David Gepner

I study algebraic topology; more specifically, homotopy theory and its interactions with algebraic geometry, algebraic K-theory, and higher category theory.

- On the motivic
spectra representing algebraic cobordism and algebraic
K-theory, with V. Snaith. Documenta Mathematica, Vol. 14
(2009), 359-396.

We elucidate the structure of the motivic spectra representing Voevodsky's algebraic cobordism and Weibel's homotopy-invariant algebraic K-theory. Our main result is that, over a base scheme S, inverting the canonical Bott classes in BGL = colim_n BGL_n and BG_m we obtain MGL and KGL, respectively, generalizing and reproving a famous theorem of Snaith in topology. As corollaries we deduce that both are canonically E_\infty motivic spectra with a universal "mapping out of" property, and as a further application we construct a motivic Conner-Floyd isomorphism.

- Twists of K-theory
and TMF, with M. Ando and A. Blumberg, in "Superstrings,
Geometry, Topology, and C*-algebras", Proceedings of Symposia in
Pure Mathematics, Vol. 81, AMS, Providence, 2010.

We study twisted generalized cohomology from the point of view of stable homotopy theory and infinity-category theory. In particular, we explain the relationship to twisted K-theory via Frebholm bundles and show that, in contrast to the case of K-theory, elliptic cohomology admits twists by degree four cohomology classes, and more generally by maps to the four-stage Postnikov system BO<0...4>. We also discuss Poincare duality and Umkehr maps in this setting.

- A universal
characterization of higher algebraic K-theory, with A.
Blumberg and G. Tabuada. Geometry and Topology 17 (2013),
733-838.

We show that (nonconnective) algebraic K-theory, viewed as a functor from small stable infinity-categories to spectra, is the universal additive (respectively, localizing) invariant. To prove these results we construct and study varieties of noncommutative motives over the sphere, and show that algebraic K-theory is the functor corepresented by the motive of the sphere. The technical backbone of the paper relies on a comparison which exhibits the infinity-category of small stable infinity-categories as the localization of spectrally-enriched categories obtained by inverting Morita equivalences. As an application we obtain a complete classification of trace maps, i.e. natural transformations from K-theory to THH and TC (or other pro-localizing theories), as well as a canonical construction of the cyclotomic trace.

- An
infinity-categorical approach to R-line bundles R-module Thom
spectra, and twisted R-homology, with M. Ando, A.
Blumberg, M.J. Hopkins and C. Rezk. Journal of Topology (2013).

We develop a generalization of the theory of Thom spectra using the language of infinity categories. This language allows us to develop a clean treatment of parametrized homotopy theory which avoids the pathology encountered in model-dependent settings. For any associative ring spectrum R, we associate a Thom spectrum to a map from a space X to the classifying space BGL_1(R) of the A_infinity space GL_1(R), which classifies local systems of free rank one R-modules (a.k.a. R-line bundles). We use our R-module Thom spectrum to define the twisted R-(co)homology of an R-line bundle over X and to obtain a generalized theory of orientations in this context. We conclude with an abstract characterization the Thom spectrum which allows us to compare to the classical case (i.e. for R the sphere spectrum).

- Actions of
Eilenberg-MacLane spaces on K-theory spectra and uniqueness of
twisted K-theory, with B. Antieau and J. Gómez.
Trans. Amer. Math. Soc. 366 (2014), no. 7, 3631-3648.

We classify homotopy coherently associative actions of K(Z,2) on the complex K-theory spectrum KU, as well as the analogue for real K-theory KO. We prove the uniqueness of twisted K-theory in both the real and complex cases using the computation of the K-theories of Eilenberg-MacLane spaces due to Anderson and Hodgkin. As an application of our method, we give some vanishing results for actions of Eilenberg-MacLane spaces on K-theory spectra.

- Units of ring
spectra and Thom spectra via rigid infinite loop space theory,
with M. Ando, A. Blumberg, M.J. Hopkins and C. Rezk, 2009.
Journal of Topology (2014).

In this paper we develop the basic theory of generalized Thom spectra using an explicit point-set version of loop space theory involving spaces and spectra with actions of the linear isometries operad. We extend the theory of Thom spectra and the associated obstruction theory for orientations in order to support the construction of the string orientation of tmf, the spectrum of topological modular forms. We also develop the analogous theory of Thom spectra and orientations for associative ring spectra. Our work is based on a new model for the Thom spectrum as a derived smash product.

- Uniqueness of the
multiplicative cyclotomic trace, with A. Blumberg and G.
Tabuada. Advances in Mathematics 260 (2014), 191-232.

Making use of the theory of noncommutative motives, we characterize the topological Dennis trace map as the unique multiplicative natural transformation from algebraic K-theory to THH and the cyclotomic trace map as the unique multiplicative lift through TC. Moreover, we prove that the space of operadic structures on algebraic K-theory is contractible and that the algebraic K-theory functor from small stable infinity categories to spectra is lax symmetric monoidal, so that E_n-ring spectra give rise to E_{n-1}-ring algebraic K-theory spectra. The key technical ingredient is a version of a multiplicative Morita theory.

- Brauer groups and
étale cohomology in derived algebraic geometry, with B.
Antieau. Geometry and Topology 18 (2014), no. 2, 1149-1244.

In this paper we study Azumaya algebras and Brauer groups in derived algebraic geometry. We establish various fundamental facts about Brauer groups in this setting, and we provide a computational tool which we use to compute the Brauer group in several examples. In particular, we show that the Brauer group of the sphere spectrum vanishes, and we use this to prove two uniqueness theorems for the stable homotopy category. Our key technical results include the local geometricity, in the sense of Artin infinity-stacks, of the moduli space of perfect modules over a smooth and proper ring spectrum, the étale local triviality of Azumaya algebras over connective derived schemes, and a local to global principle for the algebraicity of stacks of stable categories.

- Enriched
infinity-categories via nonsymmetric infinity-operads,
with R. Haugseng. Advances in Mathematics 279 (2015), 575-716.

We set up a general theory of weak or homotopy-coherent enrichment in an arbitrary monoidal infinity-category. Our theory has many desirable properties which render it useful even when the theory in question admits a model (as sometimes occurs in examples of interest, e.g. differential-graded categories, spectral categories, (infinity, n)-categories, etc.). We construct our theory via a non-symmetric version of Lurie's infinity-operads, and develop their basic theory, following Lurie's treatment of symmetric infinity-operads. Lastly, we present some applications, including the identification of algebras as a coreflective subcategory of pointed enriched infinity-categories, and a proof of a strong version of the Baez-Dolan stabilization hypothesis.

- Universality of
multiplicative infinite loop space machines, with M. Groth
and T. Nikolaus. Algebraic & Geometric Topology 15 (2015)
3107--3153.

We establish a canonical and unique tensor product for commutative monoids and groups in an infinity-category which generalizes the ordinary tensor product of abelian groups. Using this tensor product we show that E_n-(semi)ring objects in give rise to E_n-ring spectrum objects by group completion. In the case of the infinity-category of spaces this produces a multiplicative infinite loop space machine which can be applied to algebraic K-theory of rings and connective ring spectra. A central theme is the stability of algebraic structures under basechange; for example, Ring(D\otimes C)=Ring(D)\otimes C. We conclude with a treatment of these algebraic structures from the perspective of Lawvere algebraic theories.

- K-theory of
endomorphisms via noncommutative motives, with A. Blumberg
and G. Tabuada. Trans. Amer. Math. Soc. 368 (2016) 1435--1465.

The K-theory of endomorphisms KEnd, as well as the closely-related K-theory of automorphisms KAut, are refinements of algebraic K-theory which capture important arithmetic information of rings and schemes. In this paper we prove a conjecture of Almkvist from the 1970's on the classification of natural transformations of the KEnd functor: specifically, we show that the set of such transformations bijects with the product of the integers with the multiplicative group of rational functions with unit constant term. Moreover, KEnd itself splits as a copy of algebraic K-theory plus a new spectrum-valued invariant which refines the rational Witt vectors (i.e. the dense subring of the Witt vectors consisting of those power series which are ratios of polynomial functions with unit constant term). - Univalence in locally
cartesian closed infinity-categories, with J. Kock. To
appear in Forum Mathematicum.

We develop the basic theory of locally cartesian localizations of presentable locally cartesian closed infinity-categories and show that univalent families, in the sense of Voevodsky, form a poset equivalent to the poset of bounded local classes, in the sense of Lurie. In particular, infinity-topoi admit a hierarchy of "universal" univalent families and n-topoi admit univalent families classifying (n-2)-truncated maps. We show that univalent families are preserved and detected by right adjoints to locally cartesian localizations and use this to exhibit canonical univalent families in infinity-quasitopoi. We also exhibit some exotic examples of univalent families, illustrating that a univalent family in an n-topos need not be (n-2)-truncated. Lastly, we show that any presentable locally cartesian closed infinity-category is modeled by a combinatorial type-theoretic model category, and conversely that the infinity-category underlying a combinatorial type-theoretic model category is presentable and locally cartesian closed.

- On localization
sequences in the algebraic K-theory of ring spectra, with
B. Antieau and T. Barthel. Preprint, 2014, submitted for
publication.

In this paper we identify the fiber term in the algebraic K-theory of a localization of ring spectra as the algebraic K-theory of the endomorphism algebra spectrum of a Koszul-type complex. The original theorem along these lines is Quillen's famous localization sequence relating the algebraic K-theory of a number field to the algebraic K-theory of its ring of integers and residue fields, was generalized by Blumberg-Mandell (using a strong form a dévissage) to a localization sequence relating the algebraic K-theory of the integers with that of connective and periodic topological K-theory. This lead Rognes to expect the existence of analogous localization sequences at all chromatic heights; however, we show using trace computations that this fails for all higher heights and give a conceptual explanation as to why. - Parametrized spectra,
multiplicative Thom spectra, and the twisted Umkehr map,
with M. Ando and A. Blumberg. Preprint, 2015, submitted for
publication.

- Lax colimits and
free fibrations in infinity-categories, with R. Haugseng
and T. Nikolaus. Preprint, 2015, submitted for publication.

We define and discuss lax and weighted colimits of diagrams in infinity-categories and show that the cocartesian fibration associated to a functor to the infinity-category of infinity-categories is given by its lax colimit. A key ingredient, of independent interest, is a simple characterization of the free cartesian fibration associated to a a functor. As an application of these results, we prove that lax representable functors are preserved under exponentiation, the total space of a presentable cartesian fibration is presentable, generalizing a theorem of Makkai and Paré, and that pseudofunctors of (2,1)-categories give rise to functors of infinity-categories via the Duskin nerve.

- Differential
function spectra, the differential Becker-Gottlieb transfer,
and applications to differential algebraic K-theory,
with U. Bunke. Preprint, 2016, submitted for publication.

We develop differential algebraic K-theory for rings of integers in number fields and construct a cycle map from geometrized bundles of modules over such a ring to the differential algebraic K-theory. We also treat some of the foundational aspects of differential cohomology, including differential function spectra and the differential Becker-Gottlieb transfer. We then state a transfer index conjecture about the equality of the Becker-Gottlieb transfer and the analytic transfer defined by Lott. In support of this conjecture, we derive some non-trivial consequences which are provable by independent means.

- Brauer groups and
Galois cohomology of commutative ring spectra, with T.
Lawson. Preprint, 2016, submitted for publication.

In this paper we develop methods for studying Azumaya algebras over nonconnective commutative ring spectra. We construct and classify these algebras and their automorphisms using Goerss-Hopkins obstruction theory and give descent-theoretic tools, applying results of Lurie to show that a finite Galois extension of rings in the sense of Rognes becomes a homotopy fixed-point equivalence on Brauer spaces. For even-periodic ring spectra, we find that the algebraic Azumaya algebras whose coefficient ring is projective are governed by the Brauer-Wall group of pi_0(E), recovering a result of Baker-Richter-Szymik. We deduce that the algebraic Azumaya algebras over Lubin-Tate spectra have either 4 or 2 Morita equivalence classes, and all algebraic Azumaya algebras over KU are Morita trivial. Using our descent results and an obstruction theory spectral sequence, we show that there exists a unique exotic, non-algebraic Azumaya KO-algebra which becomes Morita-trivial after basechange to KU.

We introduce a general theory of parametrized objects in the setting of infinity categories. Although spaces and spectra parametrized over spaces are the most familiar examples, we establish our theory in the full generality of objects of an infinity-category parametrized by objects of an infinity-topos. We obtain a coherent functor formalism describing the relationship of the various adjoints of base-change and its symmetric monoidal structure, as well as fiberwise constructions of twisted Umkehr maps for twisted cohomology theories using a geometric fiberwise construction of Atiyah duality. Finally, to exhibit the algebraic structures on generalized Thom spectra and twisted (co)homology, we characterize the generalized Thom spectrum as a categorification of the well-known adjunction between units and group rings.

- Homotopy
topoi and equivariant elliptic cohomology, Ph.D. thesis,
2006.

We show that rational equivariant elliptic cohomology, as defined by Grojnowski, Ando-Greenlees, etc., is orientated with respect to complex string representation spheres. In order to do so, we give a new construction of rational equivariant elliptic cohomology via derived algebraic geometry. This is a baby case of a much more ambitious program (which we hope to carry out in future work) involving orientability, and more generally calculation, of Lurie's integral elliptic cohomology for suitable smooth and proper Lie groupoids with appropriate tangential structure.

- Homotopy theory
of orbispaces, with A. Henriques, 2007.

The purpose of this paper is to introduce a homotopy theory of topological Artin stacks which has the feature that the usual G-equivariant unstable categories naturally embed fully faithfully into the homotopy theory of stacks over the classifying stack BG. The resulting infinity-category has since been dubbed the "global unstable homotopy category", and has many interesting features. The technical apparatus underlying our construction is a fibrant replacement endofunctor of topological groupoids has the curious feature that the presheaf the groupoids represented by a topological groupoid is a stack if and only if the topological groupoid is fibrant. This allows us to build stacks cellularly, up to homotopy, and deduce that the underlying infinity-category is freely generated under colimits by the orbit stacks.

- Fall 2015:
MATH 35100, Linear Algebra
(Practice Exams: 1a,
solutions,
1b;
2a,
2b,
solutions;
3a,
solutions,
3b)

- Fall 2015: MATH 69700, Homotopy Theory.
Recommended texts: A. Hatcher,
*Algebraic topology*;

J.P. May,*A concise course in algebraic topology*;

E. Riehl,*Categorical homotopy theory*.