The second is a generalization, due to the speaker, of a linking "number" for arbitrary manifolds in an arbitrary manifold. This was generalized by Koschorke to higher-order linking of arbitrary spheres in Euclidean space, and the speaker generalized this to arbitrary manifolds. These higher-order invariants are also relative invariants in the same way the linking number is, and they admit a number of interesting geometric interpretations.

Along the way, we will observe that the classical linking number is related to the stable homotopy groups of spheres, whereas the higher-order generalizations are related to a certain filtration of the unstable homotopy groups of spheres. We conjecture that similar results hold for all primes. Martin Frankland. University of Illinois at Urbana-Champaign. It is a classic fact that any graded group abelian above dimension 1 can be realized as the homotopy groups of a space.

However, the question becomes difficult if one includes the data of primary homotopy operations, known as a Pi-algebra. When a Pi-algebra is realizable, we would also like to classify all homotopy types that realize it. Using an obstruction theory of Blanc-Dwyer-Goerss, we will describe the moduli space of realizations of certain 2-stage Pi-algebras. This is better than a classification: The moduli space provides information about realizations as well as their higher automorphisms.

This represents joint work with K. Hardie and N. Hardie, H. Marcum and N. This is joint work with Crichton Ogle. New topological invariants in Morse Novikov theory bar codes and Jordan cells. They are Bar Codes and Jordan Cells. A more subtle invariant like Reidemeister torsion is related to the Jordan cells.

This last extension is more elaborate and will be discussed later. We also introduce a Topological Rigidity Conjecture and we show that the above result is a consequence of it. Indira Chatterji. We discuss a question by Nori, which is to determine when a discrete Zariski dense subgroup in a semisimple Lie group containing a lattice has to be itself a lattice. This is joint work with Venkataramana. There is a theory of random graphs due to Erdos and Renyi. Associated to any group and a graph there is a notion of its graph product.

So, there also is a notion of a random graph products of groups. We compute the cohomological invariants of random graph products. This is joint work with Matt Kahle. The Hochschild chain and cochain complexes and the cyclic complex of an associative algebra serve as noncommutative analogs of classical geometric objects on a manifold, such as differential forms and multivector fields. These complexes are known to possess a very nontrivial and rich algebraic structure that is analogous to, and goes well beyond, the classical algebraic structures known in geometry.

In this talk, I will give a review of the subject and outline an approach that is based on an observation that differential graded categories form a two-category up to homotopy. Given a fibration p, we can ask when p is fiber homotopy equivalent to a topological fiber bundle with compact manifold fibers; assuming that the fibration p does admit a compact bundle structure, we can also ask to classify all such bundle structures on p. Similarly, given a map f between compact manifolds, we can ask when f is homotopic to a topological fiber bundle with compact manifold fibers, and assuming that the map f does fiber, we can ask to classify all of the different ways to fiber f.

In this talk, we will begin by describing the space of all compact bundle structures on a fibration, which is nonempty if and only if p admits a compact bundle structure. We will then show that, as long as we are willing to stabilize by crossing with a disk, the obstructions to stably fibering a map f are related to the space of bundle structures on the fibration p associated to f. The surprise will be that a lift can often be found in the topological case. Examples will be given realizing the obstructions. Courtney Thatcher. We consider free actions of large prime order cyclic groups on products of spheres.

The equivariant homotopy type will be determined and the simple structure set discussed. The cohomology classes, Pontrjagin classes, and the set of normal invariants will also be discussed. Rose-Hulman Institute of Technology. The deRham cohomology of a Poisson manifold comes equipped with a canonical filtration. For symplectic manifolds this filtration is well understood and can be computed from the cup product action of the cohomology class represented by the symplectic form.

In this talk we will discuss said filtration for the total space of sympletic fiber bundles. The latter constitute a class of Poisson manifolds closely related to the topology of the sympletic group of the typical fiber. New topological invariants bar codes and Jordan cell at work part I. Bar codes and Jordan cells provide a new type of linear algebra invariants which can be used in topology.

In this lecture I will describe some joint work with S Haller. New topological invariants bar codes and Jordan cell at work part II. The proof is totally algebraic and does not rely on general results involving categorical invariants. Cleveland State University. The LS category of a space X is a numerical invariant that measures the complexity of a space.

While it is usually very hard to compute explicitly, there are estimates and approximating invariants that help us to understand category better. A big problem is to understand the effect of the fundamental group on category. Using an interpretation of this 1-category given by Svarc, we have also been able to refine Bochner's bound on the first Betti number in the presence of non-negative Ricci curvature.

Finally, the 1-category forms a bridge between the theorems of Yamaguchi and Kapovitch-Petrunin-Tuschmann on manifolds with almost non-negative sectional curvature. Surgery on nullhomologous tori and smooth structures on 4-manifolds. Grigori Avramidi. I will describe some recent results on isometry groups of aspherical Riemannian manifolds and their universal covers. The general theme is that topological properties of an aspherical manifold often restrict the isometries of an arbitrary complete Riemannian metric on that manifold. These topological properties tend to be established by using a specific "nice" metric on the manifold.

I will illustrate this by explaining why on an irreducible locally symmetric manifold, no metric has more symmetry than the locally symmetric metric. I will also discuss why moduli space is a minimal orbifold and relate this phenomenon to symmetries of arbitrary metrics on moduli space.

I will discuss the geometry of certain abelian-by-cyclic groups and show how to establish the optimal top-dimensional isoperimetric inequality that holds in these groups. This is joint work with Noel Brady. We also explain that there is, nevertheless, a way of using the Higson compactification to prove the Novikov conjecture for a large class of groups. Dave Constantine. Group actions and compact Clifford-Klein forms of homogeneous spaces. I will also make some remarks on a conjecture of Kobayashi on the scarcity of compact forms.

We will describe decompositions of finitely presented groups, using descriptions of smooth and of symplectic four-manifolds. We will exhibit various ways of obtaining similar decompositions of finitely presented groups into graphs, via descriptions of smooth 4-manifolds into Lefschetz fibrations. We distill this data into invariants, by considering the minimal number of edges these graphs may have. These ideas are related to the minimal euler characteristic of symplectic four-manifolds and the minimal genus of a Lefschetz fibration, seen as group invariants.

The study of fundamental groups of random two dimensional simplicial complexes calls attention to the small subcomplexes of such objects. Such subcomplexes have fewer triangles than some multiple of the number of their vertices. One gets that this condition with constant less than two on a connected complex and all of its subcomplexes implies that it is homotopy equivalent to a wedge of circles, spheres and projective planes.

This analysis yields parameter regimes for vanishing, hyperbolicity and Kazhdanness of these groups. For clique complexes of random graphs there is a similar problem involving complexes with fewer edges than thrice the number of their vertices resulting in similar results on the fundamental groups of their clique complexes.

This is based on joint work with Hoffman and Kahle. We make an analogous definition for spaces. In contrast to the classical notion, the abelian duality property imposes some obvious constraints on the Betti numbers of abelian covers. On the other hand, using a result of Brady and Meier, we find that right-angled Artin groups are abelian duality groups if and only if they are duality groups: both properties are equivalent to the Cohen-Macaulay property for the presentation graph.

Building on work of Davis, Januszkiewicz, Leary and Okun, hyperplane arrangement complements are both duality and abelian duality spaces. These results follow from a slightly more general, cohomological vanishing theorem, part of work in progress with Alex Suciu and Sergey Yuzvinsky. This is joint work with Mike Davis and Tadeusz Januszkiewicz.

It is thus naturally a module over the ring of Laurent polynomials in two variables. The other development is the construction of braid representations from quantum groups. We also show irreducibility of this representation over the fraction field of the ring of Laurent polynomials. Time permitting we will discuss relations to other types of braid group representations that may shed light on this curious connection, as well as reducibility issues at certain choices of parameters. We take an obstruction-theoretic approach to the question of algebraic structure in homotopical settings.

At its heart, this is an application of the Bousfield-Kan spectral sequence adapted for the action of a monad T on a topological model category. We will present examples from rational homotopy theory illustrating the obstructions to rigidifying homotopy algebra maps to strict algebra maps, and explain in a precise way how the edge homomorphism of this obstruction spectral sequence measures the difference between up-to-homotopy and on-the-nose T-algebra maps. New topological invariants for a continuous nonzero complex valued function.

The first two are continuous assignments with respect to compact open topology, the last is locally constant on the space of continuous functions with compact open topology. For free co-compact actions there are additional restrictions, but no general sufficient conditions are known. The talk will survey this problem and its connection to the Farrell-Jones assembly maps in K-theory and L-theory. Smith, in the first half of the 20th century, developed homological tools to study actions of finite p-groups on topological spaces.

This talk will review the classical theory, give applications to actions on aspherical manifolds, and extend the theory to give restrictions on periodic knots. A few characterizations of amenability and property A are given, including Johnson's cohomological characterization of amenability and the recent work of Brodzki, Nowak, Niblo, and Wright which characterizes property A in a similar manner.

These characterizations play a major role in the relative versions of these properties. We define the notion of a group having relative property A with respect to a finite family of subgroups. Many characterizations for relative property A are given. This result leads to new classes of groups that have property A. Specializing the definition of relative property A, an analogue definition of relative amenability for discrete groups are introduced and similar results are obtained. Recent joint work with Ayelet Lindenstrauss describing this spectrum, when one is working in an analytic range in the sense of Goodwillie's calculus of functors will be discussed.

Sharp vanishing thresholds for cohomology of random flag complexes. The random flag complex is a natural combinatorial model of random topological space. In this talk I will survey some results about the expected topology of these objects, focusing on recent work which gives a sharp vanishing threshold for kth cohomology with rational coefficients. This recent work provides a generalization of the Erdos-Renyi theorem which characterizes how many random edges one must add to an empty set of n vertices before it becomes connected.

This is topology seminar, so I will assume that people know what homology and cohomology are, but I will strive to make the talk self contained and define all the necessary probability as we go. Ruben Sanchez-Garcia. For each discrete group G one can find a universal G-space with stabilizers in a prescribed family of subgroups of G. These spaces play a prominent role in the so-called Isomorphism Conjectures, namely the Baum-Connes and the Farrell-Jones conjectures. We will discuss the former conjecture in more detail and describe its topological side: the equivariant K-homology of the universal space for proper actions.

Miami University. These groups account for 73 isomorphism types of three-dimensional crystallographic groups, out of types in all. This result was one of the earliest uses of the cup product in singular cohomology. I will describe some joint work with D. The basic idea is to use motivic cohomology instead of singular cohomology. This leads into the broader subject of computations in motivic homotopy theory.

We review some old problems in algebraic topology—namely, the classification of finite-dimensional modules over subalgebras of the Steenrod algebra, and related classification problems in representation theory and finite CW complexes--and some old techniques in deformation theory--namely, the use of Hochschild 1- and 2-cocycles with appropriate coefficients to classify first-order deformations of modules and algebras, respectively. Then we work out how one has to adapt these old methods to solve these old problems, ultimately using some modern technology: a deformation-theoretic interpretation of twisted nonabelian higher-order Hochschild cohomology.

This is joint work with Tam Nguyen Phan. Moreover, it leads to new estimates for the number of commutators necessary. Christopher Davis. Computing Abelian rho-invariants of links via the Cimasoni-Florens signature. The solvable filtration of the knot concordance group has been studied closely since its definition by Cochran, Orr and Teichner in Recently Cochran, Harvey and Leidy have shown that the successive quotients in this filtration contain infinite rank free abelian groups and even exhibit a kind of primary decomposition.

Unfortunately, their construction relies on an assumption of non-vanishing of certain rho-invariants. By relating these rho-invariants to the signature function defined by Cimasoni and Florens in , we remove this ambiguity from the construction of Cochran-Harvey-Leidy. Loyola University, New Orleans. We present results concerning the existence of nontrivial homotopy spheres and also discuss the determination of the smallest dimensional Euclidean spaces in which they smoothly embed.

The ecological niche of a species is the set of environmental conditions under which a population of that species persists. This is often thought of as a subset of "environment space" — a Euclidean space with axes labeled by environmental parameters.

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This talk will explore mathematical models for the niche concept, focusing on the relationship between topological and ecological ideas. We also describe applications of machine learning to develop empirical models from data in the field. These lead to novel questions in computational topology, and we will discuss recent progress in that direction. This is joint with John Drake in ecology and Edward Azoff in mathematics.

We compute the motivic slices of hermitian K-theory and higher Witt-theory. The corresponding slice spectral sequences relate motivic cohomology to Hermitian K-groups and Witt groups, respectively. Pierre-Emmanuel Caprace. After discussing some generalities on the conjecture and some of its consequences, I will focus on the two special cases alluded to in the title. This was joint work with Christophe Pittet Univ. Often the answer to a topological question is, by its own admission, nonconstructive, but even when the answer is constructive, serious difficulties can arise in carrying out that construction.

We will consider a couple cases like this.

As an approachable, low-dimensional example, we decompose surfaces as a square complex with a fixed number of squares meeting at a vertex. As a high-dimensional example, we consider the possible Pontrjagin numbers of highly connected manifolds. For knots invariant under a finite order cyclic symmetry, Seifert, Murasugi and others developed relations constraining the Alexander polynomial of such knots.

We develop similar constraints using the transfer method of generating functions is applied to the ribbon graph rank polynomial. We develop conditions for the Jones polynomial of links which admit a periodic homeomorphism, by applying the above result and the work of Dasbach et al showing that the Jones polynomial is a specialization of the ribbon graph rank polynomial. The original Hennings TQFT is defined for quasitriangular Hopf algebras satisfying various nondegeneracy requirements. The ultimate goal is to apply this construction to the Dijkgraaf-Pasquier-Roche twisted double of the group algebra, and then show that the resulting TQFT is equivalent to a more geometric one, described by Freed and Quinn.

Aspherical manifolds obtained by gluing locally symmetric manifolds. Aspherical manifolds are manifolds that have contractible universal covers. I will explain how to construct closed aspherical manifolds by gluing the Borel-Serre compactifications of locally symmetric spaces using the reflection group trick. I will also discuss rigidity aspects of these manifolds, such as whether a homotopy equivalence of such a manifold is homotopic to a homeomorphism. Anh T. We consider the AJ conjecture that relates the A-polynomial and the colored Jones polynomial of a knot.

Using skein theory, we show that the conjecture holds true for some classes of two-bridge knots and pretzel knots. The Emmy Noether Mathematical Institute. Kauffman gives a state sum formula for the Alexander polynomial of a knot using states in a lattice that are connected by his clock moves. We show that this lattice is more familiarly the graph of perfect matchings of a bipartite graph obtained from the knot diagram by overlaying the two dual Tait graphs of the knot diagram.

We prove structural properties of the bipartite graph in general and mention applications to Chebyshev or harmonic knots obtaining the popular grid graph and to discrete Morse functions. This talk is accessible to those without a background in knot theory. Basic graph theory is assumed. I will give a historical overview of completions in topology and homotopy theory starting with the work of D.

Sullivan, together with motivation and applications of these constructions, including H. I will then describe a variation of these completion ideas for the enriched algebraic-topological context of homotopy theoretic commutative rings that arises naturally in algebraic K-theory, derived algebraic geometry, and algebraic topology. I will finish by describing some recent results on completion in this new context, which are joint with M.

Conjecturally, the amount of torsion in the first homology group of a hyperbolic 3-manifold must grow rapidly in any exhaustive tower of covers see Bergeron-Venkatesh and F. In contrast, the first betti number can stay constant and zero in such covers. Here "exhaustive" means that the injectivity radius of the covers goes to infinity. In this talk, I will explain how to construct hyperbolic 3-manifolds with trivial first homology where the injectivity radius is big almost everywhere by using ideas from Kleinian groups.

I will then relate this to the recent work of Abert, Bergeron, Biringer, et. This is joint work with Jeff Brock. Morava E-theory is an important cohomology theory in chromatic homotopy theory. Joint with Tobias Barthel. To study the exponential law for function spaces with the compact-open topology, R. Brown introduced a topology for product set, which is finer than the product topology, and showed the exponential law for any Hausdorff spaces. The method was improved by P. Booth and J. Tillotson, making use of test maps, and they removed the Hausdorff condition for spaces.

The product space they used is called the BBT-product. If we use any class of exponentiable spaces, then we can define a topology for function spaces which enables us to prove the exponential law with the BBT-product for any spaces. We can apply the result to based spaces and we get various good results for homotopy theory.

For example, we can prove a theorem of pairings of function spaces without imposing conditions on spaces and base points. If we look at the techniques carefully, we find that the results can also be applied to study group actions on function spaces. The centralizer of the BBT-product has good properties for homotopy theory. With Vyacheslav Krushkal having with D.

We hope that these matroids and the perspective will prove useful in the study of complexes. University of Southampton. Michael A. In joint work with Andrew Blumberg, we construct a category of cyclotomic spectra that is something like a closed model category and which has well-behaved mapping spectra. We show that topological cyclic homology TC is the corepresentable functor on this category given by maps out of the sphere spectrum, verifying a conjecture of Kaledin. We will discuss a few reduced forms for homotopy types of 1-dim Peano continua.

For 1-dim continua this always gives a minimal deformation retract, or core. In a core 1-dim continuum, the points which are not homotopically fixed form a graph. Furthermore, this can be homotoped to an "arc reduced" continuum, where the non-homotopically fixed points are in fact a union of arcs. In a recent paper with Tim Cochran and Arunima Ray, we show that for many patterns this map is injective. I will approach this result from a different perspective, namely by showing that satellite operators really come from a group action. In , Levine studied homology cylinders over a surface modulo the relation of homology cobordism as a group containing the mapping class group.

We show that this group also contains satellite operators and acts on an enlargement of knot concordance. In doing so we recover the injectivity result. I will also present some preliminary results on the surjectivity of satellite operators on knot concordance. This is joint work with Arunima Ray of Rice University. Then there are two natural questions:. Given a nonclassical setting for homotopy theory, such as equivariant spectra or motivic spectra, what analogue of the chromatic convergence theorem might hold?

We give an answers to each of these two questions. Second, we get conditions under which a chromatic completion theorem can hold for motivic and equivariant spectra: one needs a chromatic cover to exist in those categories of spectra. Explicit class field theory and stable homotopy groups of spheres. We present an alternative to Morse-Novikov theory which works for a considerably larger class of spaces and maps rather than smooth manifolds and Morse maps.

One explains what Morse-Novikov theory does for dynamics and topology and indicates how our theory does almost the same for a considerably larger class of situations as well as its additional features. Two knots in the 3-sphere are said to be concordant if they cobound a locally flat, properly embedded annulus in the product of the 3-sphere and the unit interval.

The notion of concordance originates from Fox and Milnor, and it is related with other 3- and 4-dimensional topological properties such as homology cobordism and topological surgery theory. In this talk, I will discuss various relationships between concordance and Seifert forms or the Alexander polynomial of knots. It implies, for example, the Borel conjecture of topological rigidity of closed aspherical manifolds and the Novikov conjecture of homotopy invariance of higher signatures. By the work of A. Bartels, W. Lueck and C. This includes for example fundamental groups of nonpositively curved closed Riemannian manifolds.

In this talk, after outlining the general strategy for proving FJIC, I will talk about the progress that I have made concerning the above question. We also discribe a procedure for obtaining Gromov's result from the discrete version. Wouter van Limbeek. In this talk I will discuss the problem of classifying all closed Riemannian manifolds whose universal cover has nondiscrete isometry group.

I will exhibit some of these, and discuss progress towards a classification. As an application, I will characterize simply-connected manifolds with both a compact and a noncompact finite volume quotient. The Baumslag-Solitar groups are a particular class of two-generator one-relation groups which have played a surprisingly useful role in combinatorial and geometric group theory.

They have provided examples which mark boundaries between different classes of groups and they often provide a test-cases for theories and techniques. In this talk, I will illustrate the proof of the Farrell-Jones conjecture for them. This is a joint work with my advisor Tom Farrell. We will explain why a generic metric on a smooth four manifold with second Betti number at least three has the exponential growth property, i. Time permitting, we shall explain related topological consequences.

I will provide alternative definitions and methods of calculations for the Alexander Polynomial of a knot and ultimately a generalization of this invariant to all odd dimensional manifolds with large fundamental group. The generalization is a "rational function" on the variety of complex rank K representations of the fundamental group. We show how recent results of Dundas-Goodwillie-McCarthy can be used to give efficient proofs of i a Fundamental Theorem for the K-theory of connective S-algebras, ii an integral localization theorem for the relative K-theory of a 1-connected map of connective S-algebras, iii a generalized localization theorem for the p-complete relative K-theory of a 1-connected map of connective S-algebras.

Following Weibel, we define homotopy K-theory for general S-algebras, and prove that the corresponding NK-groups of the sphere spectrum are non-trivial. We will give a brief summary of this program, along with recent results of Blumberg-Mandell and how they fit into some deep conjectures of Rognes. As time permits, we will add some conjectures to the list. Stratos Prassidis. University of the Aegean. We show that quasi-toric manifolds are topologically equivariant rigid with the natural torus action.

### Upcoming events

The proof of the rigidity is done in three steps. First we show that for the manifold equivariantly homotopy equivalent to the quasi-toric manifold the action of the torus is locally standard it resembles the standard action of the torus on the complex space. The second step is that the manifold is equivariantly homeomorphic to the standard model of such actions.

The final step is based on the topological rigidity of the quotient space which is a manifold with corners. This is joint work with Vassilis Metaftsis. In a loose analogy with Teichmuller space, there is a moduli space of AdS 3-manifolds with a given fundamental group. This space is not entirely understood—for instance, we do not know how many connected components it has—-but we do know a fair amount.

We know much less about how the geometry of the manifolds varies across the moduli space.

## MAT -- Algebraic Topology

It follows that two simple convex polytopes are combinatorially equivalent if and only if they are diffeomorphic as manifolds with corners. On the other hand, by a result of Akbulut, for each n greater than 3, there are smooth, contractible n-manifolds with contractible faces which are combinatorially equivalent but not diffeomorphic. Applications are given to rigidity questions for reflection groups and smooth torus actions.

Foozwell and H. Rubinstein in analogy with the classical Haken manifolds of dimension 3, using the the theory of boundary patterns developed by K. Haken manfolds in all dimensions are aspherical and, in general are amenable to proofs by induction on the length of a hierarchy and on dimension.

As such they provide a a context to explore the classical Euler characteristic conjecture for closed aspherical manifolds, which we are doing in some joint work with M. Effie Kalfagianni. Geometric structures and stable coefficients of Jones knot polynomials. Quasi-isometries of graph manifolds do not preserve non-positive curvature. In this talk, we will see the definition of high dimensional graph manifolds and see that there are examples of graph manifolds with quasi-isometric fundamental groups, but where one supports a locally CAT 0 metric while the other cannot.

We will use properties of the Euler class as well as various results on bounded cohomology. We compute the Heegaard-Floer link homology of algebraic links in terms of the multivariate Hilbert function of the corresponding plane curve singularities. A new version of lattice homology is defined: the lattice corresponds to the normalization of the singular germ, and the Hilbert function serves as the weight function.

The main result of the paper identifies four homologies: a the lattice homology associated with the Hilbert function, b the homologies of the projectivized complements of local hyperplane arrangements cut out from the local algebra by valuations given by the normalizations of irreducible components, c a certain variant of the Orlik—Solomon algebra of these local arrangements, and d the Heegaard--Floer link homology of the local embedded link of the germ.

In this talk I will propose a refinement of the Betti numbers provided by a continuous real valued map. These refinements consist of monic polynomials in one variable with complex coefficients, of degree the Betti numbers. A number of remarkable properties of these polynomials will be discussed. If the map is a Morse function the polynomials can be calculated in terms of critical values of the map and the number of trajectories of the gradient of the Morse function between critical points. Farrell Nil-groups are generalizations of Bass Nil-groups to the twisted case.

### 1st Edition

They mainly play role in 1 The twisted version of the Fundamental theorem of algebraic K-Theory 2 Algebraic K-theory of group rings of virtually cyclic groups 3 as the obstruction to reduce the family of virtually cyclic groups used in the Farrell-Jones conjecture to the family of finite groups. These groups are quite mysterious.

Farrell proved in that Bass Nil-groups are either trivial or infinitely generated in lower dimensions. We indeed derived some structural results for general Farrell Nil-groups. As a consequence, a structure theorem for an important class of Farrell Nil-groups is obtained. This is a joint work with Jean Lafont and Stratos Prassidis. K-theory is the machine that turns symmetric monoidal categories into spectra. My ultimate goal will be to discuss our proof that symmetric monoidal 2-categories model connective spectra.

Amherst College. Of particular interest is the identity functor on the category of based topological spaces which, it turns out, has an interesting calculus. In this talk, I want to describe how the nth polynomial approximation to the identity functor can be given the structure of a monad. I will also discuss analogous results for the identity functor on other categories.

In the case of algebras over an operad of spectra , the monad structures on the polynomial approximations to the identity are derived from work of Harper and Hess. Then I'll talk about the fact that there are more periodicity operators in chromatic motivic homotopy theory than in the classical story. In particular, I will describe a new non-nilpotent self map. Rational homotopy theory, fibrations and Maurer-Cartan higher products. It is well known that every even positive degree cohomology class of a finite dimensional cell complex pulls back to zero in the total space of some fibration over the cell complex, where the fibre is finite dimensional.

For odd degree cohomology classes, however, there are obstructions. In this talk, I will talk about how to characterize these obstructions by using the rational homotopy theory. For all finite dimensional connected cell complexes, we give a complete description, in term of Maurer-Cartan higher products, of the subspace of rational cohomology classes that pull back to zero under fibrations with finite dimensional fibre. In terms of physics, this describes the existence of potentials. The key role is played by Poincare's cohomology rule and the generalized Stokes integral theorem.

Disclaimer: the following is probably a special case of the more abstract notions in other answers above like this one. But I think it still worthwhile to expand on it a bit. There is a notion of duality in symmetric monoidal categories. All endomorphisms of dualisable objects have traces, represented as endomorphisms of the unit object. Typical examples:.

For modules over a ring, strongly dualisable means finitely generated projective. The trace of an endomorphism is the usual trace. Note that endomorphisms of infinite-dimensional vector spaces can have a trace, but for example, the identity typically hasn't. Similar dualities exist in other tensor categories, for example, categories of representations. In the stable homotopy category of spaces, all finite CW complexes finite CW spectra have a strong dual, the Spanier-Whitehead dual.

The trace of an endomorphism is its Lefschetz number , represented as an endomorphism of the sphere spectrum. Umbral compositional inversion is a type of duality related to multiplicative and compositional inversion of functions and to matrix inversion, which is very useful in deriving algebraic relations and other identities among polynomial sequences important in number theory, special functions, and enumerative combinatorics as well as operator calculi. This implies that the pair of lower triangular matrices comprised of the coefficients of these polynomial sequences are an inverse pair.

There are many interesting and useful dual operators associated with these UIPs e. Interweaving the two types of inversions, multiplicative and compositional, a simple formula for the Bernoulli polynomials and their associated base numbers, the Bernoulli numbers, in terms of the Stirling numbers is easily derived in my blog post Compositional Inverse Operators and Sheffer Sequences. The duality between projective modules and injective modules, also the duality between divisible abelian groups and free abelian groups.

Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. The concept of Duality Ask Question. Asked 8 years ago. Active 2 years, 7 months ago. Viewed 11k times. I am very interested in getting help with the following goal: Collect an annotated list of various notions of duality that occur in mathematics, with the ultimate aim of describing the notions in a way that makes it easier to recognize and intuitively build connections between the various notions of duality. Some additional context I got thinking about this question after reading the following amazing paper: The concept of duality in convex analysis, and the characterization of the Legendre transform , by Shiri Artstein-Avidan and Vitali Milman , where the authors talk about duality in more abstract terms though, largely in the setting of convex analysis.

Also it is a case where closing a question Survit's earlier memorable big list question was beneficial. I've also wondered about this for some time. Several concepts of duality are discussed, along with their interactions. Thanks for the link! Michael Thaddeus. Is that the first use of the term "dual" in mathematics? Gerald Edgar. While there are possibilities for answering this question mathematically, I think the poster would benefit from talking with or reading history of science type researchers first.

Could you expand it a bitwould be very helpful. It was posed as a problem by J. Sylvester in and first solved by Melchior by proving the dual problem. Projective duality, i. Joel David Hamkins. Dualizing infinite strings of quantifiers which is not generally valid is a way of expressing determinacy of infinite games. David Roberts. So we have at least Duality pairing Dualizing object Maximal fixed subcategories of an adjunction Arrow reversal Then we could look at any relations between these mechanisms, such as between 2 and 3, maps into a dualizing object form the functors for an adjunction.

Atiyah in his talk Duality in Mathematics and Physics says "Fundamentally, duality gives two different points of view of looking at the same object. At a more philosophical level though, one could ask the rhetorical question: why care about duality?

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Web-sources: 1 , 2 ; 3 ; 4 3 Linear programming duality This is an operation to move from a linear programming problem to a dual problem which have the same solution. It is used in e. Recently it also got a lot of attention from people studying the complex Monge-Ampere equation on manifolds and related concepts of energy. Link 4 above is broken. Igor Khavkine. For example, one can cite Koszul duality of quadratic algebras due to Priddy which is related to inversion of formal power series. The dual of an order is the inverse relation of the order less-than vs. To add to the answer of David Roberts: Conjunction and implication both with one argument fixed are adjoints Existential and universal quantification are adjoints to a certain form of substitution Strongest postconditions and weakest liberal preconditions in programming language semantics Sets of models and sets of formulae A lattice and its image under a closure operator In settings with a notion of time, there are temporal dualities from the interaction of the past and the future.

Boolean algebras and Stone spaces [Stone] Distributive lattices and Priestley spaces [Priestley] Heyting algebras and Esakia spaces [Esakia] Topological representations of arbitrary lattices [Urquhart] Extensions of Stone and Priestley duality to lattices with operators Dualities arising in Modal logic [Goldblatt] One 'analogy between analogies' is that of a dualising object.

Noah Stein. Nov 30 '17 at Denis Serre. Distribution theory can be regarded as an extension of linear operations on functions by continuity to more general objects distributions.

Hormander's book expresses this view in many ways. Anyhow, it's a good example. It's a great duality, e. Please expand if you have a few moments. BTW, to those big in categories: do they have a notion of probability? For example imagine a category of extendable computer programs whose meaning is never finally instantiated, and a notion of complexity for them.

Just curious What do proofs have to do with it? XII : " It turns out that cohomology and homology have their roots in the rules for electrical circuits formulated by Kirchhoff in Electrical Circuits as a Paradigm in Homology and Cohomology : pg. Cohomology describes the physics of the circuit i. There exists a crucial duality relation between homology and cohomology which reflects the influence of the geometry of an electrical circuit on its physics based on the duality relation The Electromagnetic Field and the de Rham Cohomology : pg. Tom Copeland. However, could you expand for the non cognoscenti?

For an Appell Sheffer sequence, defined by the e. Sign up or log in Sign up using Google.

## Koszul duality in algebraic topology

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