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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)
Title: Iwasawa Theory for THH
Abstract: For applications to Iwasawa theory, I explain a computation of the inverse limit over transfer maps of motivically filtered THH as one goes up the cyclotomic tower. This computation has connections to height 1 cyclotomic extensions and (conjecturally) E_\infty complex orientations of KU.
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 (Hodson 203): 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 (Hodson 203): Sam Miller (University of Georgia, Athens, GA)
Title: Permutation twisted cohomology
Abstract: Let G be a finite group and k a field of positive characteristic p. The so-called "derived category of permutation kG-modules" perhaps surprisingly controls the (usually wild) representation theory of G over k. Moreover, this category admits incarnations in equivariant topology and motivic geometry via cohomological Mackey functors, Artin motives, or modules in equivariant spectra. Balmer—Gallauer recently deduced the tensor-triangular geometry of this category, which as a set, is built up of p-local cohomological support varieties. In this talk, after a brief introduction to tensor-triangular geometry, we'll describe some extended "permutation twisted cohomology" theories which allow us to describe how these support varieties are glued together to form a (Dirac) scheme for any group.
Nov 2 (3 - 4pm): Mengwei Hu (Johns Hopkins University, Baltimore, MD)
Title: Codimension-4 Degenerations of Nilpotent Orbit Closures and Symmetric Pairs
Abstract: Nilpotent orbits in gl_n(C) are classified by partitions, and the singularities of their closures can be studied using transverse Slodowy slices. Classical work of Kraft and Procesi describes the singularities arising from minimal degenerations of nilpotent orbit closures through simultaneous row and column removals. In this talk, we study codimension-4 degenerations, where intermediate orbits may occur, through the realization of nilpotent Slodowy slices as Nakajima quiver varieties. We give explicit determinantal-type presentations for the nilpotent Slodowy slices and describe their intersections with the symmetric space for the symmetric pair (gl_p+q,gl_p+gl_q). This study is motivated by the geometric classification of Harish-Chandra modules over quantizations of nilpotent orbits.
Nov 2 (Hodson 203): Yuchen Wu (University of California, San Diego, CA)
Title: Motivic stable stems of fields and number rings
Abstract: Motivic stable stems of fields and number rings Abstract: For any commutative ring $R$, the motivic stable homotopy groups (or motivic stable stems) $pi_{**}(1_R)$ are an intricate mixture of the classical stable homotopy groups and the arithmetic of $R$ itself. Recently, using Galois approximation methods, Bachmann–Burklund–Xu made major progress in determining these groups, leading to a theoretical description of the $(ell,eta)$-completed motivic stable stems of all fields $k$ with $mathrm{char}(k) eq ell$ for every odd prime $ell$, and at $ell = 2$ for all such fields containing a square root of $-1$, together with an efficient method for computing the motivic stable stems of all finite fields from those of $mathbb{C}$.
In this talk I will survey these results, and discuss work in progress, joint with Zhouli Xu, towards two extensions: a theoretical description of the motivic stable stems of all fields $k$ with $mathrm{char}(k) eq 2$ at the prime $2$, including those without a square root of $-1$, and a categorical assembly theorem which leads to an effective method for computing the motivic stable stems of all local/global fields and their rings of integers from those of $mathbb{C}$ and $mathbb{R}$, illustrated by explicit examples.
Nov 9 (Hodson 203): Bowen Yang (Center of Mathematical Sciences and Applications, Harvard University, Cambridge, MA)
Title: Quantum Cellular Automata as Negative K-theory
Abstract: Quantum cellular automata (QCA) are locality-preserving automorphisms of quantum lattice systems. Their classification groups, the QCA groups, have been studied both for their intrinsic interest and for their close connection to topological phases of matter. In joint work with Mattie Ji, we construct spaces of QCA over an arbitrary commutative ring R, extending the usual theory over C. As the lattice dimension varies, these spaces assemble into a nonconnective spectrum. Its negative homotopy groups recover the QCA groups, and its nonnegative homotopy groups recover the algebraic K-theory of Azumaya R-algebras. So the QCA spectrum is a natural nonconnective delooping of this K-theory, with the QCA groups playing the role of its negative K-groups. I will end with a recent result: the group π₋₂ of the spectrum, which classifies three-dimensional QCA, is not finitely generated.
Nov 16 (Hodson 203): Milton Lin (NSF-Simons National Institute for Theory and Mathematics in Biology, Chicago, IL)
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Nov 30 (Hodson 203): Jackson Van Dyke (Technical University of Munich, Munich, Germany)
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