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Workshop on “Higher Structures” at Penn State

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Space-time Coordinates

December 1-4, 2022 at Penn State, University Park, PA

Conference room: McAllister Building, room 114

The workshop will start around 2:00pm on Thursday December 1 and end around noon on Sunday December 4.

The nearest airport is State College/University Park (airport code: SCE). It is served by United, American, and Delta.

See bottom of this page for hotel information.

Speakers

Organizers

Schedule

Conference room: McAllister Building, room 114

     
THU 12/1 14:00 — 15:00 Eugenio Landi
  coffee break  
  15:30 — 16:30 Ralph Kaufmann
FRI 12/2 09:00 — 10:00 Seokbong Seol
  coffee break  
  10:30 — 11:30 Yuri Berest
  11:30 — 12:00 Yun Liu
  lunch break  
  14:00 — 15:00 Artur Tsobanjan
  coffee break  
  15:30 — 16:30 Manuel Rivera
SAT 12/3 09:00 — 10:00 Victor Turchin
  coffee break  
  10:30 — 11:00 Jingfeng Xia
  11:00 — 12:00 Ajay Ramadoss
  lunch break  
  14:00 — 15:00 Jim Stasheff
  coffee break  
  15:30 — 16:30 Martin Bojowald
SUN 12/4 09:00 — 09:30 Song Gao
  09:30 — 10:00 Erick Duque
  coffee break  
  10:30 — 11:30 Markus Pflaum

Abstracts


Yuri Berest (Cornell)

Spaces of Quasi-invariants and Homotopy Lie Groups

Quasi-invariants are natural geometric generalizations of classical invariant polynomials of finite reflection groups. They first appeared in mathematical physics in the early 1990s, and since then have found applications in many other areas: most notably, representation theory, algebraic geometry and combinatorics.

In this talk, I will explain how the algebras of quasi-invariants can be realized topologically as (equivariant) cohomology rings of certain spaces naturally attached to compact connected Lie groups. Our main result is a generalization of a well-known theorem of A. Borel that realizes the algebra of invariant polynomials of a Weyl group $W$ as the cohomology ring of the classifying space $BG$ of the corresponding Lie group $G$. Replacing equivariant cohomology with equivariant $K$-theory gives multiplicative (exponential) analogues of quasi-invariants, and, in fact, quasi-invariants can be defined for an arbitrary generalized (complex oriented) cohomology theory. Most interesting perhaps is the fact that the spaces of quasi-invariants can be also constructed for some non-Coxeter ($p$-adic) pseudo-reflection groups, in which case the compact Lie groups are replaced by $p$-compact groups — remarkable homotopy-theoretic objects a.k.a. homotopy Lie groups. Topological realization of rings of quasi-invariants raises natural questions about topological (‘higher algebraic’) refinements of basic properties of these rings and related structures. Time permitting, I will discuss some of these questions in the context of stable homotopy theory.

(Based on joint work with Ajay C. Ramadoss.)

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Martin Bojowald (Penn State)

Hypersurface deformation structures and space-time models

Deformations of spacelike hypersurfaces in space-time have recently been shown to form an $L_\infty$ algebroid. Space-time models for gravitational field theories, however, are commonly derived and analyzed using a Poisson bracket. The discrepancy may have implications for physics questions such as the nature of black-hole singularities.

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Erick Ivan Duque Gonzalez (Penn State)

Covariant spacetime models in modified gravity: Spherical symmetry

General covariance is the underlying mathematical principle of general relativistic theories with physical implications that are currently still being discovered more than a century after its conception. We will discuss how the canonical formulation of general covariance leads to several conditions on the constraints of modified theories of gravity which, in turn, determine modified spacetimes; in particular, these conditions can be solved exactly for the most general Hamiltonian constraint for spherically symmetric models up to second order derivatives of the gravitational variables in vacuum. The allowed modifications are strongly restricted and include some previous studies on inverse-triad corrections and holonomy-modifications motivated by loop quantum gravity with black hole solutions free of singularity. We also discuss a local, off-shell partial Abelianization of the constraint algebra intended as first step to facilitate quantization which imposes some conditions on the general constraint’s free parameters with implications on the spacetime structure including a natural relativistic extension of the modified Newtonian dynamics approach to the dark matter problem.

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Song Gao (University of Notre Dame)

Coisotropicity of fixed points under torus action on the variety of Lagrangian subalgebras

I will talk about my recent study of coisotropic subalgebras of Lie bialgebras. Given a complex semisimple Lie algebra $\mathfrak{g}$ with adjoint group $G$, the set of coisotropic subalgebras of $\mathfrak{g}$ form an algebraic variety, which is called the variety of coisotropic subalgebras. Let $H$ be a fixed maximal torus of $G$. I will introduce my results on fixed points of $H$-action on the variety of coisotropic subalgebras. Approaches of toric varieties and algebraic groups will be used.

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Ralph Kaufmann (Purdue)

Algebraic structures on the Tate-Hochschild complex

There are a set of operations on the Tate-Hochschild complex of a Frobenius algebra which were defined by Rivera and Wang. These fit into a larger family of operations which contains a bibracket. A related bibracket also appears in the work of Iyudu, Kontsevich, Vlassoupolous. The larger family comes from cellularizations of moduli spaces and their partial compactifications. Combinatorially the cells are given by graphs. Such graphs act on Hochschild cochains providing algebraic string topology operations. These can be dualized to act on Hochschild chains. Using a connecting differential the action descends to the Tate-Hochschild complex. We explain these constructions and give a graphical calculus for them.

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Eugenio Landi (Penn State)

The Topological Half of the Grothendieck-Hirzebruch-Riemann-Roch Theorem

The HRR theorem famously states that the holomorphic Euler characteristic of X with coefficients in a holomorphic vector bundle V equals $\int_X\operatorname{ch}(V)\operatorname{td}(X)$. This can be rewritten as two theorems: the first one, analytical, identifying $\chi(X,V)$ with the K-theoretic pushforward of V to the point, while the second, purely topological, identifying the pushforward with the integral. The same can be said for the GHRR theorem and pushforwards along proper holomorphic maps between holomorphic manifolds. I will focus on the second half, introducing orientations and pushforwards in cohomology and explaining how the presence of the Todd class is natural and expected.

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Yun Liu (Cornell)

Generalized Spaces of Quasi-invariants

Abstract: In a recent work, Yu. Berest and A. C. Ramadoss [BR] formulated and studied the realization problem for rings of quasi-invariants of finite reflection groups in terms of classifying spaces of compact Lie groups and their classical homotopy-theoretic generalizations: finite loop spaces and p-compact groups. The main tool used in [BR] is the fiber-cofiber construction introduced in topology by T. Ganea.

In the first part of this talk, which is based on joint work with Yu. Berest and A. C. Ramadoss, we will describe a natural generalization of the fiber-cofiber construction in the context of abstract homotopy theory. Then, in the second part, we will show how to apply this generalized fiber-cofiber construction to obtain analogues of spaces of quasi-invariants associated with classical sphere fibrations. The rational cohomology rings of these generalized spaces of quasi-invariants have nice algebraic properties that are similar to those of classical quasi-invariants. Time permitting we will describe these properties in a number of examples and discuss potential applications.

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Markus Pflaum (Colorado Boulder)

Deformation Quantization and Homological Reduction of a Lattice Gauge Model

In the first part of the talk we explain the method of homological reduction and its quantized version. We then show that quantized homological reduction can be applied to construct deformation quantizations of certain singular symplectic spaces, namely singular symplectic quotients, where the coefficients of the moment map define a complete intersection. In the second part of the talk examples are discussed, among others one where the singularity type is worse than an orbifold singularity and a lattice gauge model. We also present a few new results on equivariant analytic structures on symplectic manifolds and the analyticity of the moment map.

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Ajay Ramadoss (Indiana Bloomington)

Representation homology, derived character maps and symmetric homology

We give a new construction (in terms of functor homology) of the derived character maps between the cyclic homology and representation homology of (group algebras of) simplicial groups. For one dimensional representations, these derived character maps factor through symmetric homology. In this case, we show that these maps are of topological origin: they are induced (on homologies) by natural maps of (iterated) loop spaces that are well studied in homotopy theory. Using this topological interpretation, we show that the symmetric homology of any associative algebra (over a field of characteristic $0$) is isomorphic to its one-dimensional representation homology. As a consequence, we compute the symmetric homology of universal enveloping algebras. We use these results to settle two conjectures of Fiedorowicz and Ault over fields of characterictic $0$. 

This is joint work with Yuri Berest.

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Manuel Rivera (Purdue)

An algebraic model for the free loop space

I will describe an algebraic construction that models the passage from a topological space to its free loop space, without imposing any restrictions on the underlying space. The input of the construction is a “curved coalgebra” over an arbitrary ring R equipped with additional structure and satisfying certain properties. The output is an R-chain complex equipped with a “rotation” operator. The construction is a modified version of the coHochschild complex of a conilpotent coalgebra and is invariant with respect to a notion of weak equivalence for coalgebras drawn from Koszul duality theory. When this construction is applied to a suitable model for the coalgebra of chains of an arbitrary simplicial set X (possibly non-simply connected and non-fibrant) one obtains a chain complex that is quasi-isomorphic to the singular chains on the free loop space of the geometric realization of X. The construction is as small as it can be. Time permitting, we also discuss the relationship with Ed Brown’s twisted tensor product model in terms of the holonomy of the free loop space fibration given by the conjugation action of (a suitable model for) the based loop space on itself. This model for the free loop space is potentially useful in studying invariants arising in string topology of non-simply connected manifolds, some of which are able to distinguish homotopy equivalent non-homeomorphic manifolds.

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Seokbong Seol (KIAS)

Formal exponential maps and the Atiyah class of dg manifolds

Exponential maps arise naturally in the contexts of Lie theory and smooth manifolds. The infinite jets of these classical exponential maps are related to Poincaré-Birkhoff-Witt isomorphism and the complete symbols of differential operators.

We will investigate the question on how to extend these maps to dg manifolds. As an application, we will show there is an $L_\infty$ structure on the space of vector fields in connection with the Atiyah class of a dg manifold.

In particular, for the dg manifold arising from a foliation, we induce an $L_\infty$ structure on the deRham complex associated with the foliation. As a special case, it is related to Kapranov’s $L_\infty$ structure on the Dolbeault complex of a Kähler manifold.

This is a joint work with Mathieu Stiénon and Ping Xu.

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Jim Stasheff (Penn)

From Lie algebra crossed modules to tensor hierarchies

Gauging procedures in supergravity produced what are known as tensor hierarchies. These rely on a pairing - called an embedding tensor - between a Leibniz algebra and a Lie algebra. The combination forms a triple called a Lie-Leibniz triple, of which Lie algebra crossed modules are particular cases. Here we show that any Lie-Leibniz triple induces a differential graded Lie algebra - its associated tensor hierarchy - whose restriction to the category of Lie algebra crossed modules is the canonical assignment of a 2-term differential graded Lie algebra to a Lie algebra crossed module. This construction of such tensor hierarchies we find clearer and more straightforward than previous derivations and suggests the existence of further well-defined Leibniz gauge theories.

This is a joint work with Sylvain Lavau.

slides

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Artur Tsobanjan (King’s College Wilkes-Barre)

Quantum dynamical reduction

Canonical approaches to reconciling quantum mechanics with general relativity run into the so-called problem of time (amongst other issues). The structural reason for this problem is that general relativity, formulated as a dynamical theory, is invariant under arbitrary time reparameterization. This particular issue can be studied in isolation using “toy models” that possess time-reparameterization invariance without other complications. In canonical formulation on a symplectic phase space such systems invariably possess a Hamiltonian constraint: their Hamiltonian is constrained to vanish and therefore can be re-scaled, making the time parameterization of the associated dynamical orbits arbitrary. Such a system appears to call for a symplectic reduction by the symplectic flow of the Hamiltonian constraint. In my talk I will explain why what we need instead is a special kind of “dynamical” reduction, accomplished by selecting a “clock” function to keep track of time. The main interest here is a precise mathematical formulation of the quantum counterpart to the dynamical reduction, and I will focus on one recent proposal for such a construction. However, as we will see, the problem remains open: even in the simplest models, this version of quantum dynamical reduction (and others that it generalizes) does not seem to fully capture the time-reparameterization freedom of its classical counterpart.

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Victor Turchin (Kansas State)

Rational homotopy theory invariants of embedding spaces

Abstract: Manifold functor calculus was invented by Goodwillie and Weiss in 1990s in order to study embedding spaces. A deep result by Goodwillie, Klein and Weiss asserts that the method is applicable whenever the codimension is $>2$. Later, around 2011 the calculus was translated to the language of operads by Boavida and Weiss as well as independently by Arone and me. Together with the (relative) formality of the little discs operad this gave an efficient way to study spaces of embeddings of manifolds in Euclidean space. The most advanced and technically most difficult results in this direction are contained in my joint work with Fresse and Willwacher “On the rational homotopy type of embedding spaces of manifolds in $\mathbb{R}^n$” [+]. For a manifold $M$ and an ambient dimension $n$, we define a graph-complex $L_\infty$ algebra $\operatorname{HGC}{M,n}$, such that the rational homotopy type of each component of the embedding space is computed as a twist of $\operatorname{HGC}{M,n}$ by some Maurer-Cartan element. The corresponding MC element modulo gauge relations MC/~ under some codimension restrictions happens to be a finite-to-one invariant of embeddings. This invariant is conjectured to be related to many interesting geometric invariants, such as linking number (in case $M$ has several components), higher linking Milnor invariants, Boéchat-Haefliger invariants of embeddings of 4-folds in $\mathbb{R}^7$, Whitney-Skopenkov invariants of 3-folds in $\mathbb{R}^6$, Vassiliev and TQFT invariants of long $(k-2)$-knots in $\mathbb{R}^k$, etc. In my talk I will describe the graph-complex $\operatorname{HGC}_{M,n}$ and will state the main results of [+]. Some examples will then be considered.

(Joint work with B. Fresse and T. Willwacher.)

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Jingfeng Xia (Temple)

Exploration of Grothendieck-Teichmueller(GT) shadows related to the full modular group

In this talk, we will introduce $GT$-shadows and the groupoid $GTSh$. We will see that the groupoid $GTSh$ is highly disconnected. Consider an onto group homomorphism $\psi$ from the Artin Braid group $B_3$ to the full modular group $SL_2(\mathbb{Z})$, where \(\psi(\sigma_1)=\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}, \qquad \psi(\sigma_2)=\begin{bmatrix} 1 & 0 \\ -1 & 1 \end{bmatrix},\) and the standard projection $P_q: SL_2(\mathbb{Z}) \longrightarrow SL_2(\mathbb{Z}/q\mathbb{Z})$ for positive integer $q$. We get a map of posets from the divisibility poset of positive integers to the poset $NFI_{PB_3}(B_3)$ of finite index normal subgroups of $B_3$ contained in $PB_3$. We will consider a certain subposet of $NFI_{PB_3}(B_3)$, and we will describe the connected components of certain elements of that subposet in $GTSh$.

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Hotel Information

We have blocked a number of rooms at Hyatt Place ($109+tax/night) and Sleep Inn ($69+tax/night).

Hyatt Place ($109+tax/night, cutoff date for reservations is November 18)

Your group block for PSU Department of Math has been created and is open for your guests to make reservations for 12/01/2022. Reservations can be made by either calling Central Reservations at 1-888-492-8847 and referencing the group name in addition to the code G-DMAT or going online at https://statecollege.place.hyatt.com. When booking online, your guests will need to type in group code G-DMAT to ensure that they receive the group discount under the “Corporate or Group Code” option from the drop-down menu (which is listed as the last one) beneath the dates.

Individual reservations do not go through Hyatt Sales, so please use the phone number above or the website for booking.

The cutoff date for booking reservations is 11/18/2022.

Sleep Inn ($69+tax/night)

https://www.choicehotels.com/pennsylvania/state-college/sleep-inn-hotels/pa421

To make a reservation, please call the Sleep Inn Hotel directly at (814) 235-1020 and say your room is under the PSU Math Seminar group and you will get the special rate $69+tax/night.