4/19/2022 – Eric Jovinelly: Extreme Divisors on \(M_{0,7}\) and Differences over Characteristic 2
The cone of effective divisors controls the rational maps from a variety. We study this important object for \(M_{0,n}\), the moduli space of stable rational curves with \(n\) markings. Fulton once conjectured the effective cones for each nwould follow a certain combinatorial pattern. However, this pattern holds true only for \(n < 6\). Despite many subsequent attempts to describe the effective cones for all \(n\), we still lack even a conjectural description. We study the simplest open case, \(n = 7\), and identify the first known difference between characteristic \(0\) and characteristic \(p\). Although a full description of the effective cone for \(n = 7\) remains open, our methods allowed us to compute the entire effective cones of spaces associated with other stability conditions.
4/14/2022 – Will Dudarov (UW): Colored Gelfand-Tsetlin Patterns and Symmetric Lattice Models
In the search for a bijective version of the proof presented in Weyl Group Multiple Dirichlet Series by Brubaker, Bump, and Friedberg, which states that two definitions of the p-parts of a multiple Dirichlet series given a Type A root system are equal, our group, mentored by Ben Brubaker himself, tackled defining new combinatorial objects: colored symmetric lattice models and their corresponding colored Gelfand-Tsetlin patterns. In this talk, we will discuss the main problem, which is motivated by representation theory and number theory, how our definitions tackle this problem, and our conjecture towards a bijective proof. In particular, we conjecture a generalization of the Schützenberger involution to these new classes of combinatorial objects.
4/5/2022 – Gurutam Thockchom: Operads and First-Order Theories
Operads of algebraic topology and first-order theories of mathematical logic both give ways to describe the structure of objects. In this talk I define an operad and an algebra over an operad, and explore some correspondences between operads and theories in first-order logic. I will also go over some obstructions to a complete correspondence between these ideas, and the cost of resolving one of these obstructions. To conclude, I will discuss the role operads might play in a certain generalization of first-order logic.
4/4/2022 – Misha Gekhtman: Five Glimpses of Cluster Algebras
Cluster algebras were introduced by Fomin and Zelevinsky 20 years ago and have since found exciting applications in many areas including algebraic geometry, representation theory, integrable systems and theoretical physics. I will use examples to explain a definition of a cluster algebra and then sketch several applications of the theory, including Somos-5 recursion, pentagram map and generalizations of Abel’s pentagon identity.
3/31/2022 – David Galvin: Stirling numbers and the normal order problem
The Stirling numbers of the second kind, introduced in 1730, arise in many contexts —combinatorial, analytic, algebraic, probabilistic… I’ll introduce these versatile numbers, and describe some of their interpretations and applications.
The standard combinatorial interpretation of the Stirling numbers involves set partitions, and this interpretation has a natural generalization to graphs. I’ll discuss an application of this generalization to a problem coming from the Weyl algebra (the algebra on alphabet \(\{x, D\}\) with the single relation \(Dx = xD+1\)). This is joint work with J. Hilyard and J. Engbers.
3/24/2022 – Felix Janda: Counting covers of a torus
I will outline a connection (observed by Dijkgraaf) between counts of ramified covers of a \(2\)-torus, Fourier expansions of modular forms, and mirror symmetry.
3/23/2022 – Philippe Mathieu: Extensions of the Abelian Turaev-Viro construction and U(1) BF theory to any finite dimensional smooth oriented closed manifold
In 1992, V. Turaev and O. Viro defined an invariant of smooth oriented closed \(3\)-manifolds consisting of labelling the edges of a triangulation of the manifold with representations of \(U_q(\mathrm{sl}_2(\mathbb{C}))\) (\(q\) being a root of unity), associating a (quantum) \(6j\)-symbol to each tetrahedron of the triangulation, taking the product of the \(6j\)-symbols over all the tetrahedra of the manifold, then summing over all the admissible labelling representations. It is commonly admitted that this construction is a regularization of a path integral occurring in quantum gravity, the so-called “Ponzano-Regge model”, which is a kind of \(\mathrm{SU}(2)\) BF gauge theory. A naive question is: Is it possible to define an abelian version of this invariant? If yes, is there a relation with an abelian BF gauge theory? These questions were answered positively in 2016, and the corresponding Turaev-Viro invariant is built from \(\mathbb{Z}/k\mathbb{Z}\) labelling representations (the equivalent of \(6j\)-symbols being “modulo \(k\)” Kronecker symbols) while the associated gauge theory is a particular \(\mathrm{U}(1)\) BF theory (with coupling constant \(k\)). This \(\mathrm{U}(1)\) BF theory can be straightforwardly extended to any finite dimensional closed oriented manifold, and so can be the Turaev-Viro construction built from \(\mathbb{Z}/k\mathbb{Z}\) labelling representations. A natural question is thus: Are these extensions still related? I will answer this question during the talk.
3/17/2022 – Anish Chedalavada (UI Chicago): An introduction to tensor-triangular geometry
In this talk, I will discuss a perspective on tensor-triangulated categories that has evolved in the last few decades which provides a unifying theme behind the chromatic picture of the stable homotopy category, stable categories of modules in modular representation theory, and derived categories of quasicoherent sheaves on a coherent scheme. Heuristically, this is based on an analogy between tt-categories and commutative rings, with thick tensor ideals playing the mediating role between the tt-category and topological spaces. Time permitting, I will also (god forbid) attempt a computation and point to some interesting questions from algebraic geometry that are still waiting to be generalized.
3/3/2022 – Jeff Diller: Dynamics of rational maps: a best case example and reasonable hopes
A map \(f : \mathbb{C}^n \to \mathbb{C}^n\) is rational if its component functions are themselves rational. A result of Gromov from the 70s suggests that the complexity of the dynamics (i.e. behavior of iterates) of \(f\) should be closely tied to the way the degrees of the components grow under iteration. I’ll lead with a particularly good example of this connection, present Gromov’s result and, time permitting, say some things about the proof and a program for further understanding the dynamics of a rational map.
2/24/2022 – Minh Chieu Tran: o-minimal method and generalized sum-product phenomena
I will discuss a joint work with Yifan Jing and Souktik Roy where we show that for a bivariate polynomial \( P(x,y) \in \mathbb{R}[x,y] \setminus (\mathbb{R}[x] \cup \mathbb{R}[y]) \) to exhibit small expansion on a finite set \( A \subseteq \mathbb{R} \), we must have \( P(x,y) = f(\gamma u(x) + \delta u(y)) \) or \( P(x,y) = f(u^m(x) u^n(y)) \) for some univariate \( f, g, u \in \mathbb{R}[t] \setminus \mathbb{R} \), constants \( \gamma, \delta \in \mathbb{R}^{\neq 0} \), and \( m, n \in \mathbb{N}^{\geq 1} \). This yields an Elekes-Ronyai type structural result for symmetric nonexpanders, resolving a question mentioned by de Zeeuw. Our result uses o-minimal/semialgebro geometric techniques to replace algebraic geometric techniques, which are only applicable to more special cases.
2/17/2022 – Pavel Mnev: On the Fukaya-Morse A-infinity category
I will sketch the construction of the Fukaya-Morse category of a Riemannian manifold \(X\) — an A-infinity category (a category where associativity of composition holds only “up-to-homotopy”) where the higher composition maps are given in terms of numbers of embedded trees in \(X\), with edges following the gradient trajectories of certain Morse functions.
I will give simple examples and explain different approaches to understanding the structure and proving the quadratic relations on the structure maps — (1a) via homotopy transfer, (1b) effective field theory approach, (2) topological quantum mechanics approach. (Maybe a subset of this, depending on time.) The talk is based on a joint work with O. Chekeres, A. Losev and D. Youmans, arXiv:2112.12756.
2/10/2022 – Alex Himonas: Analysis of the Korteweg-de Vries (KdV) equation
The KdV equation is one of the most ubiquitous models in mathematics and physics. It was first derived by Boussinesq in 1877 in his effort to demystify Russell’s observation of what he called the “great wave of translation” in a Union Canal near Edinburgh in 1834. Korteweg and de Vries rediscovered the KdV equation in 1895 and confirmed that it has traveling wave solutions (solitons). The solving of the KdV initial value problem, with nice data, was initiated by Gardner, Greene, Kruskal, and Miura in 1967 by recognizing its remarkable integrability properties. Its solving for rough (\(L^2\)) data was accomplished by Bourgain in 1993 using novel ideas from classical and harmonic analysis. In 1996, Kenig, Ponce, and Vega advanced these ideas and solved the KdV with data of regularity below \(L^2\). It is interesting that these ideas are also needed if one studies KdV solutions with the nicest possible data (analytic). In this talk we will try to present the key points of this remarkable KdV story.
1/27/2022 – Lorenzo Riva: You could have formulated the Cobordism Hypothesis
In this talk we will try to study manifolds in their totality: all n-manifolds, or all manifolds up to a fixed dimension, or all manifolds equipped with a certain structure. It turns out that these collections form specific algebraic structures that are fully characterized by the Cobordism Hypothesis, formulated by Baez-Dolan and proven by Lurie. In particular, we will talk about building manifolds by inductively gluing cobordisms, some category theory, and the algebraic restrictions imposed by a topological field theory, which is a representation of such an “algebra of manifolds”.
The talk should be accessible to everyone, but in particular it will interesting for topologists, algebraists, and mathematical physicists.