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8:30am | Registration and Coffee & Tea! |
Registration will be on the 2^{nd} floor of Skye Hall, near the main stairwell and elevators. Coffee and tea will be available in Skye 282 in the morning and throughout the conference. |
9:00am | Functors as Representations: Syntax and Semantics |
Christian Williams (UC Riverside) | |
A basic paradigm of logic, computer science, and other formal sciences is that of theory and model. A theory gives the abstract rules of a system, and these are realized in a particular model. In category theory, this is formalized by a functor from a “syntax” category to a “semantics” category. We overview examples of the ways that this idea provides a unifying perspective of many topics, and extensions which allow us to encapsulate complex concepts in computer science. |
9:25am | Monoidal Grothendieck Construction |
Joe Moeller (UC Riverside) | |
The Grothendieck construction gives an equivalence between indexed categories and fibrations of categories. We lift this equivalence to monoidal settings. We obtain equivalences between monoidal indexed categories and monoidal fibrations of categories. We demonstrate the value of this lift by giving applications in algebra and network theory. |
9:50am | Hybrid Subgroups of Complex Hyperbolic Isometries |
Joseph Wells (Arizona State University) | |
Given a lattice $\Gamma$ in a simple Lie group $G$ not locally isomorphic to $\mathrm{SL}(2,\mathbb{R})$, Mostow’s celebrated rigidity theorem tells us that all discrete, faithful representations $\Gamma \rightarrow G$ with finite-covolume image are conjugate. Geometrically, this implies that finite-volume real (or complex) hyperbolic manifolds of real dimension at least 3 are uniquely determined by their fundamental group. As such, there has been considerable interest in understanding lattices, and in particular the mysterious non-arithmetic lattices, in real and complex hyperbolic isometry groups.
In the 1980’s Gromov and Piatetski-Shapiro presented a technique called “hybridization” wherein one starts with two arithmetic real hyperbolic lattices and uses them to produce new real hyperbolic lattices (and notably, non-arithmetic lattices). It has been asked whether there exists an analogous hybridization technique for complex hyperbolic lattices. In this talk I’ll present a potential candidate hybridization technique and some recent results for both arithmetic and nonarithmetic lattices in $\mathrm{PU}(2,1)$. Some of this is joint work with Julien Paupert. |
10:15am | On Smooth Semiample Complete Intersections over Finite Fields |
Tom Grubb (UC San Diego) | |
In 2004 Poonen introduced the method of the closed point sieve in order to calculate the probability that a hypersurface of sufficiently high degree intersects smoothly with a given projective variety $X$. This asymptotic probability is given in terms of the zeta function of $X$, the product formula for which verifies the local heuristics one would expect assuming smoothness is independent between different closed points of $X$. This result was followed by a surge of results applying probabilistic techniques to algebro-geometric constructions over finite fields, leading to new information on arithmetic dynamics, arithmetic statistics, and existential results in the vein of the probabilistic method. Accordingly, work has been done to extend Poonen’s original result into different settings. Bucur and Kedlaya did this by calculating the probability that a complete intersection of hypersurfaces is smooth, Lindner extended results to toric varieties, and Erman and Matchett Wood provided a semiample generalization of Poonen’s results. The goal of this work is to combine the results of Bucur and Kedlaya and of Erman and Matchett Wood by calculating the probability that a complete intersection is smooth in Erman and Matchett Wood’s semiample setting. We show that this probability may be calculated as a product over local factors determined by the fibers of the map associated to the linear series of the semiample divisor in question. We further extend our results to allow the requirement that the complete intersection has smooth intersection with a subvariety of $X$, so long as the subvariety satisfies a mild Altman-Kleiman type restriction. In both settings we analyze this probability and show that it is asymptotically nonzero. |
10:40am | Coffee & Tea Break in Skye 282 |
11:00am | Young Diagrams |
Professor John Baez | |
Young diagrams are simple combinatorial structures that show up in a myriad of applications. Among other things they classify conjugacy classes in symmetric groups, irreducible representations of symmetric groups, irreducible representations of the groups $\mathrm{SL}(n,\mathbb{F})$ for any field $\mathbb{F}$ of characteristic zero, and irreducible complex representations of the groups $\mathrm{SU}(n)$. All these facts are tightly connected, and the central idea is that Young diagrams are irreducible objects in the category of “Schur functors”. These are functors that know how to act on the category of representations of any group, and other similar categories as well. |
Noon | Lunch in Skye 282 |
1:30pm | Towards Algebraic Deformation Invariance of Plurigenera |
Iacopo Brivio (UC San Diego) | |
In 1998 Y.T. Siu proved that, if $X\longrightarrow T$ is a smooth projective family of complex manifolds, then the plurigenera of the fibers $h^0(X_t,\omega_{X_t}^{\otimes m})$ are independent of $t$. Results of this type are of great importance in higher dimensional algebraic geometry, as they are necessary for the construction of moduli spaces. Siu’s proof relies heavily on the Ohsawa-Takegoshi extension theorem, which does not apply over fields other than $\mathbb{C}$. Using recent results on singularities of log Calabi-Yau fibrations and the Minimal Model Program we can give an algebraic proof of Siu’s theorem whenever the generic fiber of the relative Iitaka fibration of $X$ over $T$ has a good minimal model. We also outline a possible extension of this result for low dimensional smooth families in positive and mixed characteristic. |
1:55pm | Stratified Noncommutative Geometry, Genuine Representation Theory, and Higher Category Theory |
Aaron Mazel-Gee (University of Southern California) | |
I will present a new theory of stratified noncommutative geometry, which gives a means of decomposing and reconstructing (higher) categories. It generalizes the theory of recollements, which e.g. from a closed-open decomposition of a scheme (or of a topological space) gives a decomposition of its category of quasicoherent (resp. constructible) sheaves. It encompasses a version of adelic reconstruction, which is itself an elaboration of the classical “arithmetic fracture square” that reconstructs an abelian group from its $p$-completions and its rationalization. It also applies to chain complexes or spectra with “genuine” $G$-action, which are the relevant objects for equivariant Poincare duality. This represents forthcoming joint work with David Ayala and Nick Rozenblyum. |
2:50pm | Coffee & Tea Break in Skye 282 |
3:10pm | Generalized Petri Nets |
Jade Master (UC Riverside) | |
Petri nets are a powerful diagrammatic language which can be used to represent processes which can be performed in sequence and in parallel. There are many variants of Petri nets that are used to model computation, chemical reactions, and contracts. Generalized Petri nets, or $Q$-Nets, fit some of the more popular variants under a common framework. The construction of $Q$-Nets is functorial which allows for a categorical exploration of the relationships between different kinds of $Q$-Nets. As a justification of this definition, and the usefulness of $Q$-Nets, we show how they present free categories whose morphisms represent processes which are closed under composition and an arbitrary set of algebraic rules. |
3:35pm | Frobenius-Schur Indicators and the Classification of Fusion Categories Via Categorification |
Henry Tucker (UC San Diego) | |
Tensor categories provide a “categorification” of rings: equalities of elements are replaced with isomorphisms of objects, with tensor product and abelian structure replacing multiplication and addition, respectively. The Grothendieck ring of a tensor category is precisely this ring data with the morphisms forgotten. Classification of tensor categories is an important problem in representations for groups, quantum groups, and Hopf algebras, the Jones invariant theory of Murray-von Neumann subfactors, knots and quantum invariants for 3-manifolds, and axiomatizations for conformal field theory. The categorification problem is precisely the question of which tensor categories have a given Grothendieck ring. The special case of this case for fusion categories (= finite semisimple tensor categories) is the motivation for our study of categorical Frobenius-Schur indicators: these are categorical invariants generalized from the classical Frobenius-Schur indicators for complex representations of finite groups. We describe some fusion rings (Grothendieck rings of fusion categories) where the categorical Frobenius-Schur indicators are enough to distinguish all the possible associated fusion categories. |
4:00pm | TQFTs as a Source of Mathematical Connections |
Dr Andrew Manion | |
Since the 1980s, topological quantum field theories (TQFTs) have been a fascinating area of study involving ideas from physics, topology, category theory, representation theory, and other sources. Along with James Dolan, John Baez (one of the leading lights of the study of TQFTs) formulated a conjecture classifying “fully extended” TQFTs which has been central to our understanding of the subject; in the past decade proofs have been outlined by Lurie and by Ayala-Francis. I will give a brief and incomplete history of TQFTs, focusing on the motivating examples as well as the structure posited by the Baez-Dolan cobordism hypothesis. Throughout I will try to draw some “MathConnections” between TQFTs and the topics discussed by other speakers at the conference, potentially including a bit about my joint work in progress with Raphael Rouquier relating Heegaard Floer homology and higher representation theory (time permitting). |
Here are links to the programs for previous MathConnections conferences at UCR.
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