Academic year 17/18
3 July '18
Martin Frankland: Towards the dual motivic Steenrod algebra in positive characteristic
Several tools from classical topology have useful analogues in motivic homotopy theory. Voevodsky computed the motivic Steenrod algebra and its dual over a base field of characteristic zero. Hoyois, Kelly, and Ostvaer generalized those results to a base field of characteristic $p$, as long as the coefficients are mod $\ell$ with $\ell \neq p$. The case $\ell = p$ remains conjectural.
In joint work with Markus Spitzweck, we show that over a base field of characteristic $p$, the conjectured form of the mod $p$ dual motivic Steenrod algebra is a retract of the actual answer. I will sketch the proof and possible applications. I will also explain how this problem is closely related to the Hopkins-Morel-Hoyois isomorphism, a statement about the algebraic cobordism spectrum MGL.
26 June '18
Marco Zambon: Deformations of geometric structures
We will survey results (some of them classical) on the space of small deformations of a given geometric structure, i.e. on the space of nearby structures of the same kind. Often this space is parametrized by the Maurer-Cartan elements of an associated differential graded Lie algebra or, more generally, of an L-infinity algebra. We will mainly focus on deformations of foliations, but also discuss briefly other examples such as symplectic forms and complex structures.
19 June '18
Christian Lessig: Divergence Free Polar Wavelets
Divergence free vector fields play an important role in many systems in science and engineering, making their analysis and numerical representation important in fields ranging from climate science to medical imaging. We introduce a tight frame of divergence free wavelets that resolves the different scales and orientations that occur in these vector fields. Our wavelets are thereby divergence free in the ideal, analytic sense, have an intuitive correspondence to natural phenomena, closed form expressions in frequency and space, a multi-resolution structure, and fast transforms. Our construction also provides well defined directional selectivity that, among other things, models the behavior of solenoidal vector fields in the vicinity of boundaries. With suitable window functions, this provides (up to a logarithmic factor) an optimal approximation rate for piecewise continuous divergence-free vector fields in two dimensions. We demonstrate the numerical practicality and efficiency of our construction for the representation of solenoidal vector fields and the simulation of the Navier-Stokes equation.
15 May '18
Damian Sawicki: Metric spaces defined by group actions
We will discuss an over 20-year-old construction, which is currently undergoing a renaissance that has yielded spectacular results in metric geometry (a construction of a continuum of different super-expander graphs or counterexamples to the coarse Baum-Connes conjecture) and certain applications to the dimension theory of dynamical systems.
The construction associates to a group action on a compact manifold (or a Cantor
set) an unbounded metric space, "warped cone", whose geometry reflects dynamical and ergodic properties of the action.
8 May '18
Michał Jarząbek: Spectral numerical methods for fluid simulation, part 2
I will present a construction of a frame of divergence-free wavelets that can be used to represent velocity of an incompressible fluid in a numerical simulation.
24 Apr '18
Michał Jarząbek: Spectral numerical methods for fluid simulation
I will introduce the Navier-Stokes equations as a mathematical model of fluid dynamics and review some numerical methods used to compute approximate solutions. The focus will be on spectral methods, and especially on a construction of a frame of divergence-free wavelets that can be used to represent the fluid's velocity in a numerical simulation.
17 Apr '18
Urtzi Buijs: Quillen rational homotopy theory reloaded
With the idea in mind of defining a (co)-representable realization functor for differential graded Lie algebras we have developed a homotopy theory extending classical Quillen rational homotopy. In this talk I will explain some important constructions to define this functor as the cosimplicial Lie model for the n-simplex and present some applications in different fields.
Joint work with J. Carrasquel, Y. Félix, A. Murillo and D. Tanré
10 Apr '18
Janusz Przewocki: Algebraic properties of the canonical homomorphism from singular to Milnor-Thurston homology 3
Milnor-Thurston homology theory is a version of singular homology where chains are defined to be measures instead of finite linear combinations of simplices. There is a canonical homomorphism from singular to Milnor-Turston homology. We investigate behaviour of this homomorphism for spaces with bad local properties.
In the first part of the talk we will remind basic notions. Then the next part will be devoted to investigating algebraic properties of the canonical homomorphism for some class of wild spaces.
27 Mar '18
Janusz Przewocki: Algebraic properties of the canonical homomorphism from singular to Milnor-Thurston homology 2
Milnor-Thurston homology theory is a version of singular homology where chains are defined to be measures instead of finite linear combinations of simplices. There is a canonical homomorphism from singular to Milnor-Turston homology. We investigate behaviour of this homomorphism for spaces with bad local properties.
The first part of the talk will be a continuation of my previous talk on this topic (a short introduction with basic notions and motivations will be provided). The rough ideas to construct a kernel-element for the canonical of the Hawaiian Earring will be presented.
The in the next part of the talk we will focus on proving injectivity of the canonical homomorphism for one particular class of wild topological spaces.
13 Mar '18
David Recio-Mitter: Motion planning on surfaces
One of the main problems in robotics is that of motion planning. It consists of finding an algorithm which takes pairs of positions as an input and outputs a path between them. It is not always possible to find such an algorithm which depends continuously on the inputs. Studying this problem from a topological perspective, in 2003 Michael Farber introduced the topological complexity of a space, which measures the minimal (unavoidable) discontinuity of all motion planners on a given topological space. The topological complexity TC(X) turns out to be a homotopy invariant of the space X.
The nth unordered configuration space of a topological space X is the space of all n-point subsets of X. In this talk we will determine (or narrow down to a few values in some cases) the topological complexity of the unordered configuration spaces of aspherical surfaces (including surfaces with boundary and non-orientable surfaces). In particular we will see an explicit motion planner. Perhaps surprisingly, the disc turns out to be the hardest case.
This is joint work with Andrea Bianchi.
23 Jan '18
Manuel Amann: Orbifolds with all geodesics closed
The concept of a Riemannian orbifold generalises the one of a Riemannian manifold by permitting certain singularities. In particular, one is able to speak about several concepts known from classical Riemannian geometry including geodesics. Whenever all geodesics can be extended for infinite time and are all periodic, the orbifold is called a Besse orbifold — in analogy to Besse manifolds. A classical result in the simply-connected manifold case states that in odd dimensions only spheres may arise as examples of Besse manifolds.
In this talk we shall illustrate that the same holds for Besse orbifolds, namely that they are actually already manifolds whence they are spheres. The talk is based on joint work in progress with Christian Lange and Marco Radeschi.
16 Jan '18
Wacław Marzantowicz: Estimates of covering type and the number of vertices of minimal triangulations
The covering type of a space X is defined as the minimal cardinality of a good cover of a space that is homotopy equivalent to X. We derive estimates for the covering type of X in terms of other invariants of X, namely the ranks of the homology groups, the multiplicative structure of the cohomology ring and the Lusternik-Schnirelmann category of X. By relating the covering type to the number of vertices of minimal triangulations of complexes and combinatorial manifolds, we obtain, within a unified framework, several estimates which are either new or extensions of results that have been previously obtained by ad hoc combinatorial arguments. Moreover, our methods give results that are valid for entire homotopy classes of spaces.
Joint work with Dejan Govc and Petar Pavešić
14 Dec '17
Robert A. Wolak: Geometric theory of navigation - Finsler's geometry
The beginnings of the rigorous mathematical approach to the navigation problems can be attributed to E. Zermelo: E. Zermelo, Uber das Navigationsproblem bei ruhender oder veranderlicher windverteilung. Z. Angew. Math. Mech. 11 (1931), No. 2, 114–124. In the talk, we will give a short review of modern methods of solving these problems. The main emphasis will be put on geometrical methods involving Finsler metrics. The theoretical part will be complemented with practical examples.
12 Dec '17
Janusz Przewocki: Algebraic properties of the canonical homomorphism from singular to
Milnor-Thurston homology
Milnor-Thurston homology theory is a version of singular
homology where chains are defined to be measures instead of finite linear
combinations of simplices. We investigate behaviour of this theory for
spaces with bad local properties.
The question of injectivity of the canonical homomorphism from singular
theory leads to interesting questions about commutator length in free
groups. Some results obtained in this direction and some still open
problems will be presented.
28 Nov '17
Łukasz Michalak: The Reeb graph of a function on a manifold with finitely many critical points
The Reeb graph R(f) of a function f on a manifold M is the quotient space obtained by contracting the connected components of the level sets of function f to the points. For closed manifold and function with finitely many critical points this space is homeomorphic to finite graph, i.e. to one-dimensional finite CW-complex. In this talk we discuss the following problem: which graph can be realized as the Reeb graph of a function on given manifold. The main tool we use is Morse theory and handle decomposition of a manifold.
First we consider the number of cycles in the Reeb graph. We investigate the maximal number of cycles R(M) in Reeb graph of a function on a manifold M with finitely many critical points and we show that it is maximized by simple Morse functions. We also prove that for every number k no greater than R(M) there exists Morse function f on manifold M which Reeb graph has exactly k cycles. It turns out that R(M) is equal to corank of fundamental group of manifold M, i.e. to the maximal rank of free group onto which there exists epimorphism from fundamental group of M.
We also give generalized version of Sharko and Masumoto-Saeki theorem: for any good oriented finite graph G there exist some manifold M of a given dimension and Morse function f on M such that the Reeb graph of function f is isomorphic to G. After that we can answer the initial question in the case of surfaces.
7 Nov '17
Maria Marchwicka: Generic trace of proteins
Amino acids linked by peptide bonds form a polypeptide chain. This chain is folded. Its unites, amino-acids, that are far from each other in the linear sequence can be placed in close neighborhood (be in contact). Information about it is represented as a protein contact map. Based on this map we construct a fatgraph model. Polypeptides can by classified by genus of this fatgraphs. Moreover, genus can be thought as function of polypeptide length. Changes in genus correspond with structure motifs along the chain.
In my talk I will introduce main ideas of a project I was involved in. I will show how to get information about protein structure from Protein Data Base and how to establish a reasonable contact map. Then, I will discuss how to compute genus for a given contact map - I will sketch some algorithms and their implementations (Python). Finally, I will briefly talk about results of my work.
17 Oct '17
Takao Satoh: On the Andreadakis conjecture of the automorphism groups of free groups
In the mapping class group of a surface, there are two descending central filtrations of the Torelli group. One is called the Johnson filtration, which is defined by using the actions of the mapping class group on the nilpotent quotients of the fundamental group of the surface. The other is the lower central series of the Torelli group. Due to Johnson and Morita, it is known that they are different by certain "obstructions" coming from topological reasons.
Here, we consider a similar situation for the automorphism group of a free group. The group of automorphisms which act on the abelianization of the free group is called the IA-automorphism group. This group has two descending central filtrations. One is called the Andreadakis-Johnson filtration, and the other is the lower central series of the IA-automorphism group. Andreadakis showed that they are equal if the rank of the free group is two, and conjectured that they coincide in general. Recently, Bartholdi showed that this conjecture is not true if the rank is three.
In this talk, we will talk about a combinatorial group theoretic approach to this problem, and some recent results.
10 Oct '17
Piotr Mizerka: Smith sets of finite groups
We are interested in investigating finite group actions on spheres with two fixed points. In particular we want to know whether the action around one fixed point has to be the same as around the second. For this purpose we use a notion of the Smith set of the group. It gives us a full answer about all possible actions we are interested in.
A useful tool for establishing Smith sets are representation rings and their variations. They are obtained via the Grothendieck construction. The point is that these are purely algebraic objects and our knowledge about them is better. We use this machinery to find the answer for some groups we are interested in.
We make a survey on the most important results concerning computations of Smith sets. We quote the essential theoretical results. We apply these results to GAP computations and present the outputs for particular groups. In addition, we point out some unproved conjectures worth investigating. One of these conjectures is the Dovermann-Suh conjecture, which asserts that if the Smith set of a finite group is trivial, then it is also trivial for all of its subgroups. This is already known not to be true in general, however we present computations of groups for which the Dovermann-Suh conjecture holds.