Online Seminar
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04182024
Xueqing Wen: Mirror symmetries for parabolic Hitchin systems, from classical to global II
Abstract: This is the second lecture about SYZ and topological mirror symmetries for parabolic Hitchin systems in type B/C. One of the key step to establish mirror symmetries for Hitchin systems is to understand the generic fibers of the Hitchin map, and the most powerful tool is the so called BNR correspondence which relates Higgs bundles to rank one sheaves on the spectral curves. However, in parabolic case, the spectral curves usually have singularities, which may cause many difficulties but also many interesting phenomena. In this talk, I will focus on the formal disk around the singular point, and define the notion of local Higgs bundles, which can be seem as not only the restriction of parabolic Higgs bundle to the formal disk but also an infinitesimal enhancement of nilpotent Lie algebra elements. The main results in this part are some decomposition theorems of the local Higgs bundles in type B/C. As a byproduct, we found a new geometric interpretation of the Lusztig’s canonical quotient(arXiv:2403.07552).
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04102024
Yaoxiong Wen: Mirror symmetries for parabolic Hitchin systems, from classical to global, 1
Abstract: This is the first of three lectures about Strominger-Yau-Zaslow and topological mirror symmetries for parabolic Hitchin systems. In this talk, we will explain the significance of nilpotent orbit closures in parabolic Hitchin systems, i.e., the classical level. Furthermore, we establish a new perspective of Springer duality for special nilpotent orbits of types B and C. This is based on the joint work with B.Wang and X. Wen (arXiv:2403.07552).
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04072024
Omar Kidwai: Donaldson-Thomas invariants for the Bridgeland-Smith correspondence
Abstract: We describe the calculation of (refined) Donaldson-Thomas invariants for a certain class of 3-Calabi-Yau triangulated categories whose space of stability conditions can be interpreted as a space of quadratic differentials on a Riemann surface. This category is slightly different from the usual one discussed by Bridgeland and Smith, which in particular allows us to recover a nonzero invariant in the case where the quadratic differential has a second-order pole, in agreement with predictions from the physics literature. Based on joint work with N. Williams.
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03152024
Abstract.In this talk, we will review the current study of mirror symmetry for flag varieties. We will focus more on the construction of Landau-Ginzburg model, and discuss a folklore mirror symmetry expectation on the eigenvalues of the first Chern class, using concrete examples of flag varieties of Lie type A. This is based on a work in progress joint with Konstanze Rietsch, Mingzhi Yang and Chi Zhang.
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03082024
Mauro Porta: Homotopy theory of Stokes data and derived moduli
Abstract: Stokes data are the combinatorial counterpart of irregular meromorphic connections in the Riemann-Hilbert correspondence. In dimension 1, they have been studied by Deligne and Malgrange. In higher dimensions the situation is intrinsically more complicated, but the solution of Sabbah's resolution conjecture by Mochizuki and Kedlaya unlocked a much deeper understanding of the Stokes phenomenon. In recent work with Jean-Baptiste Teyssier, we develop a framework to define and study Stokes data with coefficients in any presentable stable infinity category. As an application, we construct a derived moduli stack parametrizing Stokes data in arbitrary dimensions, extending all previously known representability results. One fundamental input is a finiteness theorem for the stratified homotopy types of algebraic and compact real analytic varieties that we obtained in collaboration with Peter Haine and that extends finiteness theorems for the underlying homotopy types of Lefschetz–Whitehead, Łojasiewicz and Hironaka.
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02222024
Xin Wang: Universal structures for Gromov-Witten invariants
Abstract: In this talk, we discuss two kinds of universal structures for higher genus Gromov-Witten invariants of any target varieties. One is certain explicit universal equations from Hodge integrals. Another one is certain basic structures from topological recursion relations on the moduli space of curves. This is based on joint work with Felix Janda.
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02152024
Hiroshi Iritani: Decompositions of quantum cohomology under blowups
Abstract. It is an interesting question how (analytic continuation of) quantum cohomology is related to birational geometry. In particular, we expect that quantum cohomology decomposes in a certain way for extremal contractions. In this talk, I will explain such a decomposition for blowups: quantum cohomology of the blowup of X along Z will decompose into QH(X) and (codim Z-1) copies of QH(Z). The idea of the proof comes from Teleman's conjecture on quantum cohomology of symplectic reductions.
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01112024
Minh-Tam Trinh:Triply-Graded Link Homology and the Hilb-vs-Quot Conjecture
Abstract: Any complex algebraic plane curve singularity defines a link up to isotopy, via its intersection with a surrounding 3-sphere. The Oblomkov–Rasmussen–Shende conjecture describes the a, q, t-graded Khovanov–Rozansky homology of the link in terms of the punctual Hilbert schemes of the singularity. The t = –1 limit is a theorem of Maulik. However, no prior results managed to incorporate all three gradings, except in very simple examples. Oscar Kivinen and I introduce a different kind of Quot scheme, and prove that an analogue of the ORS conjecture for these Quot schemes does hold, with all three gradings, for many singularities of the form y^n = x^d. This motivates a further conjecture about how these new Quot schemes are related to the Hilbert schemes by a motivic substitution of variables. I have proven it for singularities y^3 = x^d with d coprime to 3. This, combined with the previous result, establishes the ORS conjecture for the same class of singularities. If time permits, I will mention how ORS relates to Gromov–Witten theory.
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01042024
Filippo Viviani: On the classification of fine compactified Jacobians of nodal curves
Abstract: We study the problem of characterizing fine compactified Jacobians of nodal curves that can arise as limits of Jacobians of smooth curves. The answer is given in terms of a new class of fine compactified Jacobians, that we call fine V-compactified Jacobians, and that is strictly larger than the class of fine classical compactified Jacobians, as constructed by Oda-Seshadri, Simpson, Caporaso and Esteves. We give several characterizations of fine V-compactified Jacobians. Furthermore, we show that most of the known properties of fine classical compactified Jacobians extend to fine V-compactified Jacobians: the relation to the Neron models of Jacobians, the autoduality property, the Fourier-Mukai equivalences, the perverse filtration of their cohomology, the relation to Mumford models of Jacobians.
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12202023
Changping Fan:Stability manifolds of Kuznetsov components of prime fano threefolds
Abstract.I will present work done in collaboration with Zhiyu Liu and Songtao Kenneth Ma. Let X be a cubic threefold, quartic double solid or Gushel--Mukai threefold, and Ku(X) be its Kuznetsov component. We show that a stability condition on Ku(X) is Serre-invariant if and only if its homological dimension is at most 2. As a corollary, we prove that all Serre-invariant stability conditions on Ku(X) form a contractible connected component of the stability manifold.
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12122023
Alessandro Chiodo:Mumford's formula on the universal Picard stack
I will present work done in collaboration with David Holmes. We construct a derived pushforward of the rth root of the universal line bundle over the Picard stack of genus g prestable curves carrying a line bundle. We prove a number of basic properties, and give a formula in terms of standard tautological generators. After pullback, our formula recovers formulae of Mumford, of Pagani-Ricolfi-van Zelm and my Grothendieck-Riemann-Roch formula for r-spin curves. We apply these constructions to prove a conjecture expressing the double ramification cycle and several generalizations.
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12072023
Rahul Pandharipande:Log intersection theory of the moduli space of curves
Abstract: I will discuss definitions, motivations, and some results concerning the log intersection theory of the moduli space of curves.
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11242023
Ce Ji: Toward a generalization of the Witten conjecture from spectral curve
Abstract: Over decades of development of the Witten conjecture, Many enumerative geometries are proven to be related to integrable hierarchies. Simultaneously, such theories can also be reconstructed from topological recursion, an algorithm producing multi-differential forms from the underlying spectral curve. In this talk, we propose a generalization of the Witten conjecture from spectral curve, which produce descendent potential functions related to certain reductions of (multi-component) KP hierarchy. Proof for genus zero spectral curve with one boundary will be sketched, which can be applied to deduce the rKdV integrability of deformed negative r-spin theory, conjectured by Chidambaram--Garcia-Falide--Giacchetto.
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11162023
Abstract: Crepant resolutions have inspired connections between birational geometry, derived categories, representation theory, and motivic integration. In this talk, we prove that every variety with log-terminal singularities admits a crepant resolution by a smooth stack. We additionally prove a motivic McKay correspondence for stack-theoretic resolutions. Finally, we show how our work naturally leads to a generalization of twisted mapping spaces. No prior knowledge of stacks will be assumed. This is joint work with Jeremy Usatine.