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Title: Integral Transformation of KZ-Type Equations and the Construction of Braid Group Representations Speaker: Haru Negami, Chiba University, Japan Time: 2:00pm–3:00pm Place: CMC 108 Sponsor: W. Ma
This work establishes a correspondence between the multiplicative middle convolution of Knizhnik-Zamolodchikov (KZ)-type equations [2], an integral transformation that reproduces the KZ-type equation, and the Katz-Long-Moody construction [3], an algebraic method for constructing infinitely many representations of the braid group \(B_n\).
Braid group representations play a pivotal role in mathematics, with applications in topology, representation theory, and mathematical physics [1]. The KZ equation [4], central to conformal field theory, is connected to Appell-Lauricella hypergeometric series, Selberg integrals, and other areas. The KZ-type equation is a certain generalization of the KZ equation. The fundamental group of the domain of the \(n\)-variable KZ-type equation corresponds to the pure braid group \(P_n\). Thus, the monodromy representation of the KZ-type equation is the anti-representation of \(P_n\). Haraoka’s convolution method can be interpreted as constructing anti-representations of \(P_n\) from existing ones.
Katz’s middle convolution technique, extended by Dettweiler and Reiter, enhances the analytic framework by constructing differential equations with specified monodromy representations. On the algebraic side, the Katz-Long-Moody construction systematically generates representations of \(B_n\) from those of \(F_n B_n\), extending Long’s foundational methods [5]. This unified approach combines algebraic and geometric perspectives to study the braid group.
Finally, the potential applications of this method to other fields, such as knot theory and quantum computing, are also discussed.
References
Title: On Gromov's Positive Scalar Curvature conjecture for RAAGs Speaker: Alexander Dranishnikov, University of Florida Time: 2:00pm–3:00pm Place: CMC 108
Gromov's PSC conjecture for a discrete group \(G\) states that the universal cover of a closed PSC \(n\)-manifold with the fundamental group \(G\) has the macroscopic dimension less than or equal to \(n-2\). We prove Gromov's conjecture for right-angled Artin groups (RAAGs).
This is joint work with Satya Howladar.
Title: Simple knots in lens spaces that surger to \(S^1\times S^2\) Speaker: Shiyu Liang, University of Texas at Austin Time: 2:00pm–3:00pm Place: CMC 108
A knot in a lens space is said to be spherical if it admits Dehn surgery yielding \(S^1\times S^2\). We classify spherical simple knots and thereby confirm the completeness of the list by Baker, Buck, and Lecuona using rational Seifert surfaces and Morse functions. Additionally, we show that the homology classes of spherical knots are determined by simple knots, analogous to Greene's work in the context of the Berge Conjecture (i.e., surgeries yielding \(S^3\)).
Title: Some fivebrane bordism groups at the prime 2 Speaker: Hassan Abdallah, Wayne State University Time: 2:00pm–3:00pm Place: CMC 108
Given a smooth \(n\)-dimensional manifold \(M\), a tangential G-structure on \(M\) is a lift of the classifying map of the tangent bundle of \(M\) to \(BG\), the classifying space of \(G\). If \(G=\mathrm{SO}(n)\) or \(G=\operatorname{Spin}(n)\), this gives the familiar notion of an orientation or spin structure. A higher analog of orientations and spin structures is that of a fivebrane structure, named based on its relationship to ideas from physics. While the bordism groups of orientable and spin manifolds are well-understood, very little is known about the bordism groups of fivebrane manifolds. The fivebrane bordism groups are isomorphic, via the Pontryagin-Thom construction, to the homotopy groups of the Thom spectrum \(MO<9>\), which has connections to deep ideas in stable homotopy theory. In this talk, I will present new calculations of some fivebrane bordism groups at the prime 2 using the Adams spectral sequence.
Title: Legendrian Representatives of Knots Realizing the Thurston-Bennequin Bound Speaker: Zhenkun Li Time: 2:00pm–3:00pm Place: CMC 108
In this talk, we investigate the Legendrian representatives of knots in contact 3-manifolds, particularly focusing on knots that achieve the three-dimensional Thurston-Bennequin bound. We show that any non-trivial knot in the standard tight contact structure on \(S^3\) that realizes this bound necessarily admits a Legendrian representative with non-negative Thurston-Bennequin invariant. Our approach combines techniques from suture Floer homology, convex surface decomposition, and open book decompositions, bridging knot Floer homology and contact geometry. This is a join work with Shunyu Wan.
Title: Mathematical Representations of DNA: Revealing Evolutionary and Environmental Signals in the Genome Speaker: Lila Kari, University of Waterloo Time: 2:00pm–3:00pm Place: CMC 109 Sponsor: N. Jonoska
Mathematical representations of DNA, such as Chaos Game Representation (CGR) of DNA sequences, provide a powerful lens for uncovering hidden evolutionary and environmental signals in genomes.
In this study, we used this approach alongside machine learning techniques to analyze microbial genomes, and discovered that extreme environmental pressures can sometimes override traditional taxonomic influences in the genome. By examining genomic signatures across a diverse dataset of extremophiles, we identified unexpected similarities between highly divergent microbial species, suggesting that shared environmental conditions drive convergent genomic adaptations. These patterns persist across the genome, reinforcing the idea that environmental pressures shape DNA composition beyond protein-coding regions. Our findings provide compelling evidence that environment-driven genomic signatures can transcend taxonomic boundaries, offering new perspectives on microbial evolution.
By integrating mathematical models with computational methods and biological insights, this research challenges conventional views on microbial evolution and deepens our understanding of how extreme environments shape life at the molecular level.
Title: The maximum connectivity for domains admitting global projective connections Speaker: Razvan Teodorescu Time: 2:00pm–3:00pm Place: CMC 108
We consider a domain in the plane, with finite connectivity n and well-defined geodesic curvature on each boundary component. Assume a projective connection can be defined globally, transforming covariantly under conformal mapping, such that on each boundary component it reduces to a Riemannian metric. Then the maximal connectivity \(n=2\) and it corresponds to a family of domains solving an extremal conformal modulus problem.
Title: The (fractional) Dehn twist coefficient and infinite-type surfaces Speaker: Hannah Turner, Stockton University Time: 2:00pm–3:00pm Place: CMC 108
The fractional Dehn twist coefficient (FDTC) is an invariant of a self-map of a surface which is some measure of how the map twists near a boundary component of the surface. It has mostly been studied for compact surfaces; in this setting the invariant is always a fraction. I will discuss work to give a new definition of the invariant which has a natural extension to infinite-type surfaces and show that it has surprising properties in this setting. In particular, the invariant no longer needs to be a fraction — any real number amount of twisting can be achieved! I will also discuss a new set of examples of (tame) big mapping classes called wagon wheel maps which exhibit irrational twisting behavior. This is joint work in progress with Diana Hubbard and Peter Feller.
Title: Spectral invariants and positive scalar curvature on ribbon homology 4-dimensional cobordism Speaker: Minh Nguyen, WUSTL Time: 1:00pm–2:00pm Place: CMC 120
Information about sectional curvature and Ricci curvature tends to make the underlying manifold “rigid” topologically. This is not the case for scalar curvature, e.g., obstruction to existence of positive scalar curvature (psc) is often via some topological invariants. In this talk, we use the Chern-Simons-Dirac functional to define an \(R\)-filtration on monopole Floer homology \(HM(Y)\) of a rational homology 3-sphere \(Y\). We define a numerical quantity \(\rho\) (spectral invariant) that measures the non-triviality of \(HM(Y)\). It turns out that \(\rho\) is an invariant of \(Y\) with a geometric structure. Using \(\rho\), we give an obstruction to psc on ribbon homology cobordsim between 2 rational homology spheres.
Title: A point-line incidence structure Speaker: Brian Curtin Time: 2:00pm–3:00pm Place: CMC 108
We introduce an incidence structure consisting of a set of points and two multisets of families of parallel classes of lines which satisfy \((R-1)\) two lines of different types meet in exactly one point and \((R-2)\) each family of parallel lines partitions the points. We describe basic properties of these objects and present some related combinatorial objects. We will discuss how these results are analogous to known results for other point-line incidence structures, such as those related to Latin squares.
Title: In Search for Hochschild cohomology of quandle algebras Speaker: Mohamed Elhamdadi Time: 2:00pm–3:00pm Place: CMC 108
Hochschild cohomology of associative algebras appears in many areas of mathematics such as algebraic geometry, topology, functional analysis and K-theory. Hochschild 1-cocycles are derivations and form Lie algebras.
We will review Hochschild cohomology and then introduce derivations over quandle algebras. This can be seen as an initiation to a general Hochschild cohomology theory of quandle algebras which are non-associative.