Observance
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Title: Persistence of unknottedness of clean Lagrangian intersections
Speaker: Johan Asplund [Stony Brook University]
Abstract: Let L and K be two Lagrangian spheres in a six dimensional compact symplectic manifold that intersect cleanly along a circle. Assuming that the clean intersection is an unknot in both L and K, we will explain how the clean intersection must remain unknotted after any Hamiltonian isotopy of L and K that is supported near their union. This is based on joint work with Yin Li.
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General Public Lecture
Monday April 21, at 5:00pm
Location: Della Pietra Family Auditorium – 103
Reception at 4:15pm, Simons Center Lobby
Title: Where Does Mathematics Come From?
Abstract:
Title: The Fukaya Algebra over The Integers
Speaker: Mohamad Rabah [Stony Brook University]
Abstract: Given a closed, connected, relatively-spin Lagrangian submanifold in a closed symplectic\r\nmanifold, we associate to it a curved, gapped, filtered An,K-algebra over the Novikov ring\r\nwith integer coefficients. Under certain conditions, such algebra can be extended to an Ainfinity\r\nalgebra. To illustrate our framework, we give a proof of the Quantum Lefschetz\r\nHyperplane Theorem in the Kaehler case, and associate virtual fundamental classes to the\r\nmoduli spaces used in local Gromov-Witten theory, in the symplectic case.
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Title: Hilbert's sixth problem: derivation of the Boltzmann and fluid equations (I and II)
Abstract: We present recent works with Zaher Hani and Xiao Ma, in which we derive the Boltzmann equation from the hard sphere dynamics in the Boltzmann-Grad limit, for the full time range in which the (strong) solution to the Boltzmann equation exists. This is done in the Euclidean setting in any dimension d at least 2, and in the periodic setting in dimensions 2 and 3. As a corollary, we also derive the corresponding fluid equations from the the hard sphere dynamics. This resolves Hilbert's Sixth Problem pertaining to the derivation of hydrodynamic equations from colliding particle systems, via the Boltzmann equation as the intermediate step.
In the first lecture we review the scope of the problem, the set up and historical backgrounds.
In the second lecture we discuss the main ideas involved in the proof, and some of the key steps.
The two lectures are independent from each other. They will cover similar contents but with emphasis put on different aspects.
Title: Curvature, Macroscopic Dimensions, and Symmetric Products of Surfaces
Speaker: Luca di Cerbo [University of Florida]
Abstract: Abstract: In this talk, I will present a detailed study of the curvature and symplectic asphericity properties of symmetric products of surfaces. I show that these spaces can be used to answer nuanced questions arising in the study of closed Riemannian manifolds with positive scalar curvature. For example, symmetric products of surfaces sharply distinguish between two distinct notions of macroscopic dimension introduced by Gromov and Dranishnikov. As a natural generalization of this circle of ideas, I will also address the Gromov–Lawson and Gromov conjectures in the Kaehler projective setting and draw new connections between the theories of the minimal model, positivity in algebraic geometry, and macroscopic dimensions.
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Special Talk for High School and Undergraduate Students
Title: Math is the Source Code of Human Mind
Title: Hodge theory of hypersurface singularities
Speaker: Daniel Brogan [Stony Brook University]
Abstract: We discuss the connection between Hodge theory and singularities of complex projective algebraic varieties. First, we consider the variation of Hodge structure associated with the complete linear system of hypersurfaces in projective space of some fixed degree. This extends to a complex of pure Hodge modules. Using Nori’s connectivity theorem, we compute the cohomology sheaves of this Hodge module, yielding various geometric corollaries. Second, we discuss secant varieties and secant bundles. Using a version of the decomposition theorem for semismall maps, we compute the intersection cohomology of various secant varieties for smooth curves in projective space embedded via a sufficiently positive line bundle. Finally, we focus on secant varieties of rational normal curves. These are defined by the vanishing of minors of a certain Hankel matrix. We prove that these secant varieties are rational homology manifolds. We also compute the nearby and vanishing cycle sheaves for the determinant of a generic Hankel matrix.
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Title: Hilbert's sixth problem: derivation of the Boltzmann and fluid equations (I and II)
Abstract: We present recent works with Zaher Hani and Xiao Ma, in which we derive the Boltzmann equation from the hard sphere dynamics in the Boltzmann-Grad limit, for the full time range in which the (strong) solution to the Boltzmann equation exists. This is done in the Euclidean setting in any dimension d at least 2, and in the periodic setting in dimensions 2 and 3. As a corollary, we also derive the corresponding fluid equations from the the hard sphere dynamics. This resolves Hilbert's Sixth Problem pertaining to the derivation of hydrodynamic equations from colliding particle systems, via the Boltzmann equation as the intermediate step.
In the first lecture we review the scope of the problem, the set up and historical backgrounds.
In the second lecture we discuss the main ideas involved in the proof, and some of the key steps.
The two lectures are independent from each other. They will cover similar contents but with emphasis put on different aspects.
Title: Hilbert's sixth problem: derivation of the Boltzmann and fluid equations II
Speaker: Yu Deng [University of Chicago]
Abstract: (note the special date and time!)
We present recent works with Zaher Hani and Xiao Ma, in which we derive the Boltzmann equation from the hard sphere dynamics in the Boltzmann-Grad limit, for the full time range in which the (strong) solution to the Boltzmann equation exists. This is done in the Euclidean setting in any dimension $d\geq 2$, and in the periodic setting in dimensions $d\in\{2,3\}$. As a corollary, we also derive the corresponding fluid equations from the the hard sphere dynamics. This resolves Hilbert's Sixth Problem pertaining to the derivation of hydrodynamic equations from colliding particle systems, via the Boltzmann equation as the intermediate step.
In this second lecture we discuss the main ideas involved in the proof, and some of the key steps. The two lectures are independent from each other. They will cover similar contents but with emphasis put on different aspects.
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Title: Revisiting minimal degree curves on hypersurfaces
Speaker: Nathan Chen [Harvard University]
Abstract: I will discuss a circle of ideas related to results about minimal degree curves on general hypersurfaces. This is joint work with D. Yang.
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For more information please visit: https://scgp.stonybrook.edu/archives/45921
Title: Inscription problems and symplectic geometry
Speaker: Joshua Greene [Boston College]
Abstract: The square peg problem was posed by Otto Toeplitz in 1911. It asks whether every Jordan curve in the place contains the vertices of a square, and it is still open to this day. I will survey the approaches to this problem and its relatives using symplectic geometry. This talk is based on joint work with Andrew Lobb.
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Date: Thursday April 24 at 4:00pm
Location: SCGP Room 102
Title: New Frontiers in the Langlands Program for Riemann Surfaces
Abstract: The Langlands correspondence for Riemann surfaces (complex algebraic curves) has two different versions. One (called geometric or categorical), due to Beilinson and Drinfeld, is in terms of sheaves. It has been studied extensively for more than 3 decades, and a version of it was recently proved by a team led by Gaitsgory and Raskin. The other (called analytic) is in terms of functions, and hence it is more down-to-Earth and closer to the original Langlands correspondence for number fields. It has been developed for the last 6 years by Etingof, Kazhdan, and myself. I will start the lecture with a brief introduction to the original formulation of the Langlands correspondence. I will then explain the setup of both the geometric and the analytic versions for Riemann surfaces and connections between them.
Title: The extremal black hole threshold
Speaker: Ryan Unger [Stanford University]
Abstract: Extremal black holes are special solutions of Einstein’s equations with absolute zero temperature in the celebrated thermodynamic analogy of black hole mechanics. In this talk, I will discuss the stability problem for extremal black holes, which turns out to be closely related with the study of the black hole formation threshold. I will present a recent positive resolution of the stability problem in spherical symmetry in the presence of a real scalar field. This is based on joint work with Yannis Angelopoulos (Caltech) and Christoph Kehle (MIT).
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Title: Random Measured Laminations and Teichm\"{u}ller Space
Speaker: Tina Torkaman [University of Chicago]
Abstract: In this talk, we introduce a canonical geodesic current $KX$ for each $X \in T_g$, representing a randomly chosen simple closed geodesic on $X$. We establish results analogous to Bonahon's work on the Liouville measure. In particular, we show that the map $X \mapsto KX$ defines a proper embedding of Teichm\"{u}ller space $T_g$ into the space of geodesic currents. This embedding leads to a compactification of $T_g$ that differs from Thurston's compactification, which Bonahon's results yield via the Liouville measure. We will discuss the construction of $KX$ and its geometric properties. This is joint work with Curt McMullen.
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Title: New tools for studying homology cobordism
Speaker: Jen Hom
Abstract: We will discuss a new family of homology cobordism invariants coming from Pin(2)-equivariant Floer homology. Our construction relies on Koszul duality and constructions inspired by the concordance invariants epsilon and upsilon. This is joint work in progress with I. Dai, M. Stoffregen, and L. Truong.
For more information, please visit: https://scgp.stonybrook.edu/archives/45808
Title: 2-torsion in instanton Floer homology
Speaker: Zhenkun Li
Abstract: Instanton Floer homology, introduced by Floer in the 1980s, has become a power tool in the study of 3-dimensional topology. Its application has led to significant achievements, such as the proof of the Property P conjecture. While instanton Floer homology with complex coefficients is widely studied and conjectured to be isomorphic to the hat version of Heegaard Floer homology, its counterpart with integral coefficients is less understood. In this talk, we will explore the abundance of 2-torsion in instanton Floer homology with integral coefficients and demonstrate how this 2-torsion encodes intriguing topological information about relevant 3-manifolds and knots. This is a joint work with Fan Ye.
Title: Naturality in HF^oo
Speaker: Mike Miler Eismeier
Abstract: There is an algebraic approximation to the group HF^oo(Y; Z) called the "cup homology" of Y. They are known to have the same dimension over F_2, they can be shown to agree (up to extension problems) for b_1(Y) <= 8, and one gets the same result over Z in millions of computer calculations on examples with 9 <= b_1(Y) <= 14. One is led to guess there should be a good reason for this; perhaps the functors are naturally isomorphic. I will prove they are not by comparing them as MCG(Y)-modules for Y = Sigma_4 x S^1. To study these examples, we develop a version of the Lefschetz decomposition over Z which may be of independent interest.
Title: TBA
Speaker: Artur Avila [University of Zurich]
Abstract: TBA
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Title: Surgery Exact Triangles in Instanton Theory
Speaker: Deeparaj Bhat
Abstract: We prove an exact triangle relating the knot instanton homology to the instanton homology of surgeries along the knot. As the knot instanton homology is computable in many instances, this sheds some light on the instanton homology of closed 3-manifolds. We illustrate this with computations in the case of some surgeries on the trefoil. In particular, we show the Poincaré homology sphere is not an instanton L-space (with Z/2 coefficients), in contrast with Heegaard Floer and monopole Floer theories. Finally, we sketch the proof of the triangle inspired by the Atiyah-Floer conjecture and results from symplectic geometry.
Title: Complete Riemannian 4-manifolds with uniformly positive scalar curvature metric
Speaker: Anubhav Mukherjee
Abstract: In three dimensions, geometry plays a crucial role in classifying the topology of manifolds. Inspired by this, we set out to explore the intricate world of smooth 4-manifolds through the lens of geometry. Specifically, we aim to understand under what conditions a contractible 4-manifold admits a uniform positive scalar curvature metric. In collaboration with Otis Chodosh and Davi Maximo, we demonstrated that in certain cases, the existence of such a metric can provide insight into the topology of 4-manifolds. Moreover, by utilizing Floer theory, we identified obstructions to the existence of such metrics in 4-manifolds.
Title: Braid varieties from several perspectives
Speaker: Lenny Ng
Abstract: Braid varieties, a family of algebraic varieties associated to any positive braid, have recently emerged in several distinct areas of mathematics. They've appeared in symplectic topology through Floer theory and Legendrian contact homology, but also in algebraic geometry through flags and constructible sheaves, and in algebraic combinatorics through cluster theory. I'll discuss the surprising interrelations between these areas, and especially the way that cluster theory provides some new insight into a well-studied low-dimensional symplectic problem: classifying Lagrangian fillings of Legendrian links.
Title: Monotone A infinity category and Atiyah-Floer functor
Speaker: Kenji Fukaya
Abstract: In this talk I will explain a work in progress with A. Daemi. We formulate the notion of Monotone A infinity category which for example can be used to study monotone (Immgersed) Lagrangian Floer theory. (It uses different kind of filtration from Floer theory of more general Lagrangian submanifold.) Then I will explain Atiyah-Floer conjecture can be formulated as a functorial equivalence between certain monotone A infinity categories.
Title: The mapping class group action on the odd character variety is faithful
Speaker: Chris Scaduto
Abstract: The moduli space of holomorphic rank 2 bundles of odd degree and fixed determinant over a given Riemann surface is a symplectic manifold which has an interpretation as a certain PU(2) character variety. There is a homomorphism from a finite extension of the mapping class group of the surface to the symplectic mapping class group of this moduli space. When the genus is 2 or more, we prove that this homomorphism is injective. The proof uses Floer's instanton homology for 3-manifolds. This is joint work with Ali Daemi.
Title: Compressing surface diffeomorphisms via bordered Floer homology
Speaker: Akram Alishahi
Abstract: Let F be a closed surface, and \psi be a diffeomorphism of F. An interesting question with nice topological implications for detecting homotopy ribbon fibered knots is whether \psi extends over some handlebody with boundary F. In 1985, Casson-Long gave an algorithm for answering this question. In this talk, first we will discuss how bordered Floer homology can be used to detect whether \psi extends over a specific handlebody. Then, we will outline how to adapt ideas of Casson-Long to use bordered Floer homology to detect whether \psi extends over any compression body. This is a joint work with Robert Lipshitz.
Title: TBA
Speaker: Gioacchino Antonelli [NYU Courant]
Abstract: TBA
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