If you're interested in meeting with a visiting speaker, contact the host. (Titles and abstracts collected at the end of the page.)
Schedule!
August 19 Stepan Paul, NC State
August 26 Teemu Saksala, NC State
September 2 Adam Levine, Duke (Host: Tye)
September 9 Tye Lidman, NC State
September 16 Irina Kogan, NC State
September 23 Spyros Filippas (online), U. of Helsinki (Teemu)
September 30 Nicklas Day, NC State
October 7 Louis Esser, NC State
October 14 Oliver Gross, UC San Diego (Host: Andy SF)
October 21 Antti Kykkänen, Rice (Host: Teemu)
October 28 Shrey Aryan, MIT (Host: Peter)
November 4 Bulent Tosun, U Alabama (Host: Tye)
November 11 TBD
November 18 Miriam Kuzbary, Amherst (Host: Tye)
November 25 Thanksgiving!
If you're interested in joining, we have a Zoom option as well. Please write me to join.
Schedule!
August 19 Tye (Intro to the Volume Conjecture)
August 26 Tye (Torus decompositions of knot exteriors
September 2 Matt (Hyperbolic volume)
September 9 Dylan (Colored Jones polynomials and the Kauffman bracket)
September 16 Nico (Yang-Baxter, R-matrices, and colored Jones polynomials)
September 23 Patrick (WRT invariants - part 1)
September 30 Fang-Rong (WRT invariants - part 2)
October 14 TBD
October 21 TBD
October 28 TBD
November 4 TBD
November 11 TBD
November 18 TBD
November 25 Thanksgiving!
Stepan Paul (NC State)
Title: The Paper Klein Bottle (with other results and illustrations from a semester at IHP)
Abstract: In principle, a Klein bottle is obtained by gluing the opposite edges of a rectangle to each other with certain choices for orientation. So, what about in practice? Is it possible to literally bend/fold/roll a flat piece of paper and tape the opposite edges together to end up with a Klein bottle? This question has been explored on a number of fronts for the torus, but the two examples I’ll present here are the first known explicit path isometric map and isometric immersion of a flat Klein bottle, respectively. I’ll also show off my origami sculptures of the main examples. Finally, I'll share some other semi-related results and illustrations from my semester playing with paper (both flat and curved) at the Poincaré Institute.
Teemu Saksala (NC State)
Title: Inverse Problems for Hyperbolic PDEs: From Microlocal Analysis To Geometry
Abstract: We consider the wave equation with variable wave speed in the Euclidean space, where the initial state is a delta function. There are three special subsets of the whole space: (1) the source set where the delta initial conditions are supported, (2) the unknown set where the wave speed is not known a priori, and (3) the receiver set where the waves are measured. The inverse problem is to reconstruct the wave speed uniquely in the unknown set by sending many waves that are measured in the receiver set. We introduce and use tools (propagation of singularities and finite speed of wave propagation) from microlocal analysis to reduce this PDE based data to geometric data, whose ``usefulness'' depends on how the three sets lie in relation to each other. We give three example scenarios where this procedure leads to unique determination of the wave speed.
Adam Levine (Duke)
Title: Knot Floer homology, bordered Floer homology, surgery modules, and immersed curves
Abstract: This talk will explore the relationship between several different versions of bordered Heegaard Floer homology for 3-manifolds with torus boundary. I will begin by describing the original Ozsváth-Szabó knot surgery formula and its reinterpretation by Zemke as a new (and very powerful) version of bordered Floer homology. I will then show how this theory can recover the original Lipshitz-Ozsváth-Thurston bordered theory, which can be explained very cleanly in terms of immersed curves. This is joint work with Ian Zemke and Jonathan Hanselman.
Tye Lidman (NC State)
Title: Exotic four-manifolds and TQFTs
Abstract: A major problem in four-dimensional topology is to determine when a four-manifold has an exotic smooth structure, i.e. when it is homeomorphic but not diffeomorphic to another smooth four-manifold. We will discuss this problem and introduce a new tool for constructing exotic four-manifolds. This is joint work with Lisa Piccirillo.
Irina Kogan (NC State)
Title: Equi-affine minimal-degree moving frames for polynomial curves
Abstract: Classical equivariant moving frames play an important role in differential geometry. However, these frames associated with a polynomial curve are, in general, neither polynomial nor even rational. We develop a theory and an algorithm for constructing minimal-degree polynomial moving frames for polynomial curves in an affine space. The algorithm is equivariant under volume-preserving affine transformations of the ambient space and the parameter shifts. We show that any matrix-completion algorithm can be turned into an equivariant moving frame algorithm via an equivariantization procedure that we develop. We prove that if a matrix-completion algorithm is of minimal degree, so is the resulting equivariant moving frame algorithm. We propose a novel minimal-degree matrix-completion algorithm, complementing the existing body of literature on this topic. This is a joint work with Hoon Hong.
Spyros Filippas (U Helsinki)
Title: Recovering a matrix-valued potential in stationary spacetimes
Abstract: In this talk we consider the problem of recovering a time dependent matrix-valued potential on a general globally hyperbolic manifold from the knowledge of the source to the solution map of a wave equation including a connection 1-form term. We show that this problem can be reduced to studying the injectivity of a non-Abelian light ray transform. Under the assumption that our manifold is stationary, we then prove that injectivity for an appropriately defined Riemannian transform allows us to uniquely determine the potential. This is based on a joint work with Lauri Oksanen and Miika Sarkkinen.
Louis Esser (NC State)
Title: Weighted surfaces with maximal Picard number
Abstract: A difficult problem in algebraic geometry is to find examples of complex algebraic surfaces with maximal Picard number, i.e. those with the "most" algebraic curve classes permitted by cohomology. In this talk, we investigate this problem for surfaces embedded in a weighted projective threefold with equations of a special form, building on methods of T. Shioda. We find an unexpected connection between the automorphism group of such a surface and its Picard number, and also prove that elliptic surfaces of maximal Picard number and arbitrary geometric genus may be embedded as hypersurfaces in weighted projective space. This talk is based on joint work with Jennifer Li.