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Eric Geiger, NC State, Non-congruent non-degenerate curves with identical signatures

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This talk will focus on using the Euclidean Signature to determine whether two smooth planar curves are congruent under the Special Euclidean group. Work done by Emilio Musso and Lorenzo Nicolodi emphasizes that signatures must be used with caution by constructing 1-parameter families of non-congruent curves with degenerate vertices (curve segments of constant curvature) with identical signatures. We address the claim…

Christine Breiner, Fordham University, Harmonic branched coverings and uniformization of CAT(k) spheres

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Consider a metric space (S,d) with an upper curvature bound in the sense of Alexandrov (i.e.~via triangle comparison).  We show that if (S,d) is homeomorphically equivalent to the 2-sphere, then it is conformally equivalent to the 2-sphere.  The method of proof is through harmonic maps, and we show that the conformal equivalence is achieved by…

Vladimir Baranovsky (UC Irvine), Integral model for graph configuration spaces

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This is a report on the joint work with Matthew Levy. We use surjection operations on integral cochains tof a topological space X (described by McClure-Smith and Berger-Fresse) to describe a complex computing (co)homology of the cartesian power of X with some diagonals removed. Host: Radmila Sazdanovic ZOOM link:  https://ncsu.zoom.us/j/97278681300

Anusha Krishnan, Syracuse University, Prescribing Ricci curvature on a product of spheres

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The Ricci curvature Ric(g) is a symmetric 2-tensor on a Riemannian manifold (M,g) that encodes curvature information. It features in several interesting geometric PDEs such as the Ricci flow and the Einstein equation. The nature of Ric(g) as a differential operator -- nonlinear and degenerate elliptic -- make these equations particularly challenging. Host: Peter McGrath Instructions to join: Zoom…

Joonas Ilmavirta, Tampere University, Finland, The light ray transform

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When is a function in the spacetime uniquely determined by its integrals over all light rays? I will introduce the problem, discuss why we might care about it, and how one might go about proving such uniqueness results. Depending on time and audience interest, I can also discuss proofs and tensor tomography.   Organizer: T.…

Woden Kusner, University of Georgia, Measuring chirality with the wind

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The question of measuring "handedness" is of some significance in both mathematics and in the real world. Propellors and screws, proteins and DNA, in fact *almost everything* is chiral.  Can we quantify chirality?  Or can we perhaps answer the question:  "Are your shoes more left-or-right handed than a potato?" We can begin with the hydrodynamic…

Orsola Capovilla-Searle, Duke, Infinitely many Lagrangian Tori in Milnor fibers constructed via Lagrangian Fillings of Legendrian links

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One approach to studying symplectic manifolds with contact boundary is to consider Lagrangian submanifolds with Legendrian boundary; in particular, one can study exact Lagrangian fillings of Legendrian links. There are still many open questions on the spaces of exact Lagrangian fillings of Legendrian links in the standard contact 3-sphere, and one can use Floer theoretic…

Christian Scharrer, University of Bonn, Isoperimetric constrained Willmore tori

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In order to explain the bi-concave shape of red blood cells, Helfrich proposed to study the minimisation of a bending energy amongst closed surfaces with given fixed area and volume. In the homogeneous case, the Helfrich functional simplifies to the scaling invariant Willmore functional. Thus, for the minimisation, the two constraints on area and volume…

Andrew Manion, NC State, Heegaard Floer homology and higher tensor products

SAS 4201

I will give a brief introduction to Heegaard Floer homology and survey what's known about its "extended" structure via Lipshitz-Ozsvath-Thurston's bordered Floer homology and Douglas-Lipshitz-Manolescu's cornered Floer homology. Then I will sketch a connection between this extended structure and a more algebraic problem, the construction of tensor products for higher representations, arising from my recent work with…

Bessa Pacelli, Universidade Federale de Ceara, Fortaleza, Stochastic half-space theorems for minimal surfaces and H-surfaces of $\mathbb{R}^{3}$.

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In the 1980s Xavier proved that a complete non-planar minimal surface with bounded curvature of $\mathbb{R}^{3}$ can not lie in half-space. In 1990, Hoffman-Meeks proved that this half-space property holds for properly immersed non-planar minimal surfaces of $\mathbb{R}^{3}$ as well. And they went further, proving what is called "the strong half-space theorem" that states that…

Melisa Zhang, UGA, Constructions toward topological applications of U(1) x U(1) equivariant Khovanov homology

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In 2018, Khovanov and Robert introduced a version of Khovanov homology over a larger ground ring, termed U(1)xU(1)-equivariant Khovanov homology. This theory was also studied extensively by Taketo Sano. Ross Akhmechet was able to construct an equivariant annular Khovanov homology theory using the U(1)xU(1)-equivariant theory, while the existence of a U(2)-equivariant annular construction is still…

Demetre Kazaras, Duke University, Calculating total mass with harmonic functions

SAS 4201

The ADM mass of an isolated gravitational system is a geometric invariant measuring the total mass due to matter and other fields. I'll describe how to compute this invariant (in 3 spatial dimensions) by studying harmonic functions as well as solutions to other elliptic equations. Recent results in this context will be presented, focusing on applications to…

Larry Gu, University of Southern California, Decategorification of HFK_n(L)

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Using a definition of Euler characteristic for fractionally-graded complexes based on roots of unity, we show that the Euler characteristics of Dowlin’s “sl(n)-like” Heegaard Floer knot invariants HFK_n recover both Alexander polynomial evaluations and sl(n) polynomial evaluations at certain roots of unity for links in S^3. We show that the equality of these evaluations can…

Irina Kogan, NC State, Group Actions, Invariants, and Applications

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I will overview some important milestones in the development of the Invariant Theory from its classical times to modern days, leading into a discussion of the current progress in theory, computation, and applications. The highlights include Hilbert's basis theorem, geometric invariant theory, differential algebra of invariants, the moving frame approach, Lie's work on symmetries of…

Joseph Hoisington, University of Georgia, Calibrations and Harmonic Mappings of Rank-1 Symmetric Spaces.

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We will prove lower bounds for the p-energies of mappings of real, complex and quaternionic projective spaces to arbitrary Riemannian manifolds.  The equality cases of the results for real and complex projective space give strong characterizations of some families of energy-minimizing harmonic mappings of these spaces.  If we have enough time, we will also describe…