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Time: Mondays, 4:30 - 5:30 pm.
Room: Gilman 219.
Contact: Cesar Bardomiano Martinez and Colton Sandvik.
Faculty Contact: David Gepner, Mee Seong Im, Mikhail Khovanov, Nitu Kitchloo, Emily Riehl.
Weekly Social Events
Department Afternoon Tea: Mondays, Tuesdays, and Thursdays, 4 - 5pm, Krieger 413
Department Wine and Cheese: Wednesdays, 4 - 5pm, Krieger 413
Aug 31: Ishan Levy (Institute for Advanced Study, Princeton, NJ)
Title: What we know about the telescopic categories
Abstract: The stable homotopy category naturally breaks up into indecomposable building blocks, the telescopic (or T(n)-local) stable categories. These pieces are not well understood in general, rather much attention has been given to their computationally tractable close relatives, the K(n)-local categories. I will give an overview of some of the recent advances in understanding the T(n)-local category, explaining what we understand about its difference with the K(n)-local category, some consequences, and some of the new tools we have for working with the T(n)-local category.
This comes from projects joint with Burklund, Carmeli, Clausen, Hahn, Schlank, and Yanovski.
Sept 14: Colton Sandvik (Johns Hopkins University, Baltimore, MD)
Title: Modular Character Sheaves and Monodromic Hecke Categories
Abstract: Character sheaves provide a geometric framework for studying characters of finite groups of Lie type. Introduced by Lusztig in the 1980s, they played a central role in the classification of irreducible complex representations of these groups and in the computation of their characters. Although the definition of character sheaves can be formulated with modular coefficients, much of Lusztig's characteristic-zero machinery breaks down in this setting.
In this talk, I will give a streamlined overview of the theory of character sheaves using modern language from categorical representation theory. From this perspective, categorical traces allow one to study character sheaves through monodromic Hecke categories. I will describe two models for these categories: a geometric categorification in terms of parity sheaves and an algebraic categorification in terms of Soergel bimodules. A key advantage of these models is that they are tightly controlled by p-Kazhdan–Lusztig combinatorics. This paves the way for a combinatorial approach to the study of modular character sheaves.
Sept 21: Cesar Bardomiano Martinez (Johns Hopkins University, Baltimore, MD)
Title: Homotopical first order logic
Abstract: We will see that model categories also have logical information on their own in the following sense: Given any model category, we can associate to it a class of first-order formulas referring to the fibrant objects of the category. For example, the associated language of the category of small categories, equipped with its canonical model structure, coincides with language for categories defined by Blanc and Freyd, whose central feature is that it respects the equivalence principle.
Similarly, the language we associate to a model category respects the appropriate version of the equivalence principle: two homotopically equivalent objects satisfy the same formulas and replacing parameters by homotopically equivalent ones does not change the validity of a formula.
Finally, if M and N are two Quillen equivalent model categories, their associated languages are, suitably, equivalent.
Sept 28: Logan Hyslop (Harvard University, Cambridge, MA)
Title: Geometric Points in Tensor Triangular Geometry
Abstract: In this talk, we will discuss several alternative definitions of homological spectra. Based on forthcoming work with Tobias Barthel and Maxime Ramzi, we will discuss how to realize the underlying set of the homological spectrum as the constructible spectrum in 2-Ring^{rig} when we are rational, together with the interpretation of the image of the comparison map from the constructible spectrum to the Balmer spectrum in general.
Time permitting, we will discuss how to realize the homological spectrum as an “E_n-constructible spectrum” in general, and how this can be used to show that any tt-category with an E_n refinement (n>=4) has the property that points of its homological spectrum are “geometric” in the category of rigid E\_{n-1}-2-Rings.
Oct 5: Kush Singhal (Harvard University, Cambridge, MA)
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Oct 13 (Tuesday, Pretalk, 4:30 - 5:45 pm, Hodson 210): Dahye Cho (G-LAMP AI-Bio Convergence Research Institute, Soongsil University, Seoul, Korea), an introduction to symplectic cohomology and mirror symmetry
Title: Introduction to Symplectic Cohomology and an Example of Closed-string Mirror Symmetry for Dimer duality
Abstract: We introduce symplectic cohomology and its application to the criterion for the uniruledness of affine varieties. Symplectic cohomology can be computed via the Hochschild cohomology of the wrapped Fukaya category and the category of matrix factorizations of mirror Landau-Ginzburg model. We briefly explain an example of such a computation in the case of dimer duality. The result on the closed-string mirror of dimer duality is based on joint work with Hansol Hong, Hyeongjun Jin, and Sangwook Lee.
Oct 14 (Wednesday, 3:00 - 4:00 pm, Latrobe 120): Dahye Cho (G-LAMP AI-Bio Convergence Research Institute, Soongsil University, Seoul, Korea)
Title: G-twisted Jacobian Algebra of Non-isolated Hypersurface Singularities and G-twisted Orbifold Cup Products.
Abstract: We continue our discussion of the category of matrix factorizations of (X,W), but with a group G acting on it. We call the triple (X,W,G), a Landau-Ginzburg orbifold. After introducing G-twisted W-curved Hochschild cohomology, we cover the extended definition of G-twisted orbifold cup products on it in the case of non-isolated hypersurface singularities (W,G). We provide properties of the G-twisted orbifold cup products and their meaning in homological mirror symmetry and symplectic cohomology. This is based on joint work in progress with Sangwook Lee.
Oct 19: Luke Conners (Universität Zürich, Institute of Mathematics, Zürich, Switzerland)
Title: HOMFLY Homology for Oriented Tangles
Abstract: We introduce an extension of Khovanov and Rozansky's categorification of the HOMFLY polynomial to oriented tangles. Unlike previous such extensions, ours is invariant under the full suite of oriented Reidemeister moves, including the "misoriented" Reidemeister IIb. For oriented links L, this removes the preliminary step of presenting L as a braid closure. Our construction admits an interpretation in terms of Bar-Natan's "canopoly" framework, allowing for local computations. Time permitting, we will briefly discuss potential consequences for functoriality under (non-compact) oriented tangle cobordisms and extracting smooth 4-manifold invariants. Based on forthcoming joint work with William Ballinger.
Oct 26: Sam Miller (University of Georgia, Athens, GA)
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Nov 9: Bowen Yang (Center of Mathematical Sciences and Applications, Harvard University, Cambridge, MA)
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Nov 16: Milton Lin (NSF-Simons National Institute for Theory and Mathematics in Biology, Chicago, IL)
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Dec 7:
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