Anatoly Preygel

The bulk of the research below was carried out while I was either an NSF postdoctoral fellow at UC Berkeley, or earlier while I was a graduate student at MIT working with Jacob Lurie.

Contact Me:
email: anatoly.preygel at gmail.com

Preprints
 Mapping stacks and categorical notions of properness. With D. Halpern-Leistner One fundamental consequence of a scheme being proper is that there is an algebraic space classifying maps from it to any other finite type scheme, and this result has been extended to proper stacks. We observe, however, that it also holds for many examples where the source is a geometric stack, such as a global quotient. In our investigation, we are lead naturally to certain properties of the derived category of a stack which guarantee that the mapping stack from it to any geometric finite type stack is algebraic. We develop methods for establishing these properties in a large class of examples. Along the way, we introduce a notion of projective morphism of algebraic stacks, and prove strong h-descent results which hold in the setting of derived algebraic geometry but not in classical algebraic geometry. (arXiv)(PDF) Some remarks on shifted symplectic structures on non-compact mapping spaces We show existence of shifted symplectic structures on some stacks closedly related to mapping stacks from non-proper sources. The motivating example is the moduli stack of stable pairs on noncompact threefolds. (PDF) Thesis A copy of my thesis as turned in to MIT (with a few typos fixed..). This is a placeholder while I split, improve, and re-organize various parts of it. It contains a slightly improved version of the "Thom-Sebastiani & Duality for Matrix Factorizations" paper below, as well as an early version of what will be the "Adjoint Actions for $E_2$-algebras and the Hochschild Package of Matrix Factorizations" paper below. (PDF) Thom-Sebastiani & Duality for Matrix Factorizations The derived category of a hypersurface has an action by "cohomology operations" $k[t]$, $\deg t = -2$, underlying the $2$-periodic structure on its category of singularities (as matrix factorizations). We prove a Thom-Sebastiani type Theorem, identifying the $k[t]$-linear tensor products of these dg categories with coherent complexes on the zero locus of the sum potential on the product (with a support condition), and identify the dg category of colimit-preserving $k[t]$-linear functors between Ind-completions with Ind-coherent complexes on the zero locus of the difference potential (with a support condition). These results imply the analogous statements for the $2$-periodic dg categories of matrix factorizations. Some applications include: we refine and establish the expected computation of $2$-periodic Hochschild invariants of matrix factorizations; we show that the category of matrix factorizations is smooth, and is proper when the critical locus is proper; we show how Calabi-Yau structures on matrix factorizations arise from volume forms on the total space; we establish a version of Knörrer Periodicity for eliminating metabolic quadratic bundles over a base. (arXiv)(PDF) Adjoint Actions for $E_2$-algebras and the Hochschild Package of Matrix Factorizations We compute the $E_2$-algebra structure on $2$-periodic Hochschild cohomology of matrix factorizations purely in terms of the $E_2$-algebra structure on Hocschild cohomology of perfect complexes. More generally, we do this for the full homotopy calculus'' structures. This allows us to leverage the formality theorem for the latter to obtain a formalilty-type theorem for the former. Our two key tools are: first, the perspective on matrix factorizations as arising from a Lie algebra action on the category of perfect complexes; second, a description of the resulting Lie algebra action on Hochschild invariants as an $E_2$ "adjoint action." (Coming soon.) Algebraic cobordism of varieties with G-bundles We partially address a speculation in "Algebraic cobordism for bundles on varieties" by Lee/Pandharipande. Namely, we show that for a reductive algebraic group $\mathbf{G}$, the geometrically defined $\omega_{*,\mathbf{G}}(X)$ is what one might expect it to be, at least after inverting the finitely many "torsion primes" for $\mathbf{G}$. (arXiv)(PDF)

Publications
 A spectral incarnation of affine character sheaves. With D. Ben-Zvi and D. Nadler. To appear in Composition Math. Abstract: We present a Langlands dual realization of the putative category of affine character sheaves. Namely, we calculate the categorical center and trace (also known as the Drinfeld center and trace, or categorical Hochschild cohomology and homology) of the affine Hecke category starting from its spectral presentation. The resulting categories comprise coherent sheaves on the commuting stack of local systems on the two-torus satisfying prescribed support conditions, in particular singular support conditions as appear in recent advances in the Geometric Langlands program. The key technical tools in our arguments are: a new descent theory for coherent sheaves or D-modules with prescribed singular support; and the theory of integral transforms for coherent sheaves developed in the companion paper [BNP] below. (arXiv)(PDF) Integral transforms for coherent sheaves. With D. Ben-Zvi and D. Nadler. To appear in J. Eur. Math. Soc. Abstract: The theory of integral, or Fourier-Mukai, transforms between derived categories of sheaves is a well established tool in noncommutative algebraic geometry. General "representation theorems" identify all reasonable linear functors between categories of perfect complexes (or their "large" version, quasi-coherent sheaves) on schemes and stacks over some fixed base with integral kernels in the form of sheaves (of the same nature) on the fiber product. However, for many applications in mirror symmetry and geometric representation theory one is interested instead in the bounded derived category of coherent sheaves (or its "large" version, ind-coherent sheaves), which differs from perfect complexes (and quasi-coherent sheaves) once the underlying variety is singular. In this paper, we give general representation theorems for linear functors between categories of coherent sheaves over a base in terms of integral kernels on the fiber product. Namely, we identify coherent kernels with functors taking perfect complexes to coherent sheaves, and kernels which are coherent relative to the source with functors taking all coherent sheaves to coherent sheaves. The proofs rely on key aspects of the "functional analysis" of derived categories, namely the distinction between small and large categories and its measurement using t-structures. These are used in particular to correct the failure of integral transforms on Ind-coherent sheaves to correspond to such sheaves on a fiber product. The results are applied in a companion paper to the representation theory of the affine Hecke category, identifying affine character sheaves with the spectral geometric Langlands category in genus one. (arXiv)(PDF) Ind-coherent complexes on loop spaces and connections Contemporary Mathematics Abstract: The goal of this note is to describe a functorial relationship between $D$-modules on $X$ and quasi-coherent sheaves on the derived loop space $LX$, for $X$ an algebraic space of finite type over a char. $0$ field $k$. This is a generalization of work of Toen/Vezzosi and Nadler/Ben-Zvi to the case where $X$ is not assumed smooth, with the added motivation that it is better compatible with the usual functoriality on $D$-modules. As an application we prove a "Borel-Moore" version of a Theorem of Feigin/Tsygan, computing the periodic cyclic homology of the dg-category of bounded coherent complexes on $X$. The present document is halfway between a real paper and an announcement: we try to state all the results, to survey the broad ideas that go into them, and to indicate the key pieces and steps in the proofs. (Version updated 2017-01-28: Fix a potentially confusing typo in the statement of Theorems 5.1.3 and 5.3.4.) (PDF) Dynamics of the p-adic shift and applications. With J. Kingsbery, A. Levin, and C. Silva. Discrete Contin. Dyn. Syst. 30 (2011), no. 1, 209–218. (arXiv) On measure-preserving $C^1$ transformations of compact-open subsets of non-archimedean local fields. With J. Kingsbery, A. Levin, and C. Silva. Trans. Amer. Math. Soc. 361 (2009), 61-85. (arXiv)(PDF) Measurable dynamics of maps on profinite groups. With J. Kingsbery, A. Levin, and C. Silva. Indag. Math. (N.S.) 18 (2007), no. 4, 561-581. (arXiv)(PDF)

Some Mathematical Notes/Talks
 Matrix factorizations via group actions on categories, etc. (String Math 2011) Gives the description of the 2-periodic dg-category of matrix factorizations as a "Tate construction" for an action of $S^1$ (or its formal completion $B\hat{\mathbb{G}}_a$) on perfect complexes on the total space. Talks about how to use this to leverage known results about Hochschild invariants of perfect complexes, together with the "$E_2$ adjoint action," to understand e.g., the homotopy Gerstenhaber structure on Hochschild cohomology of matrix factorizations. (PDF) Silly Hochschild Computation Short note relating the Hochschild invariants of matrix factorizations for a non-degenerate quadratic bundle, to the Hochschild invariants of perfect complexes on the critical locus (=the base). (N.B. This computation is just as chain complexes without any higher structure.) (PDF) Koszul Fun Short note making sense of the funny statement that "$\FF_p \stackrel{L}{\otimes}_{\mathcal{E}} \FF_p = \QQ_p/\ZZ_p$" for $\mathcal{E} = \mathrm{RHom}_{\ZZ_{(p)}}(\FF_p, \FF_p)$. (PDF) Hochschild Homology Notes from a talk in the Fall 2009-2010 juvitop seminar on THH. Briefly talks about the $S^1$-action and the connection to the differential picture. (PDF) Intro to Algebraic Stacks Background talk in the 2009 geometric representation theory graduate seminar. Some motivation, definitions, and computations with BG and X//G. (PDF) Vanishing theorems and the Hodge to de Rham spectral sequence Notes from an expository talk I gave at STAGE in March 2008. The first half of a two-part talk on the Deligne-Illusie proof of Kodaira vanishing, and degeneration of the Hodge to de Rham spectral sequence, via characteristic p methods. (The second half of the talk was given by Jay Pottharst. Here are his notes.) (PDF)

CV
Education:
• Massachusetts Institute of Technology, Cambridge, MA.
Ph.D. in Mathematics, 2012.
• Harvard University, Cambridge, MA.
A.B., summa cum laude and highest honors in Mathematics, 2007.
Awards and Honors:
• 2012-2015: NSF Mathematical Sciences Postdoctoral Research Fellowship.
• 2009-2012: NSF Graduate Research Fellowship.
• 2007-2008: MIT Presidential Fellowship & Norman Levinson Fellowship.
• 2007: Wister Prize, Harvard Math Dept. award for senior with the highest record in field.
• 2006: Phi Beta Kappa, Harvard University.
• 2003: 3rd Place, Intel Science Talent Search.
Teaching Experience:
• Spring 2010: Teaching Assistant, MIT.
Lead weekly recitation of 18.06 (=linear algebra).
• 2004-2007: Course Assistant, Harvard University.
Lead weekly sections and graded problem sets for Math 25a, 25b(=Honors Linear Algebra and Real Analysis I,II), 123(=Theory of Rings and Fields), 272a(=Graduate Algebraic Topology).
Some Talks:
Other Notable Talks (Summary):
Conferences/Workshops Attended:
• CATS4: Higher categorical structures and their interactions with algebraic geometry, algebraic topology, and algebra (2012)
• Northwestern spring school on algebraic microlocal analysis (2012)
• HMS and Category Theory in Split (2011)
• MIT-RTG Geometry workshop: Fukaya categories via microlocal sheaf theory (2011)
• String Math 2011 (2011)
• Moduli spaces and moduli stacks (2011)
• Talbot workshop: Non-abelian Hodge theory (2011)
• Operator algebras and conformal field theory workshop (2010)
• West coast algebraic topology summer school: Homotopy theory of moduli spaces (2010)
• Talbot workshop: Freed-Hopkins-Teleman (2010)
• "Mayday" (2009)
• Northwestern Topological Field Theories (2009)
• Talbot workshop: Fukaya Categories (2009)
• CIRM Thematic School & Conference on Operads (2009)
• IAS/PCMI Summer Program on Analytic and Algebraic Geometry (2008)
• Homotopical Group Theory and Topological Algebraic Geometry Workshop & Conference (2008)
• Talbot workshop: Affine Lie algebras and chiral structures (2008)
Other Activities:
• 2011: Co-organizing (w/Thomas Bitoun) seminar on irregular flat connections, MIT.
Learning seminar on irregular flat connection.
• 2011: Co-organized (w/Sheel Ganatra) juvitop graduate student seminar, MIT.
Semester long graduate student "pre-talbot" seminar on non-abelian Hodge theory.
• 2010: Co-organized (w/Nick Rozenblyum) juvitop graduate student seminar, MIT .
Semester long graduate student "pre-talbot" seminar on twisted K-theory and loop group representations.
• 2009: Co-organized (w/Nick Rozenblyum) juvitop graduate student seminar, MIT.
Semester long graduate student seminar on THH.
• 2007: Undergraduate Honors Thesis, Harvard University.
Degrees of Severi Varieties.