Harvard Mathematics News
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Seminars at the Harvard Mathematics DepartmentFri, 08 Nov 2019 13:25:54 -0500SYMPLECTIC GEOMETRY SEMINAR:Yingdi Qin (Berkeley and Harvard) speaks on <b>Coisotropic branes in symplectic tori</b> on Nov 08, 19, 2:00 pm - 3:00 pm in Science Center 530:
Abstract: Homological mirror symmetry (HMS) asserts that the Fukaya category of a symplectic manifold is derived equivalent to the category of coherent sheaves on the mirror complex manifold. Without suitable enlargement (split closure) of the Fukaya category, certain objects of it are missing to prevent HMS from being true. Kapustin and Orlov conjectured that coisotropic branes should be included into the Fukaya category from a physics view point. In this talk, I will construct for linear symplectic tori a version of Fukaya category including coisotropic branes and show that the usual Fukaya category embeds fully faithfully into it.
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http://www.math.harvard.edu/1 Sun, 08 Nov 2019 14:00:00 -0400 GAUGE-TOPOLOGY-SYMPLECTIC SEMINAR:Lisa Piccirillo (Brandeis/MIT) speaks on <b>Knot traces and PL embeddings</b> on Nov 08, 19, 3:30 pm in Science Center 507:
Abstract: It remains at the forefront of 4-manifold topology to construct simple closed 4-manifolds with distinct smooth structures. Towards that end, it is of interest to construct simple 4-manifolds with boundary with very distinct smooth structures. For all genera g, we produce pairs of homeomorphic smooth 4-manifolds Z and Z' which are homotopy equivalent to a genus g surface and which have smooth structures distinguished by several formal properties: Z is diffeomorphic to a genus g knot trace and Z' is not, Z admits a smooth spine and Z' does not admit a piecewise linear spine. When g = 0, Z is geometrically simply connected and Z' is not. In particular our Z' are simple spineless 4-manifolds, which gives an alternative to Levine and Lidman’s recent solution to Problem 4.25 on Kirby’s list. This is joint work in progress with Kyle Hayden.
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http://www.math.harvard.edu/2 Sun, 08 Nov 2019 15:30:00 -0400 CMSA GENERAL RELATIVITY SEMINAR:Pei-Ken Hung (MIT) speaks on <b>Stability of the modified wave map gauge</b> on Nov 08, 19, 10:30 am - 11:30 am in Science Center 530:
Abstract: In this talk, I will discuss a wave equation for one forms in the Schwarzschild spacetime which is the linearization of a modified wave map gauge. The equation behaves like a damped wave equation and we obtain robust estimates. In particular, it allows us to show the stability of the modified wave map equation. This is on-going joint work with S. Brendle.
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http://www.math.harvard.edu/3 Sun, 08 Nov 2019 10:30:00 -0400 MATHEMATICAL PICTURE LANGUAGE SEMINAR:Yichen Huang (MIT) speaks on <b>Renyi entanglement entropy in quantum many-body systems</b> on Nov 12, 19, 3:30 pm in Jefferson 356:
Abstract: We present two concrete examples where the Renyi rather than just the von Neumann entanglement entropy is necessary in order to obtain certain insights into quantum many-body systems. In the first example, we consider systems supporting ballistic information propagation and diffusive transport. It is well known that the linear-in-time growth of the von Neumann entanglement entropy (starting from a product state) is a probe of the former. Perhaps surprisingly, we show that the Renyi entanglement entropy (with Renyi index greater than 1) grows diffusively (i.e., as a square root of time) and is consequently a probe of the latter. In the second example, we study the problem of approximating local properties of a quantum many-body state using matrix product and projected entangled pair representations in one and two dimensions, respectively. We prove that area laws for the Renyi entanglement entropy (with Renyi index less than 1) lead to nontrivial upper bounds on the bond dimension. The bounds only depend on the accuracy of the desired approximation but not the system size. References: https://arxiv.org/abs/1902.00977 https://arxiv.org/abs/1903.10048
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http://www.math.harvard.edu/4 Sun, 12 Nov 2019 15:30:00 -0400 DIFFERENTIAL GEOMETRY SEMINAR:Yu-Shen Lin (BU) speaks on <b>On the SYZ Mirror Symmetry of Log Calabi-Yau Surfaces</b> on Nov 12, 19, 4:15 pm - 5:15 pm in Science Center 507:
Abstract: The Strominger-Yau-Zaslow conjecture predicts the existence of special Lagrangians fibration in Calabi-Yau manifolds near large complex structure limit. The SYZ conjecture has been an important guiding principle for mirror symmetry and many of the implications are verified. In this talk, I will report on the recent progress on the SYZ fibration on certain log Calabi-Yau surfaces using the Lagrangian mean curvature flow and the theory of J-holomorphic curves. As a bi-product, we produce many new special Lagrangian submanifolds. I will also explain the applications in mirror symmetry, including the tropical/holomorphic correspondence for log Calabi-Yau surfaces and a mathematical realization of renormalization process of Hori-Vafa. Part of the talk is based on the joint work with T. Collins, A. Jacob and S-C. Lau, T-J. Lee.
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http://www.math.harvard.edu/5 Sun, 12 Nov 2019 16:15:00 -0400 NUMBER THEORY SEMINAR :Daxin Xu (Caltech) speaks on <b>Bessel F-isocrystals for reductive groups</b> on Nov 13, 19, 3:00 pm in Science Center 507:
Abstract: I will first review the Frobenius structure on the classical Bessel differential equation \begin{displaymath} (x\frac{d}{dx})^{2} u -xu =0, \end{displaymath} whose Frobenius traces are classical Kloosterman sums \begin{displaymath} \textnormal{Kl}(a):= \sum_{xy=a \in \mathbb{F}_p^{\times}} \exp(\frac{2\pi i}{p}(x+y)). \end{displaymath} Recently, there are two generalizations of this story (corresponding to $\GL_2$-case) for reductive groups: one is due to Frenkel and Gross from the viewpoint of the Bessel differential equation; another one, due to Heinloth, Ng\^o and Yun, uses the geometric Langlands correspondence to produce $\ell$-adic sheaves. I will report my joint work with Xinwen Zhu, where we study the $p$-adic aspect of this theory and unify previous two constructions. \end{document}
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http://www.math.harvard.edu/6 Sun, 13 Nov 2019 15:00:00 -0400 OPEN NEIGHBORHOOD SEMINAR:Ellen Eischen (U Oregon) speaks on <b>An Introduction to the Bernoulli Numbers, from Pythagoras to Present</b> on Nov 13, 19, 4:30 pm in Science Center 507:
Abstract: Consider these basic questions: What can we say about finite sums of powers of consecutive whole numbers? What can we say about whole number solutions to polynomial equations? What about factorizations into primes? What about values of the Riemann zeta function? In interesting families of examples — elementary and sophisticated, ancient and modern — "Bernoulli numbers" unify these seemingly unrelated questions. After an introduction to the Bernoulli numbers, we will explore related developments for these intertwined problems, which lead to central challenges in number theory and beyond.
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http://www.math.harvard.edu/7 Sun, 13 Nov 2019 16:30:00 -0400 CMSA QUANTUM MATTER AND QUANTUM FIELD THEORY SEMINAR:Michael Pretko (U Colorado) speaks on <b> Introduction to Fractons</b> on Nov 13, 19, 10:30 am - 12:00 pm in CMSA, 20 Garden St, G02:
Abstract: A fracton is an exotic new type of emergent quasiparticle with restricted mobility. While a single fracton is strictly immobile in isolation, they can often come together to form certain mobile bound states. In this talk, I will give a bird’s-eye overview on the current state of the field of fractons. I will begin with the theoretical formalism for fractons in terms of symmetric tensor gauge theories, which possess unusual higher moment conservation laws. I will then outline some physical realizations of fractons, such as spin models and topological crystalline defects, the latter of which arises through a novel field theory duality. Finally, I will discuss some of the most interesting phenomenology of fracton systems, such as their non-ergodic and gravitational behavior.
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http://www.math.harvard.edu/8 Sun, 13 Nov 2019 10:30:00 -0400 BRANDEIS-HARVARD-MIT-NORTHEASTERN JOINT COLLOQUIUM:Ellen Eischen (U Oregon) speaks on <b>Some congruences and consequences in number theory and beyond</b> on Nov 14, 19, 4:30 pm- 5:30 pm in Science Center Hall E:
Abstract: In the mid-1800s, Kummer observed some striking congruences between certain values of the Riemann zeta function, which have important consequences in algebraic number theory, in particular for unique factorization in certain rings. In spite of its potential, this topic lay mostly dormant for nearly a century until it was revived by Iwasawa in the mid-1950s. Since then, advances in arithmetic geometry and number theory (in particular, for modular forms, certain analytic functions that play a central role in number theory) have enabled substantial extension to congruences in the context of other arithmetically significant data, and this has remained an active area of research. In this talk, I will survey old and new tools for studying such congruences. I will conclude by introducing some unexpected challenges that arise when one tries to take what would seem like immediate next steps beyond the current state of the art.
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http://www.math.harvard.edu/9 Sun, 14 Nov 2019 16:30:00 -0400 CMSA CONDENSED MATTER/MATH SEMINAR:Michael Pretko (U Colorado) speaks on <b>Advances in Fracton Physics: Dualities, Field Theories, and Classification</b> on Nov 14, 19, 11:50 am - 1:00 pm in CMSA, 20 Garden St, G02:
Abstract: In this talk, I will give short informal explanations of three topics in the field of fractons with some interesting mathematical structure. I will begin with a detailed discussion of fracton-elasticity duality, relating the properties of tensor gauge theories to the elastic description of two-dimensional crystals. I will emphasize the role of symmetries in this duality, and also discuss how the duality extends to three dimensions, giving rise to “higher-form” analogues of fracton conservation laws. Next, I will discuss recent advances on the field theory description of fractons, showing how fractons can be obtained by gauging field theories with a “vector” global symmetry. Finally, I will describe recent work towards classifying fracton phases in terms of their fusion theory, which can be described as a module over the group ring of translations.
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http://www.math.harvard.edu/10 Sun, 14 Nov 2019 11:50:00 -0400 NUMBER THEORY SEMINAR :Michele Fornea (IAS) speaks on <b>On the arithmetic of elliptic curves over quintic fields</b> on Dec 04, 19, 3:00 pm in Science Center 507:
Abstract: Bhargava showed that 100% of quintic fields have non-solvable Galois closure. For this reason, the arithmetic of elliptic curves over such fields is beyond the reach of methods based on Heegner points. In this talk we will report on a joint work in progress with Zhaorong Jin about the p-adic variation of Hirzebruch-Zagier cycles. The aim is to establish new instances of the rank zero BSD-conjecture over quintic fields.
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http://www.math.harvard.edu/11 Sun, 04 Dec 2019 15:00:00 -0400