Algebraic Geometry and Geometric Topology Seminar
We meet weekly on Mondays at 3:00 PM (Central time), Room: Gibson 400D
Organizers: Mahir Can, Tài Huy Hà, Rafał Komendarczyk, Kalina Mincheva.
Spring 2022
January 31-
Surena Hozoori, Georgia Institute of Technology.
On Anosovity, divergence and bi-contact surgery. .I will revisit the relation between Anosov 3-flows and invariant volume forms, from a contact geometric point of view. Consequently, I will give a contact geometric characterization of when a flow with dominated splitting is Anosov based on its divergence, as well as a Reeb dynamical interpretation of when such flows are volume preserving. Moreover, I will discuss the implications of this study on the surgery theory of Anosov 3-flows. In particular, I will conclude that the Goodman-Fried surgery of Anosov flows can be reconstructed, using a bi-contact surgery of Salmoiraghi.
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Pierre Dehornoy, Université Grenoble Alpes.
The trunkenness of a volume-preserving vector field. .This work (joint with Ana Rechtman) deals with the construction of invariants for volume-preserving vector fields in the 3-sphere, up to homeo- (or diffeo- )morphism of the underlying manifold. Not that many such invariants exist, the most famous one being the helicity. It turns out that it can be recovered in many ways, and that it is the only smooth invariant. Here we construct another invariant, inspired by and connected to the trunk for knots, which has the interest of being independent of helicity.
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Robyn Brooks, Boston College.
Multiparameter Persistence and Discrete Morse Theory. .Persistent Homology is a tool of Computation Topology which is used to determine the topological features of a space from a sample of data points. In this talk, I will introduce the (multi-)persistence pipeline, as well as some basic tools from Discrete Morse Theory which can be used to better understand the multi-parameter persistence module of a filtration. In particular, the addition of a discrete gradient vector field consistent with a multi-filtration allows one to exploit the information contained in the critical cells of that vector field as a means of enhancing geometrical understanding of multi-parameter persistence. I will present results from joint work with Claudia Landi, Asilata Bapat, Barbara Mahler, and Celia Hacker, in which we are able to show that the rank invariant for nD persistence modules can be computed by selecting a small number of values in the parameter space determined by the critical cells of the discrete gradient vector field. These values may be used to reconstruct the rank invariant for all other possible values in the parameter space.
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Michael DiPasquale, University of South Alabama.
A duality for sequences and its manifestation for symbolic powers. .In this talk we present a duality for sequences of natural numbers. The sequence duality we present takes a sub-additive sequence to a super-additive sequence (and vice-versa) and inverts the coefficients of linear growth. We indicate at least one instance where this duality shows up for symbolic powers of ideals. Concretely, if an ideal defines a projective variety, its nth symbolic power consists of those polynomials which vanish to order n on the variety. The main example which we will explore is the sequence of initial degrees of symbolic powers, which is a sub-additive sequence giving rise to what is known as the Waldschmidt constant. Via apolarity (or Macaulay-Matlis duality), the sequence of initial degrees is dual (in our sense of sequence duality) to a sequence of regularities of certain ideals generated by powers of linear forms. There will be plenty of examples. This is joint work with Alexandra Seceleanu.
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Clayton Shonkwiler, Colorado State University.
Geometric Approaches to Frame Theory. .Frames are overcomplete systems of vectors in Hilbert spaces. They were originally introduced in the 1950s in the context of non-harmonic Fourier series, and came to renewed prominence in the 1980s in signal processing applications. More recently, there has been burgeoning interest in frames in finite-dimensional Hilbert spaces, with applications to signal processing, quantum information, and compressed sensing.
In this talk, I will describe some ways in which tools from the differential, Riemannian, and symplectic geometry can be applied to problems in frame theory. For example, frame spaces of interest are often closely related to adjoint or isotropy orbits of compact groups, and hence can be thought of as symplectic manifolds or isoparametric submanifolds. This is joint work with Tom Needham.
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Zvi Rosen, Florida Atlantic University.
This talk will be over Zoom.
Oriented Matroids and Combinatorial Neural Codes. .A convex neural code is a combinatorial object arising as the intersection pattern of convex open subsets of Euclidean space. In this talk, we relate the emerging theory of convex neural codes to the established theory of oriented matroids, both categorically and with respect to feasibility and complexity. By way of this connection, we prove that all convex codes are related to some representable oriented matroid, and we show that deciding whether a neural code is convex is NP-hard.
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This talk will be at 1pm, in room GI-400A.
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Daciberg Goncalves, University of Sao Paulo.
Free cyclic actions on surfaces and the Borsuk-Ulam theorem. .By 1930 Ulam posed the question: Given a continuous map $f: S^n \to R^n$ does ther exist a point $x\in S^n$ such that $f(x)=f(-x)$? The classical Borsuk-Ulam theorem say: Given a continuous map $f: S^n \to R^n$ then there exist a point $x\in S^n$ such that $f(x)=f(-x)$. This result has been generalised in many directions and it continues to be an attractive and current topic. More general one may consider the situation where the space $S^n$ is replaced by a topological space endowed with a free involution of a finite group $G$, and possibly the target replaced by a space $Y$. In this talk we first present a short survey of the results which were obtained for the case where $X$ is a closed surface and $Y$ either $R^2$ of a closed surface. Then we present some new results when $X$ is a closed surface, $G=Z_n$ the cyclic group of order $n$ and $Y=R^2$. The main result is: Theorem: Let $M$ be a compact surface without boundary, and let $\tau \colon\thinspace Z_n \times M \to M$ be a free action. Then the quadruple $(M,Z_n,\tau;R^2)$ has the Borsuk-Ulam property if and only if the following conditions are satisfied: (1) $n \equiv 2 \ mod \ 4$. (2) $M_\tau$ is non-orientable, and $(\theta_\tau)_{Ab} (\delta)$ is non trivial. At the end, we make some comments regarding the case of a general group $G$.
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G.V. Ravindra, University of Missouri-St. Louis.
Matrix representations of polynomials and Noether-Lefschetz theory. .Given a homogeneous polynomial of degree d in n variables, a century old problem due to Dickson asks if some power of this polynomial can be expressed as the determinant of a matrix with smaller degree homogeneous polynomial entries in a non-trivial way. This talk will introduce a precise version of this question and show how this question is intricately related to the geometry of hypersurfaces of degree d in projective space.
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Christopher Perez, Loyola University.
Towers and elementary embeddings in toral relatively hyperbolic groups. .In a remarkable series of papers Zlil Sela classified the first-order theories of free groups and torsion-free hyperbolic groups using geometric structures he called towers, and independently Olga Kharlampovich and Alexei Myasnikov did the same using equivalent structures they called regular NTQ groups. It was later proved by Chloé Perin that if H is an elementarily embedded subgroup (or elementary submodel) of a torsion-free hyperbolic group G, then G is a tower over H. We prove a generalization of Perin’s result to toral relatively hyperbolic groups using JSJ and shortening techniques.
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Organizational meeting.
All invited. .We will discuss the format of the seminar and perspectives for next academic year. Cookies will be provided.