59 research outputs found
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Microlocal sheaves and mirror symmetry
No full textIn this thesis, based on the work completed by the author during his time in graduate school, we explain various ways in which microlocal sheaf methods in symplectic geometry can be used to prove homological mirror symmetry. We explain how tropical methods originally developed by Mikhalkin can be used to prove homological mirror symmetry for any hypersurface in (C*)^n, and we also present a proof of homological mirror symmetry in the case of multiplicative hypertoric varieties, emphasizing the features which we expect will prove common to all K-theoretic Coulomb branches
Sustaining Phoenix : Valley of the Sun beyond desert survival
No full textabstract: This report examines the sustainability of Phoenix, a desert city that some critics say is at risk due to extreme climate, water supply, growth demands and politics
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A Morse-theoretic approach to family Floer homology
No full textThis dissertation introduces a new model of the family Floer approach to Kontsevich's homological mirror symmetry conjecture constructed via Morse theoretic technology. Homological mirror symmetry (HMS) asserts a derived equivalence between the Fukaya category of a symplectic manifold X and the category of coherent sheaves on its mirror Xˇ. On the other hand, the family Floer program gives a modern reinterpretation of the construction of a Strominger--Yau--Zaslow (SYZ) mirror, and this mirror space typically comes equipped with a functor from the Fukaya category of X into coherent sheaves on Xˇ which can be used to prove HMS as asserted.
In order to give an analogous presentation of this story, we define the Morse--Fukaya algebra A associated to a suitable class of SYZ fibrations π : X → B; this is a curved A∞-algebra determined by a Morse function on the total space X, taking coefficients in analytic functions on its rigid analytic mirror space. For an appropriate choice of Morse function, A can be understood as a (suitably deformed) algebra of Čech cochains valued in polyvector fields on Xˇ. We then construct an A∞-functor from (a suitable subcategory of) the Fukaya category of X into the category mod-A of modules over A implementing the expected correspondence. Along the way we record comparison maps which together witness invariance of our constructions under a change of auxiliary technical choices.Mathematic
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An Extension of the Kazhdan-Lusztig Equivalence
No full textWe establish an equivalence between the DG category of Iwahori–integrable affine Lie algebra representations and the DG category of representations of the “mixed” quantum group, thus confirming a conjecture made by D. Gaitsgory. This result is a tamely ramified version of an equivalence established by D. Kazhdan and G. Lusztig in the spherically integrable setting. Our proof, which utilizes the notion of factorization algebras introduced by A. Beilinson and V. Drinfeld, is independent from the original Kazhdan–Lusztig equivalence. Along the way we establish several results in the quantum local geometric Langlands program.
The work reported here was done jointly with L. Chen
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Nearby Cycles and Dualities in Geometric Langlands Program
No full textIn this thesis, we study nearby cycles on certain Vinberg-style degenerations in the geometric Langlands program. We relate them to various exotic dualities in this field, such as the (local and global) geometric second adjointness and the miraculous duality. We also prove the Deligne-Lusztig duality for automorphic sheaves, which was conjectured by Drinfeld-Wang and Gaitsgory
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Equivalence of Hecke Categories with Deeper Level Structures
No full textLet G be a reductive group of classical type. We study representations of the loop
group G((t)) and geometrizations of Hecke algebras. The representations are generalizations
of epipelagic representations. They have positive depth and appear in the representation
induced from some level group (J, ψ). The associated Hecke category is the category of
mixed Ql-sheaves on G((t)) with equivariant conditions, and we prove that this monoidal
category is equivalent to an affine Hecke category of a smaller group H (not necessarily split).
This equivalence relates certain positive depth representations of G((t)) and tamely ramified
representations of H. Therefore, it has potential applications to local geometric Langlands
program in a wildly ramified setting. The proof relies on the theory of Soergel bimodules
and a reduction step using hyperbolic localization. It can be potentially generalized to (J, ψ)
defined using Yu’s data
Local mirror symmetry via SYZ
No full textIn this note, we explain how mirror symmetry for basic local models in the Gross-Siebert program can be understood through the non-toric blowup construction described by Gross-Hacking-Keel. This is part of a program to understand the symplectic geometry of affine cluster varieties through their SYZ fibrations.14 pages. v2: Added details on polarizations, pushout of categories, and Liouville homotop
Mirror symmetry for Berglund-H\"ubsch Milnor fibers
No full textWe explain how to calculate the Fukaya category of the Milnor fiber of a
Berglund-H\"ubsch invertible polynomial, mostly proving a conjecture of
Yank{\i} Lekili and Kazushi Ueda on homological mirror symmetry. As usual, we
begin by calculating the "very affine" Fukaya category; afterwards, we deform
it, generalizing an earlier calculation of David Nadler. The main step of our
calculation may be understood as determining a certain canonical extension of a
perverse schober
The biggest estate on earth
No full textAn Author event presented by The Friends of the University of Adelaide Library
Sectoral allocation by gender of Latin American workers over the liberalization period of the 1990s
No full textThe recent restructuring of Latin American economies has renewed interest in the effects of trade liberalization, on labor markets, and on the gender division of labor. The author does not attempt to establish casuality between economic reforms, and the types of jobs that men and women hold. Instead, she provides a detailed description of the trends in male, and female formal, and informal sector participation during the economic reform period in Argentina, Brazil, and Costa Rica. The author first compares the gender composition of the formal, informal wage, and self-employment sectors in a year before reforms (1988 for Argentina, 1989 for Brazil, and Costa Rica), and a year after reforms implementation (1997 for Argentina, 1995 for Brazil and Costa Rica). Although women continued to be more likely than men to work in the informal wage sector, there is no trend of"masculinization"or"feminization"of the informal sector, or any other. Instead, in Argentina men have overtaken women as the most prevalent workers in the informal wage sector, while in Brazil, the opposite has occurred (as men move into self-employment). In Costa Rica there have been no statistical, observable changes. The author then considers the distribution across sectors within each gender group, to identify whether men, and women are more likely to select different sectors in the post-reform period relative to the pre-reform period. Among both men, and women in all three countries (except Brazilian men), workers have become more likely to hold informal wage jobs, and less likely to hold formal sector jobs. Trends in human capital accumulation explain these changes for both men, and women, while changes in gender roles, primarily in homecare and marriage, do not seem to have an effect.Health Monitoring&Evaluation,Labor Policies,Population&Development,Public Health Promotion,Environmental Economics&Policies,Health Monitoring&Evaluation,Environmental Economics&Policies,Population&Development,Banks&Banking Reform,Work&Working Conditions
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