The goal of this project is to use topological complexity, in the sense of Smale, to measure the complexity of enumerative problems in classical algebraic geometry. We present four examples. We obtain lower bounds for the Schwarz genera of covers associated with those enumerative problems. The main technical challenge is to understand cohomology classes of BPU_n, the classifying spaces of projective unitary groups. This talk is on joint works with Zheyan Wan and Xing Gu.

Sequential parametrized topological complexity is a numerical homotopy invariant of a fibration, which arose in the robot motion planning problem with external constraints. I will talk about sequential parametrized topological complexity in view of rational homotopy theory. I will give an explicit algebraic upper bound for sequential parametrized topological complexity when a fibration admits a certain decomposition, which is a generalization of the result of Hamoun, Rami and Vandembroucq on topological complexity. I will also mention a variant of the TC-generating function, which is introduced by Farber and Oprea.

Recently, the distributional versions of LS-category and sequential topological complexity were defined and studied as lower bounds to the classical versions of LS-category and sequential topological complexity. More recently, the corresponding intertwining versions were introduced, which bound their respective distributional counterparts from below. In general, the theories of these three versions differ from each other in various interesting ways. In this talk, we will first obtain some sufficient conditions for the distributional LS-category (dcat) of a closed manifold to be maximum, i.e., equal to its classical LS-category. This will give us many new computations of dcat, especially for some essential manifolds, connected sums, and 3-manifolds. Then we will look at the intertwining invariants and interpret the notion of intertwining complexity in terms of motion planning. Using cohomological lower bounds, we will also determine the intertwining category and complexity of some closed manifolds. Finally, we will end this talk by discussing a few open problems.

We define and develop a homotopy-invariant notion for the sequential topological complexity of a map f:X→Y, TC_{r}(f), that interacts with TC_{r}(X) and TC_{r}(Y) in the same way Jamie Scott's topological complexity map TC(f) interacts with TC(X) and TC(Y). Furthermore, we apply TC_{r}(f) to studying group homomorphisms ϕ:Γ→Λ. We present some computations of group homomorphisms for specific groups, including free groups, surface groups, abelian groups, nilpotent groups, and almost nilpotent groups. Additionally, we provide a characterization of the cohomological dimension of group homomorphisms. Time permitting, we will conclude the talk with a discussion of several open problems.

Covering type ct(X) of a space X is the minimal size of a good cover of a space that is homotopy equivalent to X. The concept was introduced by Karoubi and Weibel [1] and is a homotopy invariant of X. It was related to the Lusternik-Schnirelmann category and the size of minimal triangulations by Govc, Marzantowicz and Pavesic [2] (see also [3, 4]), and to systolic inequalities by Babenko and Moulla [5].

We will present basic properties and results, discuss some new directions and open problems.

References
[1]  M. Karoubi and C. Weibel. On the covering type of a space, Enseign. Math., 62 (2016) 457–474.
[2]  D. Govc, W. Marzantowicz, and P. Pavesic. Estimates of covering type and the number of vertices of minimal triangulations. Discret. Comput. Geom., 63 (2020) 31–480.
[3]  E. Borghini and E. G. Minian, The covering type of closed surfaces and minimal triangulations, J. Combin. Theory Ser. A, 166 (2019) 1–10.
[4]  D. Govc, W. Marzantowicz, and P. Pavesic, How many simplices are needed to triangulate a Grassmannian?, Top. Meth. Nonlin. Analysis, 56 (2020) 501-518.
[5]  I. Babenko and T. Moulla, The Karoubi-Weibel complexity for groups, Enseign. Math. 70 (2024) 197–230.

Crisp and Paris showed that for any Artin group, the squares of the standard generators generate the  obvious right-angled Artin subgroup. In recent work with Jankiewicz, we proposed that powers of a larger collection of naturally distinguished elements also generate a right-angled Artin group, and proved it in some cases. In more recent work, we studied variants of this conjecture for graph braid groups. I will talk about this, as well as applications to topological complexity. 

Configuration spaces of points were some of the first objects to be studied via the lens of topological complexity. Disk configuration spaces are natural generalizations of these spaces that force one to consider the geometry, not just the topology, of the underlying space. One of the simplest of these disk configuration spaces is conf(n,w), the ordered configuration space of n open unit-diameter disks in the infinite strip of width w. If n is at most w, this space is homotopy equivalent to the ordered configuration space of n points in the plane, though if n is greater than w these spaces differ drastically. In this talk we discuss the topology of conf(n,w) and calculate its topological complexity.

Abstract (joint work with Ben Knudsen):  Reinterpreting the seminal paper of Smale that launched our field, we shall show that among robots that have access to randomness, there is a price to be paid for being "digital" as opposed to being "analog".  Besides analog complexity, one also has "analog category" - which has an Eilenberg-Ganea theorem, but has surprising properties for finite groups!

(Similar notions were introduced also by Dranishnikov and Jauhari with different interpretations and motivations.)

The considerations are similar to ones involved in computing "universal" parametrised complexity for certain manifolds.  Depending on what happens during the talk, we might discuss some of this direction (which is joint also with Nick Wawrykow).

In this talk I will estimate the topological complexity of 4-dimensional spherically monotone manifolds whose Kodaira dimension is not negative infinity.

Since the inception of the original notion of Topological Complexity, many different variants of the concept have been developed through the years, to capture distinct kinds of information that may be of interest for the motion planning problem. One of those kinds of ''specific information" concerns the impact of the symmetries that often appear in the configuration spaces. Formally, those symmetries are seen as actions of groups on the base topological space X and, as such, this naturally leads to the consideration of equivariant versions of topological complexity. There are several non-equivalent approaches to the matter. In this talk we will make a very brief review of them, but we will focus on the known as Effective Topological Complexity. This variant was devised by Zbigniew Błaszczyk and Marek Kaluba as a way to reduce the complexity of the motion planning problem through the symmetries of the configuration space by means of acknowledging the physically different but functionally equivalent states of the mechanical system that may appear, and which are linked by those symmetries. We will introduce a notion of Effective Lusternik-Schnirelmann category, and we will investigate some of the properties of both effective TC and cat, in particular their relationship with the orbit projection map of the group action and some non-vanishing conditions for effective TC at stage 2. This is a joint work with Zbigniew Błaszczyk and Antonio Viruel.

In this talk I shall give some bounds for the higher topological complexities of usual unordered configuration spaces of trees as well as for the anchored configuration spaces of the circle with two anchored points.

Geodesic complexity is a variant of the topological robot motion planning problem on metric spaces. Instead of arbitrary paths one uses only length-minimizing geodesics for the motion planners. It turns out that the geodesic complexity of a closed Riemannian manifold strongly depends on the structure of the cut locus of the manifold. By introducing the notion of a fibered decomposition of the total cut locus of a Riemannian manifold it is possible to establish lower and upper bounds on geodesic complexity. This leads to interesting applications, e.g. for projective spaces and for lens spaces. This talk is based on joint work with Stephan Mescher.

In this talk, we introduce sequential parametrized motion planning. A sequential parametrized motion planning algorithm produces a motion of the system which is required to visit a prescribed sequence of states, in a certain order, at specified moments of time. The sequential parametrized algorithms are universal as the external conditions are not fixed in advance but constitute part of the algorithm’s input. We study the sequential parametrized topological complexity of sphere bundles. We use the Euler characteristic class to obtain a lower bound of the topological complexity. (Joint work with Prof. Michael Farber)

The higher topological complexity of a space X, TC_r(X), r=2,3,..., and the topological complexity of a map f, TC(f), have been introduced by Rudyak and Pavevsic, respectively, as natural extensions of Farber's topological complexity of a space. In this talk, we present a notion of higher topological complexity of a map f, TC_{r,s}(f), for 0<s<=r and 1<r, which simultaneously extends Farber-Rudyak's usual TC, and Pavesic-Scott-Wu-Murillo TC of maps. Our unified concept is relevant in the r-multitasking motion planning problem associated with a robot device when the forward kinematics map plays a role in s prescribed stages of the motion task.

The notion of topological complexity was introduced by Farber to measure the complexity of robot motion planning in a configuration space of a mechanical system. Rudyak subsequently introduced the higher analogue of this notion known as higher/sequential topological complexity. In this talk, we define the notion of higher subspace topological complexity and obtain an upper bound on the higher topological complexity of total spaces of fibrations. We use this upper bound to show that the dimensional upper bound on the higher topological complexity can be improved in the presence of group actions. Then using these results, we compute the higher topological complexity of several higher dimensional lens spaces.

We study the existence of atoroidal classes and obtain lower bounds for the topological complexity of certain connected sums of closed manifolds. As an application, we show that the topological complexity of connected sums of aspherical 4-manifolds with positive second Betti numbers attains its maximum value nine.

We establish some upper and lower bounds of the rational topological complexity for certain classes of elliptic spaces. Our techniques permit us in particular to show that the rational topological complexity coincides with the dimension of the rational homotopy for some special families of coformal elliptic spaces.

Topological Complexity (TC) addresses a foundational problem in Robotics from the 2nd half of the 20th century: Quantify the complexity of planning continuous paths through a topological space, regarded as the configuration/state space of a programmable synthetic system.

Technological breakthroughs of the 21st century brought a vast array of much more extensive notions of task, as well as the aching need for characterizing whole classes of environments (and hence, also configuration spaces) for which a synthetic system could have guaranteed performance.

However, in the Engineering world so far, these needs have been countered with little more than brute-force solutions on the one hand, or ML-based solutions offering few performance guarantees (if any), on the other.

In this talk, through surveying a few problems I have been working on, I will try to build the case that the field of “Robotic Autonomy”, for lack of a better term, is ripe for a major intervention by the TC community, whose experience can be put to use in constructing rigorous formulations of yet non-extant objects such as task-spaces, and in studying their complexity---especially in contexts with natural interactions between environment topology and combinatorics, such as cooperative motion in obstructed environments.

In this talk, we compute the LS-category for certain classes of Seifert fibered manifolds. We show that in most cases the topological complexity of a Seifert fibered manifold lies between 6 and 7. We then give lower bounds for higher TC in terms of weights of cohomology classes and show that TC_n for most classes of Seifert fibered manifolds lies between 3n and 3n+1.

Abstract: here in pdf form

We consider the sectional category of a map between finite T_0 spaces (posets) from a combinatorial viewpoint. We compute some examples of the sectional category (or number) for the McCord map, the weak homotopy equivalence on the barycentric subdivision, and the Fadell-Neuwirth fibration associated with a finite space.

Let X be compact CW complex with residually finite fundamental group G.  Given a residual chain G_n of finite index subgroups of G, what can be said about the growth of Betti numbers normalised by index in the homology of finite covers X_n corresponding to G_n?  In this talk we will explore this question and relate it to other notions such as amenable category and minimal volume entropy.  Based on joint work with Sam Fisher and Ian Leary.

A calculation for the symmetric and symmetrized topological complexity of the torus is described, thus completing the determination of symmetric and symmetrized topological complexities of closed surfaces, orientable or not. The method is based on obstruction theory and depends on the construction of an explicit resolution of the integers over the full braid group on two strings for the torus. 

I will discuss a proof of Farber's conjecture on the topological complexity of configuration spaces of graphs. The argument eschews cohomology, relying instead on group theoretic estimates for higher topological complexity due to Farber–Oprea following Grant–Lupton–Oprea.

By analogy with the classical Ganea conjecture, which has been disproved by N. Iwase, the TC-Ganea conjecture asks whether the equality TC(X x S^n)=TC(X)+TC(S^n) holds for all finite CW complexes X and all positive integers n. In a previous work in collaboration with J. González and M. Grant, we have constructed a space satisfying TC(X x S^n)=TC(X)+1 for all n>1, which proves that the TC-conjecture fails for even n. In this talk, we will use the notion of weak topological complexity to establish some sufficient conditions for a space X to satisfy TC(X x S^n)=TC(X)+TC(S^n). This is a joint work with J. Calcines.

In a recent breakthrough, Dranishnikov has computed the topological complexity of hyperbolic groups. His approach is based on a characterization of TC in terms of classifying spaces for families of subgroups due to Farber, Grant, Lupton, and Oprea. I will present a simplified proof of Dranishnikov's result by using the Lück--Weiermann construction for classifying spaces and equivariant Bredon cohomology.

I will start by describing the concept of a parametrized motion planning algorithm which allows to achieve high degree of flexibility and universality. The main part of the talk will focus on the problem of understanding the parametrized topological complexity of sphere bundles. I will explain how characteristic classes of vector bundles enter the picture and help to find a solution. I will analyse several specific examples.

Geodesic complexity is motivated by Farber’s notion of topological complexity of a space, which gives a topological description of the motion planning problem in robotics. Motivated by this, D. Recio-Mitter recently introduced geodesic complexity as an isometry invariant of geodesic spaces which formalizes the notion of efficient robot motion planning mathematically, i.e. of motion planning along shorts paths. In my talk, I will present recent work with M. Stegemeyer, in which we study the geodesic complexity of complete Riemannian manifolds. Using structure results for cut loci from Riemannian geometry, we derive lower and upper bounds for geodesic complexity from the geometry of cut loci and illustrate our results by some examples.

We consider variations of the Lusternik--Schnirelmann category, based on open covers satisfying constraints on the level of the fundamental group. Such LS-category invariants can be analysed through equivariant methods. For example, classifying spaces for families of subgroups can be used to obtain lower bounds. I will explain these methods and apply them in the case of the family of amenable subgroups.

Abstract: here in pdf form

Abstract: here in pdf form

In this talk I will revisit the concept of effectual and effective TC in the context of sequential motion planning. These invariants provide a natural context to incorporate group actions into the study of the motion planning problem. Related to these invariants, I will talk about a third version of TC that incorporates the group action into its planners, which we call orbital topological complexity. I will discuss how they relate to each other and to the TC of the quotient space. I will also present some calculations for actions of the group of order two on orientable surfaces and spheres.

Abstract: The parametrized approach to motion planning offers flexibility for variable situations. This is typically encoded in a fiber bundle with the base space parametrizing external constraints on the system. Here, the input (initial and terminal states) and the output (a path between them) of a motion planning algorithm must all be subject to the same external conditions (that is, they are in the same fiber of the bundle). We consider this parametrized motion planning problem where each of the spaces involved is the complement of a union of hyperplanes in a complex vector space. Under a combinatorial hypothesis of supersolvability, we determine the parametrized topological complexity of a fiber bundle of arrangement complements. This is joint work with Dan Cohen. 

Abstract: Lusternik-Schnirelmann category and topological complexity are particular cases of a more general notion, that we call homotopic distance between two maps. As a consequence, several properties of those invariants can be proved in a unified way and new results arise.

For instance, we prove that the homotopic distance between two  maps defined on a manifold is bounded by the sum of their relative distances on the critical submanifolds of any Morse-Bott function. This generalizes the well known Lusternik-Schnirelmann theorem (for Morse functions), and a similar result by Farber for the topological complexity. As an application, we show how navigation functions can be used to solve a generalized motion planning problem.

Abstract: here in pdf form

In this talk we will introduce the concept of Effectual Topological Complexity, which is a new version of the Topological Complexity (TC) for G-Spaces.

We will state some of its main properties, for instance, we will explain the relation between this notion with the standard version of TC and also with an effective one.

We will present some results about the Effectual TC for surfaces, in particular, we will compute the exact value of the effectual TC in the case of the torus.

Tverberg’s theorem states that any (d+1)(r-1)+1 points in R^d can be partitioned into r subsets whose convex hulls have a point in common. There is a topological version of it, which is often compared with an LS-version of the Borsuk-Ulam theorem. I will talk about a generalization of the topological Tverberg’s theorem. A key ingredient is a homotopy decomposition of a discretized configuration space.

Abstract:  I will discuss various versions of the Lusternik-Schnirelman category involving covers and fillings of 4-manifolds by various sets. In particular, I will discuss Gay-Kirby trisections, which are certain decompositions of 4-manifolds into 1-handlebodies. 

Abstract: I will discuss a recent proof of a conjecture of Farber, asserting that the ordered configuration spaces of graphs have the highest possible topological complexity generically.

Abstract: Configuration spaces of points in the plane are well studied and the topology of such spaces is well understood. But what if you replace points by particles with some positive thickness, and put them in a container with boundaries? It seems like not much is known. To mathematicians, this is a natural generalization of the configuration space of points, perhaps interesting for its own sake. But is also important from the point of view of physics––physicists might call such a space the "phase space" or "energy landscape" for a hard-spheres system. Since hard-spheres systems are observed experimentally to undergo phase transitions (analogous to water changing into ice), it would be quite interesting to understand topological underpinnings of such transitions.

We have just started to understand the homology of these configuration spaces, and based on our results so far we suggest working definitions of "homological solid, liquid, and gas". This is joint work with a number of collaborators, including Hannah Alpert, Ulrich Bauer, Kelly Spendlove, and Robert MacPherson.

Abstract: What would a robot have to do to get past a bunch of blockers to get to the end zone (or score a goal; the football is general)?  I will discuss some issues involving modelling, non-fibrations, speed, information, sensing, and things larger than points.  We will not solve these problems.  The math is related to a paper with D.Cohen and M.Farber.

Abstract: Farber posed the problem of describing the topological complexity of aspherical spaces in terms of algebraic invariants of their fundamental groups. In Part One of this talk, I’ll discuss joint work with Farber, Lupton and Oprea in which we use Bredon cohomology (a sharp tool from equivariant topology which provides the natural setting for equivariant obstruction theory) to give purely algebraic estimates of the topological complexity of aspherical spaces. In Part Two, I’ll report on joint work with Meir and Patchkoria in which we used these ideas to give a completely algebraic description of the equivariant LS-category of a group with operators. If time permits, I’ll try to describe how these two projects are related via work of Iwase and his collaborators.

Abstract: We will give sufficient conditions for the (normalized) topological complexity of a closed manifold M with abelian fundamental group to be nonmaximal, that is to satisfy TC(M)<2dim(M), and see through examples that our conditions are sharp. This generalizes for manifolds some results of Costa and Farber on the topological complexity of spaces with small fundamental group. Relaxing the condition of commutativity of the fundamental group, we also generalize Dranishnikov's results on the LS-category of the cofibre of the diagonal map $\Delta: M \to M \times M$ for nonorientable surfaces by establishing the nonmaximality of this invariant for a large class of manifolds. Joint work with Dan Cohen.

Abstract: A discriminantal variety V is the complement in C^m of the zero locus of a polynomial. Many interesting spaces arise in this way: for example both the ordered configuration space F_n(R^2) and the unordered configuration space C_n(R^2) of n points in the plane can be realised as a discriminantal variety.

Motivated by these two examples, we want to estimate TC(V). It is convenient to this purpose to look for a nice action of a torus T^s on V by "scalar multiplication", where we wish s to be as large as possible, and the stabilisers of the action to be as small as possible, say all of dimension <=t. In these hypotheses, our main results says that TC(V)<=2m-s+t. The proof is a combination of basic results of Morse theory, equivariant homotopy theory and complex geometry.

As an application we establish the upper bound in the equality TC(C_n(R^2))=2n-3, and we reprove the upper bound of the equality TC(F_n(R^2))=2n-3 originally due to Farber and Yuzvinsky.

In the end I will try to explain why the strategy of the proof does not lead immediately to an explicit motion planner on C_n(R^2) realising TC, and I will explain what obstructs the use of a similar strategy for unordered configuration spaces C_n(R^d) of Euclidean spaces of higher dimension.