Speaker: Porter Morgan
Title: Irreducible 4-manifolds with order two fundamental group
Abstract: Let R be a closed, smooth, oriented 4–manifold with order two fundamental group. The works of Freedman and Hambleton-Kreck show that R is determined up to homeomorphism by just a few basic properties. That said, there are often many different manifolds that are homeomorphic to R, but not diffeomorphic to it or each other. In this talk, we’ll describe how to construct irreducible copies of R; roughly speaking, these are smooth manifolds that are homeomorphic to R, and don’t decompose into non-trivial connected sums. We’ll show that if R has odd intersection form and non-negative first Chern number, then in all but seven cases, it has an irreducible copy. We’ll describe some of the techniques used to realize these irreducible smooth structures, including torus surgeries, symplectic fiber sums, and a novel approach to constructing Lefschetz fibrations equipped with free involutions. This is joint work with Mihail Arabadji.
Institution: UMass Amherst
Website: https://www.umass.edu/mathematics-statistics/about/directory/porter-morgan
Speaker: Kerem İnal
Title: Minimal generation of spin mapping class groups
Abstract: Given a spin structure s on a closed oriented surface S, the spin mapping class group Mod(S,s) is the subgroup of Mod(S) stabilizing s. These groups are intimately related to the spin geometry of 3- and 4-manifolds. In this talk, we will discuss our results on generating Mod(S,s) with minimal number of generators, including some generalisations to r-spin groups.
Institution: UMass Amherst
Website: https://www.umass.edu/mathematics-statistics/about/directory/kerem-inal
No Seminar
Speaker: Carlos Soto
Title: Differential Privacy over Riemannian Manifolds and Shape Space
Abstract: Motivated by the problem of statistical shape analysis, this work considers the problem of producing sanitised differentially private estimates through the K-norm Gradient Mechanism (KNG) when the data or parameters live on a Riemannian manifold. In particular, Kendall's 2D shape space is a Riemannian manifold (a projective space) with positive sectional curvature. Traditionally, KNG requires an objective function and produces a sanitized estimate by favoring values which produce gradients close to zero of the objective. This work extends KNG to consider objective functions which take on manifold-valued data. Respecting the nature of the data leads to utility gains when compared to sanitization in an ambient space, as well as removing the need for post-processing. Specifically, this work proposes sanitizing the Fréchet mean for the sphere, symmetric positive definite matrices, and Kendall’s 2D shape space under a pure differentially private framework with an application to corpus callosum data.
Institution: UMass Amherst
Website: https://carlos-soto-phd.netlify.app/bio/
No Seminar
No Seminar
Speaker: Paul Hacking
Title: Homological mirror symmetry for projective K3 surfaces
Abstract: Joint work with Ailsa Keating (Cambridge). A K3 surface is a simply connected compact complex surface with c_1=0. We prove the homological mirror symmetry conjecture of Kontsevich for projective K3 surfaces X equipped with the Fubini Study form. That is, the Fukaya category of X, with objects Lagrangian submanifolds and morphisms determined by Floer cohomology, is equivalent to the derived category of coherent sheaves of a mirror K3 surface Y, with objects complexes of holomorphic vector bundles and morphisms determined by sheaf cohomology. Our proof is based on the heuristic of Strominger--Yau--Zaslow, which asserts that mirror Calabi--Yau manifolds admit dual Lagrangian torus fibrations over a common base, and builds on prior work of Seidel, Sheridan, Lekili--Ueda, and Ganatra--Pardon--Shende.
Institution: UMass Amherst
Website: https://people.math.umass.edu/~hacking/
Speaker: Maggie Miller
Title: Seifert surfaces in 4D
Abstract: A classical fact is that any two smooth disks with the same boundary in S^3 are isotopic rel. boundary. Many more complicated Seifert surfaces bounded by more interesting knots become isotopic once you push their interiors into the 4-ball. However, it turns out that not all such surfaces become isotopic in B^4. Surprisingly, any two genus-g Seifert surfaces bounded by an alternating knot become isotopic once you push their interiors into the 4-ball. It remains unknown which other knots have this property, or in general how one might count different Seifert surfaces up to isotopy in 4D. In this talk, we’ll talk about classical constructions of Seifert surfaces, how to produce isotopies, and how to obstruct them.
This talk is about joint work with Seungwon Kim and Jaehoon Yoo, and also joint work with Kyle Hayden, Seungwon Kim, JungHwan Park, and Isaac Sundberg.
Institution: UT Austin
Website: https://web.ma.utexas.edu/users/mhm799/
Speaker: Baris Coskunuzer
Title: Topological Machine Learning
Abstract: In this talk, we will explore key techniques in topological machine learning and highlight their applications in two distinct areas. First, we will discuss computer-aided drug discovery, where Multiparameter Persistence is leveraged for graph representation learning. Second, we will examine cancer detection from histopathological images using cubical persistence. Our approach is applied to five different cancer types, achieving superior performance compared to state-of-the-art deep learning methods. The talk is designed to be accessible for advanced undergraduate students in mathematics, science, and engineering, requiring no prior knowledge of topology or machine learning.
Institution: The University of Texas at Dallas
Website: https://personal.utdallas.edu/~coskunuz/
No seminar
Speaker: Jason Liu
Title: Toric extension of complexity one spaces
Abstract: A complexity k space is a 2n dimensional symplectic manifold equipped with an effective Hamiltonian action of a torus of dimension n-k. When k = 0, these spaces are also known as toric manifolds. Given a toric manifold, by considering the action of an (n-1)-dimensional subtorus, we get a complexity one space. A natural question to ask is: given a complexity one space, is there a way to extend it to a toric manifold? In this talk, I will start by introducing some preliminary results in equivariant symplectic geometry and then discuss the sufficient and necessary conditions for the extension. This is joint with Joseph Palmer and Susan Tolman.
Institution: University of Illinois Urbana-Champaign
Website:
Speaker: Alec Payne
Title: Flexible, Non-Simply Connected Surfaces in R^3
Abstract: Given a smooth surface in R^3, a classical question in differential geometry asks whether it can be continuously deformed through a smooth, nontrivial family of isometric surfaces. If such a family exists and does not arise from rigid motions of R^3, then the surface is said to be flexible. It is an open question whether there exists a smooth closed flexible surface in R^3. In this talk, we survey this question and the general uniqueness problem for isometric immersions. We then present the first examples of smooth flexible surfaces in R^3 which are not simply connected, are not minimal, and have non-zero Gaussian curvature. We conclude with a discussion of speculative approaches to the construction of smooth, closed flexible surfaces. These results are part of upcoming work with Andrew Sageman-Furnas.
Institution: North Carolina State University, SLMSI
Website: https://sites.google.com/view/alecpayne
Speaker: Claire Dai
Title: Sectorial Decompositions of Symmetric Products and Homological Mirror Symmetry
Abstract: Symmetric products of Riemann surfaces play a crucial role in symplectic geometry and low-dimensional topology. They are key to defining Heegaard Floer homology and serve as important examples of Liouville manifolds when the surfaces are open. In this talk, I will present ongoing work on the symplectic topology of these spaces using Liouville sectorial techniques, along with examples and applications of these decompositions in the context of homological mirror symmetry.
Institution: Harvard
Website: