The Speakers

Skein Modules and Chebyshev Polynomials

Abstract:

Skein modules were introduced by Józef H. Przytycki as generalisations of all the various polynomial link invariants in the 3-sphere to arbitrary 3-manifolds. Over time, they have evolved into one of the most important objects in knot theory and quantum topology having strong ties with many fields of mathematics such as algebraic geometry, hyperbolic geometry, and the Witten-Reshetikhin-Turaev 3-manifold invariants, to name a few. Chebyshev polynomials are an important class of polynomials related to the sine and cosine functions, which have made surprising appearances, and have been rather indispensable, in the study of Kauffman bracket skein modules and algebras. In this talk we will discuss some of these connections between Chebyshev polynomials and skein modules and algebras.

Tit-for-Tat Dynamics and Market Volatility

Abstract:

We consider tit-for-tat dynamics in production markets, where there is a set of n players connected via a weighted graph. Each player i can produce an eponymous good using its linear production function, given as input various amounts of goods in the system. In the tit-for-tat dynamic, each player i shares its good with its neighbors in fractions proportional to how much they helped player i's production in the last round. This dynamic has been studied before in exchange markets by Wu and Zhang. We analyze the long term behavior of the dynamic and characterize which players grow in the long term as a function of the graph structure. At a high level, we find that a player grows in the long term if and only if it has a good self loop (i.e. is productive alone) or works well with at least one other player. We also consider a generalized damped update, where the players may update their strategies with different speeds, and obtain a lower bound on their rate of growth by finding a function that gives insight into the behavior of the dynamical system.

Coefficients of Catalan States of Lattice Crossing

Abstract:

Plucking polynomial of a plane rooted tree with a delay function α was introduced in 2014 by J.H.~Przytycki. As we show, this polynomial factors when α satisfies additional conditions. This result yields several consequences for coefficients C(A) of Catalan states C of an m x n-lattice crossing L(m,n) established in this paper. In particular, using earlier introduced Θ_A-state expansion, it is shown that C(A) can be found after computing C'(A) for a Catalan state C' obtained from C after cutting its removable arcs. Moreover, when C admits a local family of arcs λ that vertically factorizes C then C(A) factors into a product of coefficients of two Catalan states which depend respectively on λ and its complement in C. As an application, we give closed-form formulas for coefficients of Catalan states of L(m,3)

Skein modules: A braid theoretic approach

Abstract:

Skein modules were introduced independently by Przytycki and Turaev as generalizations of knot polynomials in S^3 to knot polynomials in arbitrary 3-manifolds. They are quotients of free modules over isotopy classes of links in 3-manifolds by properly chosen local (skein) relations. The Kauffman bracket skein module of a 3-manifold M, KBSM(M), is equivalent to the independent Jones polynomials for links in M up to regular isotopy. Skein modules have become very important algebraic tools in the study of 3-manifolds, since they detect geometric and topological properties of them, as for example the presence of non-separating 2-spheres and tori. The algebraic counterpart construction of the classical Kauffman bracket is the Temperley-Lieb algebra together with a unique Markov trace. Through the pioneering work of V.F.R. Jones, this algebra related knot theory to statistical mechanics, topological quantum field theories and the construction of quantum invariants for 3-manifolds (works of Witten, Reshetikhin-Turaev, Lickorish, etc). However, computing the KBSM of a 3-manifold is known to be very hard. In this talk we develop a braid theoretic approach for computing KBSM of c.c.o. 3-manifolds and we focus on the case of the lens spaces L(p, q), via the generalized Temperley-Lieb algebra of type B, TL_{1,n}.

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Knotted and unknotted flexible loops settling under gravity in a viscous fluid

Abstract:

Dynamics of elastic, knotted loops settling under gravity in a viscous fluid are studied numerically and experimentally. The loops are modelled as chains of beads, with harmonic stretching and bending potential energies between the consecutive beads and the consecutive bonds linking the beads. The Reynolds number is assumed to be much smaller than unity and the fluid flow obeys the Stokes equations. Hydrodynamic interactions between all the beads are described by the Rotne-Prager mobility matrices. In most cases, the trefoil and other torus knots attain rather flat, toroidal-like, horizontal structures while settling. They perform a swirling motion and a slower rotation. The basic features of the motion and shapes, determined numerically, are also detected in the experiments with closed flexible ball-chains sedimenting in a very viscous silicon oil. The dynamics of knotted and unknotted flexible loops are shown to differ significantly from each other. The results have been published as: M. L. Ekiel-Jezewska, M. Gruziel, K. Thyagarajan, G. Dietler, A. Stasiak and P. Szymczak, Phys. Rev. Lett. 121, 127801 (2018) and M. Gruziel-Slomka, P. Kondratiuk, P. Szymczak and M. L. Ekiel-Jezewska, Soft Matter 15, 7262 (2019).

Knots and the 24-cell

Abstract:

One of the delights (and frustrations) of mathematics is that two seemingly far apart ideas can come together and contribute to both. In this talk I will discuss how knot theory is enriched by the unlikely input from a regular polytope in 4-dimensions: the 24-cell.

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On the commutation properties of finite convolution and differential operators

Abstract:

Spectral properties of many finite convolution integral operators have been understood by finding differential operators that commute with them. We compile a complete list of such commuting pairs, extending previous work to complex-valued and non self-adjoint operators. Further, we introduce a new kind of commutation relation, called sesqui-commutation, that also has implications for the spectral properties of the integral operator. In this case as well a complete list of sequi-commuting pairs of integral and differential operators is obtained.

Why are 7-manifolds special?

Abstract:

In this talk we will explore Fomenko and Matveev's proof of Steifel's theorem (All orientable 3-manifolds are parallelizable) using knot theory. This leads to the question: For what collection of dimensions does stably parallelizable imply parallelizable? In answering this we explain why 7-manifolds are special in the context of parallelizability.

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Reconnection Number of Vortex Knots

Abstract:

Knotted vortices such as those produced by Kleckner and Irvine at the Frank Institute of the Univ of Chicago tend to transform by reconnection to collections of unknotted and unlinked circles. The reconnection number R(K) of an oriented knot of link K is the least number of reconnections (oriented re-smoothings) needed to unknot/unlink K. Putting this problem into the context of knot cobordism, we show, using Rasmussen’s Invariant that the reconnection number of a positive knot is equal to twice the genus of its Seifert spanning surface. In particular an (a,b) torus knot has R = (a-1)(b-1). For an arbitrary positive knot or link K, R(K) = c(K) - s(K) + 1 where c(K) is the number of crossings of K and s(K) is the number of Seifert circles of K. Examples of vortex dynamics will be illustrated.

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Apollonian integrality revisited

Abstract:

Apollonian disk packing is called integral if the curvatures of the disks are integers, and super-integral, if the reduced coordinates (coordinates of the disk centers, scaled by the curvatures) are integers as well. Equivalently, if the vector representations of the disks in the Minkowski space have integral values. The question is whether all integral packings can be made super-integral. We show that it is so, and as a byproduct, new interesting discrete groups, intriguing occurrence of the Fibonacci numbers, and some other features, emerge.

Quasi-decimal representation of integer (and real) numbers without the digit 0 and (–) sign

Abstract:

First, the positional (quasi-decimal) system of natural numbers without the digit 0 will be discussed. The digit X will be used, not as a substitute for 0, but for the sake of decency of the system and simplicity of correspondence. The system for natural numbers is “more” bijective than the normal system (001 = 01 = 1). On the basis of the written subtraction algorithm, the representation of the number 0 will be constructed, and in the next step, the number (–1). As a consequence, there will be a bijective system of representation of integers. The system will correctly perform multiplication, addition and subtraction. Finally, the extension of the system to real numbers will be given, which will have an ambiguous reprezentasions of even 0 itself. It will not be decided whether the ambiguity is an equivalence class of the equality relation or whether it means system inconsistency. The quasi-decimal system resembles 10-adic numbers, but in the author's intention it is a construction of real numbers, with particular emphasis on negative numbers. The motivation is to develop the formalism for the possible construction of hyperreal numbers. The presentation will not contain formal definitions and statements, but will be based on numerous examples.

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Braidings, braid equivalences and Jones-type invariants

Abstract:

We will present L-move algorithms for braiding knots and links in various diagrammatic settings and the corresponding braid equivalences up to link isotopy. Then we will explain the construction of Jones-type knot and link invariants via Markov traces on appropriate quotient algebras of braid groups.

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Dissecting innate immunity responses to viral infection at the single-cell level

Abstract:

Recognition of viral RNA initiates a signaling cascade culminating in synthesis of interferons (IFNs). Secreted IFNs, by activation of transcription factors of STAT family in surrounding cells, prompt them to prepare for viral infection. Viruses, in turn, convey non-structural proteins to impede the innate immune response. Based on results obtained using single-cell techniques, we proposed an agent (single cell)-based, stochastic, computational model and used it to explain how a infected population of cells can stratify into distinct subpopulations. The winning cells, in response to viral RNA, produce IFNβ (warning yet not infected cells), loosing cells express viral proteins that inhibit innate immune signaling. The proposed model reproduces the experimentally observed complex spatial patterns of respiratory syncytial virus (RSV) spread and dichotomous cell responses.

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Topological Invariant of Manifolds in Sławianowski’s Field Theory

Abstract:

Witten constructed topological invariants of manifolds by introducing actions depending only on smooth structures of the manifold without using a metric. He focused on an example of Chern-Simons’ theory on 3D manifolds. Sławianowski based his field theory on frame/coframe fields without a metric. A Witten-type topological invariant can be introduced in Sławianowski’s framework, using an integral over the reper fields. Such invariant behaves differently for group manifolds than for all others. It additionally distinguishes semi-simple Lie group manifolds from other group manifolds. Further study of this new invariant is needed, including possible generalizations to other field theories. (work with Robert Owczarek)

Visualizing Framing Changes of Links Through Skein Modules

Abstract:

The notion of a skein module was introduced in 1987 by Józef H. Przytycki as a generalization of polynomial link invariants in S^3 to arbitrary 3-manifolds. In this talk we give a historical introduction of skein modules, formally define this structure, and subsequently, we connect skein modules with framing changes of links in 3-manifolds. In particular, we focus on the framing skein module.

What I have learned about knot theory

Abstract:

When considering what to say at this conference, I decided to step back and describe my journey from what I was doing in physics of superfluid helium and knot theory in my PhD towards abstract knot theory which I try to do in recent years. In the humble beginnings I was using Witten's path integrals to argue about phase transition in superfluid helium, by relating it with 2D Ising model. Then, trying to get rid of non-rigorous approach of path integrals, I tried, with Goldin and Sharp, to use Fock space approach, which has some success (at least in 2D) but it is clear that we meet a wall when trying to define a Hamiltonian in terms of creation-annihilation operators, a bit like in loop quantum gravity (where the problem is even much more pronounced). This was the deciding element in switching to pure knot theory and hoping that rigorous theory will help in doing the physics. I will tell the story, including most recent results and questions that I face in this apparently never-ending journey.

Elastic Moduli Determination from Resonant Ultrasound Spectroscopy. Osmium and Uranium Oxide

Abstract:

Resonant Ultrasound Spectroscopy (RUS) is a well-established method for determination of the full tensor of elastic moduli of a solid sample in a single frequency sweep. RUS is qualitatively different from other frequency-domain techniques in that the measurements and analysis are highly redundant and sensitive to all the components of the elastic tensor. The direct problem consist of calculating resonance frequencies of an object from its density, dimensions and elastic moduli, using an energy minimization technique (a complete analytical solution does not exist). The inverse problem is more powerful and interesting, as one can go from easy-to-measure quantities (resonance frequencies) to elastic moduli. Elastic moduli are of quintessential importance for understanding fundamental properties of materials, as they relate directly to the internal energy, and they enter in the equation of state of materials. The basics of the RUS technique will be discussed, along with two case studies: (1) Osmium, a material believed to be harder than diamond, and (2) Uranium Oxide, used as nuclear fuel rods in nuclear reactors

Deterministic and Statistical Methods for Large-scale Dynamic Inverse Problems

Abstract:

Inverse problems are ubiquitous in many fields of science such as engineering, biology, medical imaging, atmospheric science, and geophysics. Three emerging challenges on obtaining relevant solutions to large-scale and data-intensive inverse problems are ill-posedness of the problem, large dimensionality of the parameters, and the complexity of the model constraints. In this talk we discuss efficient methods for computing solutions to dynamic inverse problems, where both the quantities of interest and the forward operator may change at different time instances. We consider large-scale ill-posed problems that are made more challenging by their dynamic nature and, possibly, by the limited amount of available data per measurement step. In the first part of the talk, to remedy these difficulties, we apply efficient regularization methods that enforce simultaneous regularization in space and time (such as edge enhancement at each time instant and proximity at consecutive time instants) and achieve this with low computational cost and enhanced accuracy [1]. In the remainder of the talk, we focus on designing spatial-temporal Bayesian models for estimating the parameters of linear and nonlinear dynamical inverse problems [2]. Numerical examples from a wide range of applications, such as tomographic reconstruction, image deblurring, and chaotic dynamical systems are used to illustrate the effectiveness of the described approaches.

[1] Pasha, Mirjeta and Saibaba, Arvind K and Gazzola, Silvia and Espanol, Malena I and de Sturler, Eric, Efficient edge-preserving methods for dynamic inverse problems, arXiv preprint arXiv:2107.05727, 2021.

[2] Lan, Shiwei and Li, Shuyi and Pasha, Mirjeta, Bayesian Spatiotemporal Modeling for Inverse Problems, arXiv preprint arXiv:2204.10929, 2022.

Looking at Linking Numbers

Abstract:

We shall show that often one can see the linking numbers between branch curves of covering spaces of knots.

Extreme Khovanov homology of 4-braids in polynomial time

Abstract:

Computing Khovanov homology of links is NP-hard. Thus finding homotopy type of its geometric realization is also NP-hard. We conjecture that for braid diagrams of fixed number of strings finding homotopy type of geometric realization (and its homology) has polynomial time complexity with respect to the number of crossings. The conjecture is wild open but its solution would have big impact on understanding of Khovanov homology. As a step toward a solution of the conjecture we prove the following result (it has topological and computational flavor). First we show that the Independence Simplicial Complex (ISC), I(w) associated to 4-braid diagram w (that is geometric realization of extreme Khovanov homology) is either contractible or homotopy equivalent to a sphere, wedge of 2 spheres (possibly of different dimensions), a wedge of 3-spheres at least two of them of the same dimension, or a wedge of four spheres at least three of them of the same dimension. On the algorithmic side we prove that finding the homotopy type of I(w) can be done in polynomial time with respect to the number of crossings in w. This is joint work with Marithania Silvero.

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Labeled graph framework for unique games

Abstract:

A unique game a scenario in which two players are asked to assign values a and b from a set [n] to certain randomly chosen variables x and y. Neither of the players knows which variable was chosen for the other one and they are not allowed to communicate. For each pair (x, y) of variables which may be chosen, we want the values a and b to satisfy certain constraints defined by a permutation of [n]. If the players' answers satisfy the constraints, they both win, otherwise they both lose. This type of game can be represented by a graph with permutations of [n] assigned to its edges. The properties of these labeled graphs can tell us a lot about the associated games. In particular they can be used to calculate the probability of winning certain games.

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On the uniqueness of the decomposition of manifolds, polyhedra and continua into Cartesian products.

Abstract:

There exist topological spaces which have multiple decompositions into Cartesian products. There exist such examples for continua, polyhedra and manifolds. We also have a lot of theorems where with additional conditions the decomposition is unique. This is a survey of the results in this field.

A conjecture on circle graphs and Khovanov spectrum

Abstract:

A chord graph is the intersection graph associated to a chord diagram, which can be thought as a circle together with a finite set of chords with disjoint boundary points. In this talk we present a conjecture stating that the independence complex associated to a circle graph is homotopy equivalent to a wedge of spheres. We show some advances towards the proof of the former conjecture, providing a proof for the particular case of certain families of graphs. We will also explain how this conjecture is related to Khovanov spectrum (Khovanov homotopy type introduced by Lipshitz and Sarkar) of knots and links. This is joint work with Józef H. Przytycki.

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Non-existence of Bose-Einstein condensation in Bose-Hubbard model in dimensions 1 and 2

Abstract:

We apply the Bogoliubov inequality to the Bose-Hubbard model to rule out the possibility of Bose-Einstein condensation. The result holds in one and two dimensions, for any filling at any nonzero temperature and it can be considered as complementary to the analogous, classical result known for interacting bosons in a continuum. Instead of trying to generalize the Bogoliubov inequality to particular unbounded operators, as was done previously to prove the result in a continuum, we use carefully controlled finite dimensional approximations of thermal averages and prove their convergence.