All talks will take place in David Rittenhouse Laboratory (DRL) room A1 on the first floor.  Abstracts appear below the schedule. The address for the math building, DRL is 209 South 33rd St, Philadelphia. 

Saturday, April 15

9-9:30 Donuts and coffee + Registration

9:30-10:15 Inna Zakharevich: Coinvariants, assembler K-theory, and scissors congruence

10:15-10:45 Coffee break

10:45-11:30 Bena Tshishiku: Symmetries of exotic flat and hyperbolic manifolds

11:45-12:30 Sanjeevi Krishnan: CAT(0) Pasting Schemes

12:30-2:30 Lunch

2:30-3:15 Agnes Beaudry: Homotopical framework for parametrized quantum spin systems

3:15-3:45 Coffee Break

3:45-4:30 Iva Halacheva: Homomorphic expansions, foams, and Kashiwara–Vergne groups

4:45-5:30 Zhouli Xu: The Adams differentials on the classes h_j^3

6:30-9:30  Conference Dinner 

Sunday, April 16

9-9:30 Donuts and coffee

9:30-10:15 Jesse Wolfson: Essential dimension via prismatic cohomology

10:15-10:45 Coffee break

10:45-11:30 Jose Perea: Vector bundles for data alignment and dimensionality reduction

11:45-12:30 Cary Malkiewich: Higher scissors congruence

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Agnes Beaudry: Homotopical framework for parametrized quantum spin systems

Abstract: In recent years, there has been a growing number of applications of stable homotopy theory to condensed matter physics, many of which stem from a conjecture of Kitaev that gapped invertible phases of matter should be classified by the homotopy groups of a spectrum. This gives rise to a mathematical modeling question: how do we model quantum systems in such a way that this result can be better understood, perhaps even proved? In this mostly expository talk, I'll will explain some aspects of Kitaev's conjecture and of this modeling problem. I will then explain how, from certain models for quantum systems called "quantum state types", we can naturally extract an infinite loop space using techniques well-known and well-loved by homotopy theorists. This is based on joint work with Mike Hermele, Juan Moreno, Markus Pflaum Marvin Qi and Daniel Spiegel.

Iva Halacheva: Homomorphic expansions, foams, and Kashiwara–Vergne groups

Abstract: The theory of universal finite-type invariants, or homomorphic expansions for knotted objects has developed in parallel with the exploration of its close ties to Lie theory. In the case of parenthesized braids, such invariants were shown to correspond to Drinfeld associators, and their symmetry groups to the Grothendieck-Teichmuller (GT) groups. For another such family—ribbon-knotted tubes in 4-space—or more generally welded foams, Bar-Natan and Dancso constructed a bijection between their homomorphic expansions and solutions to the Kashiwara—Vergne equations. In joint work with Dancso and Robertson, we recast the symmetry groups of this set of solutions in the topological setting, and within that setting relate them to GT groups.

Sanjeevi Krishnan: CAT(0) Pasting Schemes

Abstract: At least intuitively, it is clear that the right analogue for a Kan complex in directed homotopy theory is some notion of higher category.  We define such a higher category to be a cubical set with a notion of composition, where freely composable configurations of cubes are given by rooted finite CAT(0) cubical complexes. The combinatorial homotopy theory we get is equivalent in a strong sense to directed homotopy theory of directed topological spaces.   There are some immediate applications. Some are calculations of directed homotopy monoids and cohomology monoids.  Another is an equivalence between classical equivariant homotopy theories of diagrams.  After describing the general theory and results, we will focus on some concrete applications to compute science, and namely type theory.

Cary Malkiewich: Higher scissors congruence

Abstract: Hilbert's Third Problem asks for sufficient conditions that determine when two polyhedra in three-dimensional Euclidean space are scissors congruent. Classically, the attempts to solve this problem (in this and other geometries) lead into group homology and algebraic K-theory, in a somewhat ad-hoc way. In the last decade, Zakharevich has shown that the presence of K-theory here is not ad-hoc, but is integral to the definition of scissors congruence itself. This leads to a natural notion of "higher" scissors congruence groups, namely, the homotopy groups of an algebraic K-theory spectrum K(P). In this talk, I'll describe a surprising recent result that K(P) is actually a Thom spectrum. Its base space is the homotopy orbit space of a Tits complex, and the vector bundle is the negative tangent bundle of the underlying geometry. Using this result, we can explicitly compute the higher scissors congruence groups in new cases. Much of this is joint work with Anna-Marie Bohmann, Teena Gerhardt, Mona Merling, and Inna Zakharevich.

Jose Perea: Vector bundles for data alignment and dimensionality reduction

Abstract: Vector bundles have rich structure and arise naturally when trying to solve dimensionality reduction and synchronization problems in data science. I will show in this talk how the classical machinery (e.g., classifying maps, characteristic classes, etc) can be adapted to the world of algorithms and noisy data, as well as the insights one can gain.  Applications to computational chemistry and dynamical systems will be presented.

Bena Tshishiku: Symmetries of exotic flat and hyperbolic manifolds

Abstract: How do the Lie group actions on a manifold depend on the smooth structure? Some answers appear in work of Hsiang-Hsiang for spheres and Farrell-Jones for hyperbolic manifolds. For hyperbolic manifolds, we are still far from a complete picture. We will present some new phenomena and structural results, including a classification result for finite group actions on exotic flat and hyperbolic 7-manifolds. This is joint work with Mauricio Bustamante. 

Jesse Wolfson: Essential dimension via prismatic cohomology

Abstract: Classical resolvent problems (essential dimension, essential p-dimension, resolvent degree, . . .) ask some form of "How complex is . . . a polynomial, an enumerative problem, a branched cover, a variation of Hodge structure, . . .?"  An idea going back to Arnold is that characteristic classes should be able to detect this intrinsic complexity. However, to make this work one must show that the relevant characteristic class remains nonzero under restriction to arbitrary Zariski open subvarieties.  In this talk, we describe a new method for solving this restriction problem in many cases using prismatic cohomology.  As an application, we prove a conjecture of Brosnan that for a complex abelian variety A, the essential p-dimension of the p-isogeny cover A\to A equals dim A for all but finitely many p.  This is joint work with Benson Farb and Mark Kisin.

Zhouli Xu: The Adams differentials on the classes h_j^3

Abstract: In filtration 1 of the Adams spectral sequence, using secondary cohomology operations, Adams computed the differentials on the classes h_j, resolving the Hopf invariant one problem. In Adams filtration 2, using equivariant and chromatic homotopy theory, Hill--Hopkins--Ravenel proved that the classes h_j^2 support non-trivial differentials for j \geq 7, resolving the celebrated Kervaire invariant one problem. I will talk about joint work with Robert Burklund: In Adams filtration 3, we prove an infinite family of non-trivial d_4-differentials on the classes h_j^3 for j \geq 6, confirming a conjecture of Mahowald. Our proof uses two different deformations of stable homotopy theory – C-motivic stable homotopy theory and F_2-synthetic homotopy theory – both in an essential way.

Inna Zakharevich: Coinvariants, assembler K-theory, and scissors congruence

Abstract: For a geometry $X$ (such as Euclidean, spherical, or hyperbolic) with isometry group $G$ the scissors congruence group $\mathcal{P}(X,G)$ is defined to be the free abelian group generated by polytopes in $X$, modulo the relation that for polytopes $P$ and $Q$ that intersect only on the boundary, $[P\cup Q] = [P] + [Q]$, and for $g\in G$, $[P] = [g \cdot P]$.  This group classifies polytopes up to "scissors congruence," i.e. cutting up into pieces, rearranging the pieces, and gluing them back together.  With some basic group homology one can see that $\mathcal{P}(X,G) \cong H_0(G, \mathcal{P}(X,1))$.  Using combinatorial $K$-theory $\mathcal{P}(X,G)$ can be expressed as the $K_0$ of a spectrum $K(X,G)$.  In this talk, which is the first part of a two-part series (see Cary Malkiewich's talk for the second part!) we will generalize this formula to show that, in fact, $K(X,G) \simeq K(X,1)_{hG}$, and in fact more generally that this is true for any assembler with a $G$-action.  This is joint work with Anna Marie Bohmann, Teena Gerhardt, Cary Malkiewich, and Mona Merling.