Student Run Research Seminar Talks: (Wednesdays, Room 3106)

Co-hosted with Soyeon Kim.

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Dates	Speaker and Abstract
Oct 4th 12:10pm-1pm (Room 1147),	Tonie Scroggin Title: Cluster Structures on Braid Varieties Abstract: Given a simple algebraic group G and an element β of the positive braid monoid, we consider an affine, smooth algebraic variety X ( β ) which includes well known varieties in Lie theory such as open Richardson and positroid varieties. In this talk we will show that the coordinate ring of regular functions of any braid variety is a cluster algebra using Lusztig cycles in algebraic weaves.
Oct 11th 11am-12pm,	Yuze Luan Title: Irreducible components of the Hilbert scheme of points on non-reduced curves Abstract:We classify the irreducible components of the Hilbert scheme of n points on non-reduced algebraic plane curves, and give a formula for the multiplicities of the irreducible components. The irreducible components are indexed by partitions of n ; all have dimension n ; and their multiplicities are given as a polynomial of the parts of the corresponding partitions..
Oct 18th 11am-12pm,	Arthur Kalb Title: Leaky Rigid Lid Modes in a Nonhydrostatic Atmosphere. Abstract: We calculate the modes produced by a nonhydrostatic, incompressible, nonrotating atmosphere with a piecewise constant Brunt-V\"ais\"al\"a frequency, with consideration given to the vertical structure of these modes in the stratosphere. We find agreement with previous work done on the hydrostatic problem in the small horizontal wavenumber, hydrostatic regime. In addition to the previously derived discrete family of baroclinic modes and stratospheric mode, the absence of hydrostatic balance on shorter horizontal length scales introduces a singular perturbation of the problem which corresponds to the emergence of a barotropic mode. The frequency of this barotropic mode provides a limit to the frequency and decay rate of the high horizontal wave number baroclinic modes, thereby introducing a scale selective wave drag. The decay of the baroclinic modes can be characterized by the angle of the wavefront, and a differential operator is constructed to approximate the decay for rigid lid boundary conditions.
Oct 25th 12:10pm-1pm,	Trevor Oliveira-Smith Title: To what extent are 0-surgeries on fibered knots classifying? Or, exotic S 2 × S 2 's and where we probably shouldn't be looking for them. Abstract: It is known from the works of Gabai and Teragaitio that 0-surgery on torus knots and the figure-8 knot are characterizing. In general, 0-surgery on a fibered knot K in the 3-sphere does not characterize K. Examples of such knots are constructed using annular twisting techniques yielding non-isotopic knots which 0-surger to the same manifold. Thus, a similar surgery-theoretic characterization to Gabai and Teragaitio for general fibered knots is not possible. In the proposed talk, we conjecture a possible classification using handle-slides and the notion of a dual surgery. By treating surgeries as handle-attachements, the proposed classification has connections to the classification of smooth structures on S 2 × S 2 .
Nov 1st 12:10pm-1pm,	Addie Chan Title: Singularity Formation along the Line Bundle Mean Curvature Flow Abstract: The line bundle mean curvature flow (LBMCF) is a complex analogue of the Lagrangian mean curvature flow (LMCF) for graphs, with fixed points solving the deformed Hermitian-Yang-Mills equation. In this talk, we briefly overview the history of LMCF, and introduce the duality between LMCF and LBMCF. Next, we show two examples of singularities along LBMCF, one being a finite time singularity in the stable case, ruling out long time existence of the flow in general; and the other one being an infinite time singularity in the semi-stable case.
Nov 8th 12:10pm-1pm,	Aidan Epperly Title: Relaxations and Exact Solutions to Quantum Max Cut via Algebraic Structure of Swap Operators Abstract: The Quantum Max Cut (QMC) problem has emerged as a test-problem for designing approximation algorithms for local Hamiltonian problems. In this talk I’ll describe an approach to solving this problem using the algebraic structure of QMC, in particular the relationship between the quantum max cut Hamiltonian and the representation theory of the symmetric group. The first major result is an extension of non-commutative Sum of Squares (ncSoS) optimization techniques to give a new hierarchy of relaxations to Quantum Max Cut. This hierarchy is based on optimizations over polynomials in the qubit swap operators. This is in contrast to the “standard” quantum Lasserre Hierarchy, which is based on polynomials expressed in terms of the Pauli matrices. The second major result covered will be a polynomial-time algorithm that exactly computes the maximum eigenvalue of the QMC Hamiltonian for certain graphs, including graphs that can be “decomposed” as a signed combination of cliques. A special case of the latter are complete bipartite graphs with uniform edge-weights, for which exact solutions are known from the work of Lieb and Mattis. Our methods, which use representation theory of the symmetric group, can be seen as a generalization of the Lieb-Mattis result.
Nov 15th 12:10pm-1pm,	Neetal Neel Title: Minimal Genus Seifert surface of 11 crossing alternating knots Abstract: The Kakimizu complex of knots for crossings less than or equal to 10 has been found by O.Kakimizu in 1992. The 0 skeleton of the Kakimizu complex of 2-bridge links was found by Hatcher and Thurston, generalized by Sakuma to find the Kakimizu complex of Special Arborescent links. Jessica Banks gave a complete proof of results announced by Hirasawa and Sakuma, describing explicitly the Kakimizu Complex of a non-split, prime special alternating links. The Kakimizu complex of prime, non-split alternating links has finite vertices. We find the Kakimizu complex for all 11 crossing alternating knots. For most links we were able to use the algorithms known to us. We found the rest of the Kakimizu complex using sutured manifold theory. I will give a brief exposition of the algorithms known, for computing the Kakimizu complex of links and future questions.
Nov 22nd 12:10pm-1pm,
Nov 29th 12:10pm-1pm,	Ian Sullivan Title: Kirby Belts and the Skein Lasagna Module of $S^2\times S^2$. Abstract: The lasagna module of a chosen link homology theory is a relatively new kind of smooth $4$-manifold invariant. In general, these invariants are difficult to compute, often requiring a colimit of "symmetrized" Khovanov homology groups. In this talk, we define and investigate a certain homotopy colimit of a directed system of Khovanov homology tangle complexes. We then propose a new "local" computation tool for $4$-manifolds whose Kirby diagrams contain certain tangles, and use this technique to compute the $\mathfrak{gl_{2}}$ Khovanov homology flavored skein lasagna module of some new manifolds, including $S^2\times S^2$.
Dec 6th 12:10pm-1pm,	Brittney Marsters Title: Generating Conical Semigroups: SOC and PSD Abstract: The integer points within a convex cone form a conical semigroup. When the (pointed) cone is polyhedral, there is a unique inclusion minimal integer generating set known as the Hilbert Basis. This means that every integer point in the polyhedral cone can be represented uniquely as a positive integer combination of the elements of the Hilbert basis. Thus, it is natural to ask: do we preserve any notion of finite generation for conical semigroups when we relax the polyhedral condition? Do we have any analogue to the Hilbert Basis? In this talk, we will introduce the notion of (R,G)-finite generation and present recent results showing that the conical semigroup given by the second order cone, SOC(n), is (R,G)-finitely generated in dimensions three through ten. Lastly, we will present an analogous result for the conical semigroup given by the cone of positive semidefinite matrices, showing that it is also (R,G)-finitely generated.
Dec 13th -,	Finals Week
Dec 20th -,	Winter Break (I)
Jan 3rd -,	Winter Break (II)
Jan 10th -,	-
Jan 17th 1:10pm-2pm,	Shanon Rubin Title: How to glue categories using homotopy theory: symplectic surfaces via graphs. Abstract: In this talk I will give a broad overview of my thesis results. The story begins with an attempt to understand an open symplectic surface by retracting it to a trivalent graph. After giving the geometric motivation for this procedure with some examples, I'll explain how to turn this trivalent graph into a homotopy limit of differential-graded categories. From there, we'll discuss the general idea behind homotopy-invariant constructions, giving concrete examples from topology. Then we'll put all of this together to describe why our homotopy limit of categories is an invariant of the original symplectic surface.
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Mar 20th	Finals Week,
Mar 27th	Spring Break,
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