can be recovered purely from the tangent structure in the same way that n-jets of smooth maps are in ordinary calculus. If time permits, I will suggest how other concepts from differential geometry, such as connections, may play out into the context of functor calculus.
HHH 9th October 2018
15:30 to 16:30
Benjamin Böhme Equivariant multiplications and idempotent splittings of > G-spectra
My PhD research concerns multiplicative phenomena in > equivariant stable homotopy theory. Equivariantly with respect to a > finite group G, there are many different notions of a commutative ring > spectrum. The idempotent summands of the genuine equivariant versions > of the sphere spectrum and the topological K-theory spectra provide > natural examples of such objects. I give a complete characterization > of the best possible equivariant commutative ring structures on these > summands. As an important step in my approach, I establish a > classification of the idempotent elements in the (p-local) > representation ring of G, which may be of independent interest.
HHH 11th October 2018
15:30 to 15:45
Agnes Beaudry Determining the determinant sphere
HHH 11th October 2018
15:45 to 16:00
Julie Bergner To complete or make discrete? That is the question.
HHH 11th October 2018
16:00 to 16:15
Paolo Salvatore (Non)-formality of little discs spaces and operads
HHH 11th October 2018
16:15 to 16:30
Ulrike Tillmann Infinite loop space machines and TQFTs
HHH 15th October 2018
14:00 to 15:00
Doug Ravenel Orthogonal G-Spectra I
HHH 16th October 2018
13:45 to 14:45
Lennart Meier Real homotopy theory
Real homotopy theory has its origin in Atiyah capitalizing on the complex conjugation action on complex K-theory to obtain the $C_2$-spectrum of Real K-theory. More generally it concerns $C_2$-spectra with a Real orientation. We will discuss several examples and several results, including some joint work with Hill and Greenlees.
HHH 16th October 2018
15:15 to 16:15
Robert Oliver TBA
HHH 17th October 2018
14:00 to 15:00
Doug Ravenel Orthogonal G-Spectra II
HHH 18th October 2018
15:30 to 15:45
Peter Haine Constructible étale sheaves and analytic sheaves
HHH 18th October 2018
15:45 to 16:00
Ieke Moerdijk The 10 minute shuffle
HHH 18th October 2018
16:00 to 16:15
Sam Nariman Thurston's fragmentation and non-abelian Poincare duality
HHH 18th October 2018
16:15 to 16:30
Vesna Stojanoska The Tate sphere and duality
HHH 23rd October 2018
14:00 to 15:00
Hans-Werner Henn The exotic part of the Picard group of K(2)-local spectra at the prime 2
The exotic part of the Picard group of K(2)-local spectra at the prime 2 This is a report on joint work with A. Beaudry, I. Bobkova and P. Goerss. Exotic elements in the Picard groups Pic(n,p) of K(n)-local spectra are represented by spectra whose Morava module is isomorphic to that of the 0-dimensonal sphere. These elements form a sub-group of Pic(n,p) which for given n is trivial if p is sufficiently large (p>2 if n=1, and p>3 if n=2). These sub-groups have been explicitly known for n=1 and p=2, and for n=2 and p=3. In this talk we will give a complete description of its isomorphism type if n=p=2.
HHH 23rd October 2018
15:30 to 16:30
Lars Hesselholt K-theory of cusps
In the early nineties, the Buonos Aires Cyclic Homology group calculated the Hochschild and cyclic homology of hypersurfaces, in general, and of the coordinate rings of planar cuspical curves, in particular. With Cortiñas' birelative theorem, proved in 2005, this gives a calculation of the relative K-theory of planar cuspical curves over a field of characteristic zero. By a p-adic version of Cortiñas' theorem, proved by Geisser and myself in 2006, the relative K-groups of planar cuspical curves over a perfect field of characteristic p > 0 can similarly be expressed in terms of topological cyclic homology, but the relevant topological cyclic homology groups have resisted calculation. In this talk, I will show that the new setup for topological cyclic homology by Nikolaus and Scholze has made this calculation possible. This is joint work with Nikolaus and similar results have been obtained by Angeltveit.
HHH 25th October 2018
15:30 to 15:45
Clemens Berger Dold-Kan correspondences and involutive factorisation systems
HHH 25th October 2018
15:45 to 16:00
Paul Goerss What does it mean to be continuous?
HHH 25th October 2018
16:00 to 16:15
Fabian Hebestreit Homotopy Cobordism Categories
HHH 25th October 2018
16:15 to 16:30
Oscar Randal-Williams On the Cayley--Hamilton Theorem
HHH 30th October 2018
14:00 to 15:00
Hiro Tanaka The stack "Broken" and associative algebras
After reviewing some aspects of Morse theory, I'll talk about "Broken," the moduli stack of constant Morse trajectories (possibly broken) on a point. Surprisingly, this stack has the following property: Factorizable sheaves on it are the same thing as (possibly non-unital) associative algebras. We all know that having geometric descriptions of algebraic structures should buy us mileage; so what mileage does this property buy us? If time allows, I'll try to explain why this theorem leads to a roadmap for constructing Morse chain complexes, and in fact, for constructing the stable homotopy type of a compact manifold with a Morse function. (That is, this gives a different way to realize ideas of Cohen-Jones-Segal.) The motivation is to construct a stable homotopy type for Lagrangian Floer Theory--the latter is an important invariant in symplectic geometry and mirror symmetry. This is all joint work with Jacob Lurie.
HHH 30th October 2018
15:30 to 16:30
George Raptis Devissage theorems in algebraic K-theory
In this talk I will give an overview of old and new devissage-type results in algebraic K-theory.
HHH 1st November 2018
15:30 to 15:45
Tobias Barthel Spooky cats
HHH 1st November 2018
15:45 to 16:00
Anna Marie Bohmann On the existence of ghosts
HHH 1st November 2018
16:00 to 16:15
John Greenlees Two skeletons in a cellular cupboard
HHH 1st November 2018
16:15 to 16:30
Constanze Roitzheim Franke's Scary Models
HHH 6th November 2018
14:00 to 15:00
Carles Broto Homotopy fixed points by finite p-group actions on classifying spaces of fusion systems
HHH 6th November 2018
15:30 to 16:30
Kathryn Hess An introduction to topological coHochschild homology
HHH 8th November 2018
15:30 to 15:45
Andrew Baker Orientations old and new
HHH 8th November 2018
15:45 to 16:00
Magdalena Kedziorek Surprise!
HHH 8th November 2018
16:00 to 16:15
HHH 8th November 2018
16:15 to 16:30
Sarah Whitehouse E_r model structures
HHH 12th November 2018
14:00 to 15:00
Doug Ravenel Outlining the Proof of the Kervaire Invariant Theorem
HHH 13th November 2018
14:00 to 15:00
Tobias Barthel Stratifying categories of representations.
Stratifying categories of representations. Abstract: In this talk, we discuss some aspects of the global structure of categories of representations from the point of view of stable homotopy theory. This is joint work with Castellana, Heard, and Valenzuela.
HHH 13th November 2018
15:30 to 16:30
Anna Marie Bohmann Tools for understanding topological coHochschild homology
Hochschild homology is a classical invariant of algebras.  A "topological" version, called THH, has important connections to algebraic K-theory, Waldhausen's A-theory, and free loop spaces.  For coalgebras, there is a dual invariant called "coHochschild homology" and Hess and Shipley have recently defined a topological version called "coTHH."  In this talk, I'll talk about coTHH (and THH) are defined and then discuss work with Gerhardt, Hogenhaven, Shipley and Ziegenhagen in which we develop some computational tools for approaching coTHH.
HHH 15th November 2018
15:30 to 15:45
Scott Balchin Grout (or the story of how my coauthor stopped me from naming things)
HHH 15th November 2018
15:45 to 16:00
Michael Ching What is a pro-operad?
HHH 15th November 2018
16:00 to 16:15
Manuel Krannich A Hermitian K-group via geometry
HHH 15th November 2018
16:15 to 16:30
Mingcong Zeng Slice Differentials
HHH 19th November 2018
14:00 to 15:00
Doug Ravenel The Kervaire invariant problem at odd primes
HHH 20th November 2018
14:00 to 15:00
George Raptis Devissage theorems in algebraic K-theory
In this talk I will give an overview of old and new devissage-type results in algebraic K-theory.
HHH 20th November 2018
15:30 to 16:30
Markus Szymik Symmetry groups of algebraic structures and their homology
The symmetric groups, the general linear groups, and the automorphism groups of free groups are examples of families of groups that arise as symmetry groups of algebraic structures but that are also dear to topologists. There are many other less obvious examples of interest. For instance, in joint work with Nathalie Wahl, this point of view has led to the computation of the homology of the Higman-Thompson groups. I will survey a general context and some more geometric examples in this talk.
HHH 22nd November 2018
15:30 to 15:45
Alexander Berglund Rational models for automorphisms of fiber bundles
HHH 22nd November 2018
15:45 to 16:00
Jesper Grodal Savour thy units! (or why the p-adics sometimes beats the integers)
HHH 22nd November 2018
16:00 to 16:15
Michael Joachim On the Gromov-Lawson-Rosenberg Conjecture
HHH 22nd November 2018
16:15 to 16:30
Doug Ravenel The Pontrjagin twist
HHH 26th November 2018
14:00 to 15:00
Doug Ravenel The slice filtration and slice spectral sequence
HHH 27th November 2018
14:00 to 15:00
Claudia Scheimbauer Constructing extended AKSZ topological field theories in derived symplectic geometry
Derived algebraic geometry and derived symplectic geometry in the sense of Pantev-Toen-Vaquié-Vezzosi allows for a reinterpretation/analog of the classical AKSZ construction for certain $\sigma$-models. It is given by taking mapping stacks with a fixed target building and describes semi-classical TFTs". Using the formalism of derived algebraic geometry as a blackbox, I will sketch how this construction yields extended TFTs, which involves harnessing many higher homotopies. This is joint work in progress with Damien Calaque and Rune Haugseng.
HHH 27th November 2018
15:30 to 16:30
David Ayala Orthogonal group and adjoints
In
this talk I will articulate and contextualize the following sequence of results.

The
Bruhat decomposition of the general linear group defines a stratification
of the orthogonal group.Matrix
multiplication defines an algebra structure on its exit-path category in a
certain Morita category of categories.  In
this Morita category, this algebra acts on the category of
n-categories.

This result is extracted from a
larger program -- entirely joint with John Francis, some parts joint with Nick
Rozenblyum -- which proves the cobordism hypothesis.

HHH 29th November 2018
15:30 to 15:45
Christian Ausoni On a Greenlees spectral sequence
HHH 29th November 2018
15:45 to 16:00
Rachael Boyd Homological stability for Artin monoids
HHH 29th November 2018
16:00 to 16:15
Federico Cantero Morán Spaces of merging submanifold
HHH 29th November 2018
16:15 to 16:30
Christian Schlichtkrull Higher monoidal monomorphisms
HHH 30th November 2018
14:00 to 15:00
Doug Ravenel More about the slice filtration and slice spectral sequence
HHHW04 3rd December 2018
10:00 to 11:00
Soren Galatius H_{4g-6}(M_g)
The set of isomorphism classes of genus g Riemann surfaces carries a natural topology in which it may be locally parametrized by 3g-3 complex parameters. The resulting space is denoted M_g, the moduli space of Riemann surfaces, and is more precisely a complex orbifold of that dimension. The study of this space has a very long history involving many areas of mathematics, including algebraic geometry, group theory, and stable homotopy theory. The space M_g is not compact, essentially because a family of Riemann surface may degenerate into a non-smooth object, and may be compactified in several interesting ways. I will discuss a compactification due to Harvey, which looks like a compact real (6g-6)-dimensional manifold with corners, except for orbifold singularities. The combinatorics of the corner strata in this compactification may be encoded using graphs. Using this compactification, I will explain how to define a chain map from Kontsevich's graph complex to a chain complex calculating the rational homology of M_g. The construction is particularly interesting in degree 4g-6, where our methods give rise to many non-zero classes in H_{4g-6}(M_g), contradicting some predictions. This is joint work with Chan and Payne (arXiv:1805.10186).
HHHW04 3rd December 2018
11:30 to 12:30
Alexander Kupers Cellular techniques in homological stability 1: general theory
This is the first of two talks about joint work with S. Galatius and O Randal-Williams on applications higher-algebraic structures to homological stability. The main tool is cellular approximation of E_k-algebras, and we start with a discussion of the general theory of such cellular approximations. This culminates in a generic homological stability result.
HHHW04 3rd December 2018
14:30 to 15:30
George Raptis The h-cobordism category and A-theory
A fundamental link between Waldhausen's algebraic K-theory of spaces (A-theory) and manifold topology is given by an identification of A-theory in terms of stable homotopy and the stable smooth h-cobordism space. This important result has had many applications in the study of diffeomorphisms of manifolds. In more recent years, the theory of cobordism categories has provided a different approach to the study of diffeomorphism groups with spectacular applications. In collaboration with W. Steimle , we revisit the classical Waldhausen K-theory in light of these developments and investigate new connections and applications. In this talk, I will first discuss a cobordism-type model for A-theory, and then I will focus on the h-cobordism category, the cobordism category of h-cobordisms between smooth manifolds with boundary, and its relationship to the classical h-cobordism space of a compact smooth manifold. This is joint work with W. Steimle.
HHHW04 3rd December 2018
16:00 to 17:00
Christopher Schommer-Pries The Relative Tangle Hypothesis
I will describe recent progress on a non-local variant of the cobordism hypothesis for higher categories of bordisms embedded into finite dimensional Euclidean space.
HHHW04 4th December 2018
09:00 to 10:00
Wolfgang Lueck On the stable Cannon Conjecture
The Cannon Conjecture for a torsionfree hyperbolic group $G$ with boundary homeomorphic to $S^2$ says that $G$ is the fundamental group of an aspherical closed $3$-manifold $M$.  It is known that then $M$ is a hyperbolic $3$-manifold.  We prove the stable version that for any closed manifold $N$ of dimension greater or equal to $2$  there exists a closed manifold $M$ together with a simple homotopy equivalence $M \to N \times BG$. If $N$ is aspherical and $\pi_1(N)$ satisfies the Farrell-Jones Conjecture, then $M$ is unique up to homeomorphism.
This is joint work with Ferry and Weinberger.
HHHW04 4th December 2018
10:00 to 11:00
Thomas Willwacher Configuration spaces of points and real Goodwillie-Weiss calculus
The manifold calculus of Goodwillie and Weiss proposes to reduce questions about embedding spaces of manifolds to questions about mapping spaces of the (little-disks modules of) configuration spaces of points on those manifolds. We will discuss real models for these configuration spaces. Furthermore, we will see that a real version of the aforementioned mapping spaces is computable in terms of graph complexes. In particular, this yields a new tool to study diffeomorphism groups and moduli spaces.
HHHW04 4th December 2018
11:30 to 12:30
Alexander Kupers Cellular techniques in homological stability 2: mapping class groups
This is the second of two talks about joint work with S. Galatius and O Randal-Williams on applications higher-algebraic structures to homological stability. In it we apply the general theory to the example of mapping class groups of surfaces. After reproving Harer's stability result, I will explain how to prove the novel phenomenon of secondary homological stability; there are maps comparing the relative homology groups of the stabilization map for different genus and there are isomorphisms in a range tending to infinity with the genus.
HHHW04 4th December 2018
14:30 to 15:30
I will talk about the connection between the following concepts: manifold calculus, little discs operads, embedding spaces, problem of delooping, relative rational formality of the little discs, and graph-complexes. I will review main results on this connection by Boavida de Brito and Weiss, my coauthors and myself. At the end I will briefly go over the current joint work in progress of Fresse, Willwacher, and myself on the rational homotopy type of embedding spaces.

Co-authors: Gregory Arone (Stockholm University), Julien Ducoulombier (ETH, Zurich), Benoit Fresse (University of Lille), Pascal Lambrechts (University of Louvain), Paul Arnaud Songhafouo Tsopméné (University of Regina), Thomas Willwacher (ETH, Zurich).
HHHW04 4th December 2018
16:00 to 17:00
Nathalie Wahl Homotopy invariance in string topology
In joint work with Nancy Hingston, we show that the Goresky-Hingston coproduct, just like the Chas-Sullivan product, is homotopy invariant. Unlike the Chas-Sullivan product, this coproduct is a "compactified operation", coming from a certain compactification of the moduli space of Riemann surfaces. I'll give an idea of the ingredients used in the proof.
HHHW04 5th December 2018
09:00 to 10:00
Cary Malkiewich Periodic points and topological restriction homology
I will talk about a project to import trace methods, usually reserved for algebraic K-theory computations, into the study of periodic orbits of continuous dynamical systems (and vice-versa). Our main result so far is that a certain fixed-point invariant built using equivariant spectra can be "unwound" into a more classical invariant that detects periodic orbits. As a simple consequence, periodic-point problems (i.e. finding a homotopy of a continuous map that removes its n-periodic orbits) can be reduced to equivariant fixed-point problems. This answers a conjecture of Klein and Williams, and allows us to interpret their invariant as a class in topological restriction homology (TR), coinciding with a class defined earlier in the thesis of Iwashita and separately by Luck. This is joint work with Kate Ponto.
HHHW04 5th December 2018
10:00 to 11:00
Christine Vespa Higher Hochschild homology as a functor
Higher Hochschild homology generalizes classical Hochschild homology for rings. Recently, Turchin and Willwacher computed higher Hochschild homology of a finite wedge of circles with coefficients in the Loday functor associated to the ring of dual numbers over the rationals. In particular, they obtained linear representations of the groups Out(F_n) which do not factorize through GL(n,Z).

In this talk I will explain how viewing higher Hochschild homology of a finite wedge of circles as a functor on the category of free groups provides a conceptual framework which allows powerful tools such as exponential functors and polynomial functors to be brought to bear. In particular, this allows the generalization of the results of Turchin and Willwacher; this gives rise to new linear representations of Out(F_n) which do not factorize through GL(n,Z).

(This is joint work with Geoffrey Powell.)

HHHW04 5th December 2018
11:30 to 12:30
Fabian Hebestreit The homotopy type of algebraic cobordism categories
Co-authors: Baptiste Calmès (Université d'Artois), Emanuele Dotto (RFWU Bonn), Yonatan Harpaz (Université Paris 13), Markus Land (Universität Regensburg), Kristian Moi (KTH Stockholm), Denis Nardin (Université Paris 13), Thomas Nikolaus (WWU Münster), Wolfgang Steimle (Universität Augsburg). Abstract: I will introduce cobordism categories of Poincaré chain complexes, or more generally of Poincaré objects in any hermitian quasi-category C. One interest in such algebraic cobordism categories arises as they receive refinements of Ranicki's symmetric signature in the form of functors from geometric cobordism categories à la Galatius-Madsen-Tillmann-Weiss. I will focus, however, on a more algebraic direction. The cobordism category of C can be delooped by an iterated Q-construction, that is compatible with Bökstedt-Madsen's delooping of the geometric cobordism category. The resulting spectrum is a derived version of Grothendieck-Witt theory and I will explain how its homotopy type can be computed in terms of the K- and L-Theory of C.
HHHW04 6th December 2018
09:00 to 10:00
Alexander Berglund Rational homotopy theory of automorphisms of manifolds
I will talk about differential graded Lie algebra models for automorphism groups of simply connected manifolds M. Earlier results by Ib Madsen and myself on models for block diffeomorphisms combined with rational models for Waldhausen's algebraic K-theory of spaces suggest a model for the group of diffeomorphisms homotopic to the identity, valid in the so-called pseudo-isotopy stable range. If time admits, I will also discuss how to express the generalized Miller-Morita-Mumford classes in the cohomology of BDiff(M) in terms of these models.
HHHW04 6th December 2018
10:00 to 11:00
Johannes Ebert Cobordism categories, elliptic operators and positive scalar curvature
We prove that a certain collection of path components of the space of metrics of positive scalar curvature on a high-dimensional sphere has the homotopy type of an infinite loop space, generalizing a theorem by Walsh. The proof uses an version of the surgery method by Galatius and Randal--Williams to cobordism categories of manifolds equipped with metrics of positive scalar curvature. Moreover, we prove that the secondary index invariant of the spin Dirac operator is an infinite loop map. The proof of that fact uses a generalization of the Atiyah--Singer index theorem to spaces of manifolds. (Joint work with Randal--Williams)
HHHW04 6th December 2018
11:30 to 12:30
Ben Knudsen Configuration spaces and Lie algebras away from characteristic zero
There is a close connection between the theory of Lie algebras and the study of additive invariants of configuration spaces of manifolds, which has been exploited in many calculations of rational homology. We begin the computational exploration of this connection away from characteristic zero, exhibiting a spectral sequence converging to the p-complete complex K-theory of configuration spaces---more generally, to their completed Morava E-(co)homology---and we identify its second page in terms of an algebraic homology theory for Lie algebras equipped with certain power operations. We construct a computationally accessible analogue of the classical Chevalley--Eilenberg complex for these Hecke Lie algebras, and we use it to perform a number of computations. This talk is based on joint work in progress with Lukas Brantner and Jeremy Hahn.
HHHW04 6th December 2018
14:30 to 15:00
Manuel Krannich Contributed talk - Mapping class groups of highly connected manifolds
The group of isotopy classes of diffeomorphisms of a highly connected almost parallelisable manifold of even dimension 2n>4 has been computed by Kreck in the late 70’s. His answer, however, left open two extension problems, which were later understood in some particular dimensions, but remained unsettled in general. In this talk, I will explain how to resolve these extension problems in the case of n being odd, resulting in a complete description of the mapping class group in question in terms of an arithmetic group and the cokernel of the stable J-homomorphism.
HHHW04 6th December 2018
15:00 to 15:30
Rachael Boyd Contributed Talk - The low dimensional homology of Coxeter groups
Coxeter groups were introduced by Tits in the 1960s as abstractions of the finite reflection groups studied by Coxeter. Any Coxeter group acts by reflections on a contractible complex, called the Davis complex. This talk focuses on a computation of the first three integral homology groups of an arbitrary Coxeter group using an isotropy spectral sequence argument: the answer can be phrased purely in terms of the original Coxeter diagram. I will give an introduction to Coxeter groups and the Davis complex before outlining the proof.
HHHW04 6th December 2018
16:00 to 16:30
Csaba Nagy Contributed Talk - The Sullivan-conjecture in complex dimension 4
The Sullivan-conjecture claims that complex projective complete intersections are classified up to diffeomorphism by their total degree, Euler-characteristic and Pontryagin-classes. Kreck and Traving showed that the conjecture holds in complex dimension 4 if the total degree is divisible by 16. In this talk I will present the proof of the remaining cases. It is known that the conjecture holds up to connected sum with the exotic 8-sphere (this is a result of Fang and Klaus), so the essential part of our proof is understanding the effect of this operation on complete intersections. This is joint work with Diarmuid Crowley.
HHHW04 6th December 2018
16:30 to 17:00
Danica Kosanović Contributed talk - Extended evaluation maps from knots to the embedding tower
The evaluation maps from the space of knots to the associated embedding tower are conjectured to be universal knot invariants of finite type. Currently such invariants are known to exist only over the rationals (using the existence of Drinfeld associators) and the question of torsion remains wide open. On the other hand, grope cobordisms are certain operations in ambient 3-space producing knots that share the same finite type invariants and give a geometric explanation for the appearance of Lie algebras and graph complexes.

I will explain how grope cobordisms and an explicit geometric construction give paths in the various levels of the embedding tower. Taking components recovers the result of Budney-Conant-Koytcheff-Sinha, showing that these invariants are indeed of finite type. This is work in progress joint with Y. Shi and P. Teichner.
HHHW04 7th December 2018
10:00 to 11:00
Andre Henriques The complex cobordism 2-category and its central extensions
I will introduce a symmetric monoidal 2-category whose objects are 0-manifolds, whose 1-morphisms are 1-dimensional smooth cobordisms, and whose 2-morphisms are Riemann surfaces with boundary and cusps. I will introduce a certain central extension by ℝ₊ and explain its relevance in chiral conformal field theory. Finally, I will explain the state of my understanding on the question of classification of such extensions by ℝ₊.
HHHW04 7th December 2018
11:30 to 12:30
Sam Nariman Topological and dynamical obstructions to extending group actions.
For any 3-manifold $M$ with torus boundary, we find finitely generated subgroups of $\Diff_0(\partial M)$ whose actions do not extend to actions on $M$; in many cases, there is even no action by homeomorphisms. The obstructions are both dynamical and cohomological in nature. We also show that, if $\partial M = S^2$, there is no section of the map $\Diff_0(M) \to \Diff_0(\partial M)$. This answers a question of Ghys for particular manifolds and gives tools for progress on the general program of bordism of group actions. This is a joint work with Kathryn Mann.
HHHW04 7th December 2018
13:30 to 14:30
Ryan Budney Some prospects with the splicing operad
Roughly six years ago I described an operad that acts on spaces of long knots'. This is the space of smooth embeddings of R^j into R^n. The embeddings are required to be standard (linear) outside of a disc, and come equipped with a trivialisation of their normal bundles. This splicing operad gives a remarkably compact description of the homotopy-type of the space of classical long knots (j=1, n=3), that meshes well with the machinery of 3-manifold theory: JSJ-decompositions and geometrization. What remains to be seen is how useful this splicing operad might be when n is larger than 3. I will talk about what is known at present, and natural avenues to explore.
HHHW04 7th December 2018
14:30 to 15:30
Diarmuid Crowley Relative kappa-classes
Diff(D^n), the the space of diffeomorphisms of the n-disc fixed near the boundary has rich rational topology. For example, Weiss's discovery of surreal'' Pontrjagin classes leads to the existence of rationally non-trivial homotopy classes in BDiff(D^n).

For any smooth n-manifold M, extension by the identity induces a map BDiff(D^n) \to BDiff(M). In this talk I will report on joint work with Wolfgang Steimle and Thomas Schick, where we consider the problem of computing the image of the `Weiss classes'' under the maps on homotopy and homology induced by extension. This problem naturally leads one to consider relative kappa-classes.

Via relative kappa-classes, we show that the maps induced by extension are rationally non-trivial for a wide class of manifolds M, including aspherical manifolds (homology, hence also homotopy) and stably parallelisable manifolds (homotopy). When M is aspherical, our arguments rely on vanishing results for kappa-classes due to Hebestreit, Land, Lueck and Randal-Williams.

HHH 11th December 2018
14:00 to 15:00
Ieke Moerdijk Dendroidal spaces and mapping spaces between little cubes operads.
HHH 11th December 2018
15:30 to 16:30

The Fulton-MacPherson operads are E_n operads that are miraculously invariant under the W cofibrant resolution by Boardman-Vogt. This allows a geometric approach to several topics including factorization homology, Koszul duality of E_n-operads, cellular operadic subdivision, and E_n-algebra homology. (part of this is joint work with Michael Ching and Benoit Fresse)

HHH 13th December 2018
15:30 to 15:45
Clark Barwick The line isn’t contractible
HHH 13th December 2018
15:45 to 16:00
Daniela Egas Santander Wandering down the path to path homology
HHH 13th December 2018
16:00 to 16:15
Gijs Heuts Whitehead products in v_n-periodic homotopy groups
HHH 13th December 2018
16:15 to 16:30
Martin Palmer-Anghel Homological stability for moduli spaces of disconnected submanifolds
HHH 18th December 2018
14:00 to 15:00
Boris Botvinnik Positive scalar curvature metrics on manifolds with fibred singularities
HHH 18th December 2018
15:30 to 16:30
Birgit Richter Spaces and cochains -- yet another approach
Rationally, the homotopy type of any reasonable space is completely determined by (a minimal model of) the Sullivan cochain algebra of the space. If you want to be non-rational, then Mandell's result says that the $E_\infty$-algebra structure of the cochains determines the homotopy type. In joint work with Steffen Sagave we construct a strictly commutative model of the cochains of a space using the diagram category of finite sets and injections in order to free things up. We show that this cochain algebra determines the homotopy type of (finite type, nilpotent) spaces

HHH 20th December 2018
15:30 to 15:45
Robert Bruner Gröbner bases
HHH 20th December 2018
15:45 to 16:00
Sam Nariman Vanishing or non-vanishing of kappa_2
HHH 20th December 2018
16:00 to 16:15
Nathalie Wahl Pushing around, trying to stay normal and positive
HHH 20th December 2018
16:15 to 16:30
Anna Marie Bohmann Here Happily Have we assembled


text:
# FEATURED STORY

## Building C# objects dynamically

### Using ExpandoObject to create objects that you can add properties, methods, and events to.

Buy “C# 6.0 Cookbook” in early release.
Editor’s note: This is an excerpt from “C# 6.0 Cookbook,” by Jay Hilyard and Stephen Teilhet. It offers more than 150 code recipes to common and not-so-common problems that C# programmers face every day. In it, you’ll find recipes on asynchronous methods, dynamic objects, enhanced error handling, the Rosyln compiler, and more.

Problem

You want to be able to build up an object to work with on the fly at runtime.

Solution

Use ExpandoObject to create an object that you can add properties, methods, and events to and be able to data bind to in a user interface.

We can use ExpandoObject to create an initial object to hold the Name and current Country of a person.

Once we have added properties directly, we can also add properties to our object in a more dynamic fashion using the AddProperty method we have provided for you. One example of why you might do this is to add properties to your object from another source of data. We will add the Language property.

## Get started with functional programming in Python

### Start writing shorter and less bug-prone Python code.

It is hard to get a consistent opinion on just what functional programming is, even from functional programmers themselves. A story about elephants and blind men seems apropos here. Usually we can contrast functional programming with “imperative programming” (what you do in languages like C, Pascal, C++, Java, Perl, Awk, TCL, and most others, at least for the most part). Functional programming is not object-oriented programming (OOP), although some languages are both. And it is not Logic Programming (e.g., Prolog).

I would roughly characterize functional programming as having at least several of the following characteristics:

• Functions are first class (objects). That is, everything you can do with “data” can be done with functions themselves (such as passing a function to another function). Moreover, much functional programming utilizes “higher order” functions (in other words, functions that operate on functions that operate on functions).
• Functional languages eschew side effects. This excludes the almost ubiquitous pattern in imperative languages of assigning first one, then another value to the same variable to track the program state.
• In functional programming we focus not on constructing a data collection but rather on describing “what” that data collection consists of. When one simply thinks, “Here’s some data, what do I need to do with it?” rather than the mechanism of constructing the data, more direct reasoning is often possible.

Functional programming often makes for more rapidly developed, shorter, and less bug-prone code. Moreover, high theorists of computer science, logic, and math find it a lot easier to prove formal properties of functional languages and programs than of imperative languages and programs.

### The principles of reactive applications facilitate adaptation.

One of the fascinating things found in nature is the ability of a species to adapt to its changing environment. The canonical example of this is Britain’s Peppered Moth. When newly industrialized Great Britain became polluted in the nineteenth century, slow-growing, light-colored lichens that covered trees died and resulted in a blackening of the trees bark. The impact of this was quite profound: lightly-colored peppered moths, which historically had been well camouflaged and the majority, now found themselves the obvious target of many a hungry bird. Their rare, dark-colored sibling, who had been conspicuous before, now blended into their recently polluted ecosystem. As the birds, changed from eating dark-colored to light-colored moths, the previously common light-colored moth became the minority, and the dynamics of Britain’s moth population changed.

So what do moths have to do with programming? Read more…

## 4 ways the Raspberry Pi is being used in education

### Get inspired to create, teach, and learn with the Raspberry Pi.

The Raspberry Pi is a small computer that can be used for a variety of projects, and has been heralded as a great boon to education due to its flexibility and simplicity. While PcPro magazine noted in January of 2014 that Pi’s were “gathering dust” in classrooms, production has not ceased. The usage map is pretty impressive and the Raspberry Pi 2 was recently released.

In February of this year, the Raspberry Pi Foundation announced that they’re starting a mentoring program for people 16-21 years old. Here are four other ways that the Pi is being used in education and growing the tech community.

## Learn a C-style language

### Improve your odds with the lingua franca of computing.

You have a lot of choices when you’re picking a programming language to learn. If you look around the web development world, you’ll see a lot of JavaScript. At universities and high schools, you’ll often find Python used as a teaching language. If you go to conferences with language theorists, like Strange Loop, you’ll hear a lot about functional languages, such as Haskell, Scala, and Erlang. This level of choice is good: many languages mean that the overall state of the field is continually evolving, and coming up with new solutions. That choice also leads to a certain amount of confusion regarding what you should learn. It’s not possible to learn every language out there, even if you wanted to. Depending on the area you’re in, the choice of language may be made for you. For the overall health of your career, and to provide you the widest range of future opportunities, the single most useful language-related thing you can do is learn a C-style language.

A boring old C-style language just like millions of developers learned before you, going back to the 1980s and earlier. It’s not flashy, it’s usually not cutting edge, but it is smart. Even if you don’t stick with it, or program in it on a daily basis, having a C-style language in your repertoire is a no-brainer if you want to be taken seriously as a developer.

## .NET open source

### Microsoft .Net Team Program Manager, Beth Massi, on the open source .NET Core.

You might have heard the news that .NET is open source. In this post I’m going to explain what exactly we open sourced, why we did it, and how you can get involved.

## Defining .NET

If you’re not familiar with .NET, it’s a managed execution environment that provides a variety of services to its running applications including things like automatic memory management, type safety, native interop, and multiple modern programming languages that make it easier to build all kinds of apps, for nearly any device, quickly. The first version of .NET was initially released in 2002 and quickly picked up steam in many businesses. Today there are over 1.8 billion active installs of the .NET Framework and 6 million .NET developers in the world.

The .NET Framework consists of these major components: the common language runtime (CLR), which is the execution engine that handles running applications; the .NET Framework Base Class Libraries (BCL), which provides a library of tested, reusable code that developers can call from their own applications; and the managed languages and compilers for C#, F#, and Visual Basic. Application models extend the common libraries of the .NET Framework to provide additional libraries that developers can use to build specific types of applications, like web, desktop, mobile device apps, etc. For more information on all the components in .NET 2015 see: Understanding .NET 2015.

There are multiple implementations of .NET, some from Microsoft and others from other companies or open source projects. In this post, I’ll focus on .NET Core from Microsoft.

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