Seminars and Colloquia by Series

What the Hecke is the BMW Algebra?

Series
Geometry Topology Student Seminar
Time
Wednesday, February 12, 2025 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jake GuyneeGeorgia Tech

The Jones polynomial was first defined by Vaughan Jones as a "trace function" on an algebra discovered via operator algebras. It was discovered that the polynomial satisfies certain skein relations. The HOMFLY polynomial was discovered through both skein relations and a "lift" of the trace function on the Jones algebra to the "Hecke algebra". Another 2-variable polynomial called the Kauffman polynomial was discovered purely via skein relations. In this talk, we discuss how the process started by Jones was reversed for this polynomial. More precisely, we will show how Birman, Wenzl, and Murakami constructed the BMW algebra and a trace function that yields the Kauffman polynomial. We will discuss the significance of the Kauffman polynomial as well as some relationships between the BMW, Hecke, and Jones algebras.

Some specialized Kirby calculus constructions

Series
Geometry Topology Student Seminar
Time
Wednesday, January 29, 2025 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Sean EliGeorgia Tech

I'll talk about some specialized kirby calculus constructions: immersed surface complements and round handles. I'll prove using kirby calculus that S2xS2 minus an appropriate smooth embedded S2vS2 is diffeomorphic to R4. Maybe that is obvious, but the point is we can find nice diagrams where you see everything explicitly.

Indigenous bundles and uniformization

Series
Geometry Topology Student Seminar
Time
Wednesday, November 13, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Akash NarayananGeorgia Tech

The uniformization theorem states that every Riemann surface is a quotient of some subset of the complex projective line by a group of Mobius transformations. However, a number of closely related questions regarding the structure of uniformization maps remain open. For example, it is unclear how one might associate a uniformizing map to a given Riemann surface. In this talk we will discuss an approach to this question due to Gunning by attaching a projective line bundle to a Riemann surface and studying its analytic properties.

Mathematics you thought you knew

Series
Geometry Topology Student Seminar
Time
Wednesday, October 23, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker

Please Note: Further notes on the talk: “Mathematically, what is 5 feet divided by 2 secs?” In other words, how do we make this question mathematically rigorous? The answer was initiated by Newton, carefully explained by Hölder in 1901 using axioms of a quantity space, and finally generalized by Hassler Whitney in the 1960s. Whitney’s explanation is a bit idiosyncratic and hard to understand in terms of modern vector bundle theory. Jim Madden and I reworked it so that it makes sense in terms of tensor products of 1-dimensional vector spaces with a chosen basis element.

Mathematically, what is a 5 feet divided by 2 seconds? Is it 2.5 ft/sec? What is a foot per second? We go through several examples of basic mathematical terms you learned in elementary, middle, and high school and understand them at a deeper, graduate student level. You may be surprised to learn that things you thought you knew were actually put on very weak mathematical foundations. The goal is to learn what those foundations are so that you can bring these basic ideas into your classroom in a non-pedantic-but-mathematically sound way.

Structure of Boundaries of 3-Dimensional Convex Divisible Domains

Series
Geometry Topology Student Seminar
Time
Wednesday, April 3, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Alex NolteGeorgia Tech

I read Benoist's paper Convexes Divisibles IV (2006, Invent. Math.), and will talk about it. The main result is a striking structural theorem for triangles in the boundaries of 3-dimensional properly convex divisible domains O that are not strictly convex (which exist). These bound "flats" in O. Benoist shows that every Z^2 subgroup of the group G preserving O preserves a unique such triangle. Conversely, all such triangles are disjoint and any such triangle descends to either a torus or Klein bottle in the quotient M = O/G (and so must have many symmetries!). Furthermore, this "geometrizes" the JSJ decomposition of M, in the sense that cutting along these tori and Klein bottles gives an atoroidal decomposition of M.

Clifford Algebra: A Marvelous Machine Offered By the Devil

Series
Geometry Topology Student Seminar
Time
Wednesday, March 27, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jaden WangGeorgia Tech

Clifford algebra was first developed to describe Maxwell's equations, but the subject has found applications in quantum mechanics, computer graphics, robotics, and even machine learning, way beyond its original purpose. In topology and geometry, Clifford algebra appears in the proofs of the celebrated Atiyah-Singer Index Theorem and Bott Periodicity; it is fundamental to the understanding of spin structures on Riemannian manifolds. Despite its algebraic nature, it somehow gives us the power to understand and manipulate geometry. What a marvelous machine offered by the devil! In this talk, we will investigate the unreasonable effectiveness of Clifford algebra by exploring its algebraic structure and constructing the Pin and Spin groups. If time permits, we will prove that Spin(p,q) is a double cover of SO(p,q), complementing the belt trick talk of Sean Eli.

Virtual Knot Theory and the Jones Polynomial

Series
Geometry Topology Student Seminar
Time
Wednesday, March 13, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jacob GuyneeGeorgia Tech

Virtual knot theory is a variant of classical knot theory in which one allows a new type of crossing called a "virtual" crossing. It was originally developed by Louis Kauffman in order to study the Jones polynomial but has since developed into its own field and has genuine significance in low dimensional topology. One notable interpretation is that virtual knots are equivalent to knots in thickened surfaces. In this talk we'll introduce virtual knots and show why they are a natural extension of classical knots. We will then discuss what virtual knot theory can tell us about the both the classical Jones polynomial and its potential extensions to knots in arbitrary 3-manifolds. An important tool we will use throughout the talk is the knot quandle, a classical knot invariant which is complete up to taking mirror images.

Smooth Fine Curve Graphs

Series
Geometry Topology Student Seminar
Time
Wednesday, February 28, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Speaker
Katherine BoothGeorgia Tech

The curve graph provides a combinatorial perspective to study surfaces. Classic work of Ivanov showed that the automorphisms of this graph are naturally isomorphic to the mapping class group. By dropping isotopies, more recent work of Long-Margalit-Pham-Verberne-Yao shows that there is also a natural isomorphism between the automorphisms of the fine curve graph and the homeomorphism group of the surface. Restricting this graph to smooth curves might appear to be the appropriate object for the diffeomorphism group, but it is not. In this talk, we will discuss why this doesn’t work and some progress towards describing the group of homeomorphisms that is naturally isomorphic to automorphisms of smooth fine curve graphs.

Two-fold branched covers of hyperelliptic Lefschetz fibrations

Series
Geometry Topology Student Seminar
Time
Wednesday, February 21, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Sierra KnavelGeorgia Tech

When studying symplectic 4-manifolds, it is useful to consider Lefschetz fibrations over the 2-sphere due to their one-to-one correspondence uncovered by Freedman and Gompf. Lefschetz fibrations of genera 0 and 1 are well understood, but for genera greater than or equal to 2, much less is known. However, some Lefschetz fibrations with monodromies that respect the hyperelliptic involution of a genus-g surface have stronger properties which make their invariants easier to compute. In this talk, we will explore Terry Fuller's results from the late 90's which explore two-fold branched covers of hyperelliptic genus-g Lefschetz fibrations. We will look at his proof of why a Lefschetz fibration with only nonseparating vanishing cycles is a two-fold cover of $S^2 \times S^2$ branched over an embedded surface. The talk will include definitions, constructions, and Kirby pictures of branched covers in 4 dimensions. If time, we will discuss his results on hyperelliptic genus-g Lefschetz fibration which contain at least one separating vanishing cycles. 

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