**Yale Math Graduate Student Seminar**

**Fall 2022**

**Fall 2022**

**Fridays**

Current organizers: **Dongryul and Mengyi**

Contact: yale dot math dot grad dot student dot seminar at gmail dot com.

### 9/23, 12 pm ~ 1pm - Dongryul

Title: Drunken climber on trees in mapping class groups

Abstract: Given a pair of filling multicurves on a closed surface, Thurston ('88) introduced a PSL(2, R)-representation of a subgroup generated by the multitwists along the multicurves by thinking of the subgroup as a stabilizer subgroup (in the Mapping class group) of a hyperbolic disk isometrically embedded in the Teichmüller space. In most cases, such a stabilizer subgroup is free of rank two. We consider a random walk on the stabilizer subgroup and study asymptotic behaviors of the random walk. This talk discusses the following two main theorems, which were known for much more general settings but under finite support conditions by Joseph Maher ('11, '12), Joseph Maher and Giulio Tiozzo ('18), and François Dahmani and Camille Horbez ('18):

- Strong law of large numbers of topological entropy along the random walk under finite first-moment conditions.

- Almost every sample path consists of all but finitely many pseudo-Anosov mapping classes without any assumption on the moment.

This is based on the joint work with Hyungryul Baik and Inhyeok Choi. While the same results for much more general settings were also obtained in the sequel joint work with Hyungryul Baik and Inhyeok Choi, which was reported in Yale Geometry and Topology Seminar in 2021, the argument in the above specific situation is totally different from the general settings and is of independent interest itself.

### 9/30, 11:30 am ~ 12:30 pm - Weite

Title: From curves to sheaves: a tale of two moduli spaces

Abstract: Moduli spaces parametrize certain kinds of geometric objects, and they are central objects in algebraic geometry. In this talk, we look at two different moduli spaces: one of smooth curves of fixed genus, and the other of one-dimensional sheaves on the projective plane. The former is a classical subject, dating back to Bernhard Riemann, while the latter relatively new, motivated by enumerative geometry, mathematical physics and so on. Despite the different nature of the objects they parametrize, we will show how their ‘intersection rings’ share surprisingly similar features. This talk is based on joint work with Junliang Shen.

### 10/14, 12 pm ~ 1 pm - Asaf

Title: Geodetic Graphs: Geometry, Diameter and the Problem of Classification

Abstract: A graph is called geodetic if there is a unique shortest path between any two vertices (i.e - a geodesic). Ore (’62) sought to characterize such graphs, and despite a considerable body of work on this problem - a characterization is surprisingly elusive. In this talk we introduce the idea of viewing graphs from a geometric perspective and survey its connections to the problem of classification. In particular, we review some known classifications and constructions of geodetic graphs, and show that such graphs must be of relatively high connectivity. This talk is based on joint work with Nati Linial.

### 10/28 , 12 pm ~ 1 pm - Junior Colloquium: Prof. Sebastian Hurtado-Salazar

Title: Rigidity of group actions

Abstract: I will try to explain what rigidity in dynamics and geometry, and what type of groups (or actions) have it. The main characters of this story are cousins of SL_n(Z), the group of invertible matrices with integer coefficients and n >=3.

### 11/4, 12 pm ~ 1 pm - Junzhi

Title: Rigidity of Curve Complexes

Abstract: The curve complex is a simplicial complex encoding the intersection patterns of simple closed curves on a surface. There is a natural mapping class group action on the complex and we will discuss different rigidity results on the action. In particular, we will look at the finite rigidity of the curve complex proved by Aramayona-Leininger, i.e. one can always recover a mapping class from the translation of a finite subcomplex. Thus the curve complex serves as a good model to study the mapping class group: there is not much flexibility in the complex other than the automorphisms induced by the mapping classes. Similar rigidity phenomenon occurs when one considers other natural variations of the curve complex, such as the separating curve complex. However, such complexes are not as well-studied as the curve complex. One question is how the whole story of the curve complex looks like on other complexes. The talk is based on joint work with Bena Tshishiku.

### 11/11, 12 pm ~ 1 pm- Jonas

Title: An introduction to formal methods with applications to floating-point error estimation

Abstract: In a broad sense, formal methods are mathematically rigorous techniques used to describe, monitor, and check system properties and limitations in computer systems, along with the fields in which these techniques can generally be applied, guidelines for how these techniques can be used, and tools or software used in the implementation of these techniques. One application of formal methods is in monitoring and checking the propagation of errors in computational programs. Due to the limits of finite precision arithmetic, as one works with floating-point representations of real numbers in computer programs, round-off errors tend to accumulate under mathematical operations and may become unacceptably large. Symbolic Taylor Error Expressions is a technique used in the tool FPTaylor to soundly bound the round-off error of mathematical operators which provide a good trade-off between efficiency and accuracy.

In this presentation, I will give a general overview to formal methods, the history of formal methods, and how formal methods are applied in various fields of engineering and industry. Afterwards, I will present a formally verified implementation of Symbolic Taylor Error Expressions in a specification language, automated theorem prover, and typechecker called Prototype Verification System (PVS). I will also go over how these error expressions can be used in PRECiSA, a prototype static analysis tool that can compute verified round-off error bounds for floating-point programs. Time permitting, I will introduce other theorems, properties, and structures which I have proven in PVS for inclusion in the NASA PVS Library.

### 11/18, 12 pm ~ 1 pm - Anna

Title: Computations of the involutive concordance invariants of (1,1)-knots

Abstract: Heegaard Floer knot homology is an invariant of knots introduced by P. Ozsvath and Z. Szabo in the early 2000's which associates to a knot K a chain complex called CFK^\infty(K), and improves on classical invariants of the knot. Involutive Heegaard Floer homology is a variant theory introduced in 2015 by K. Hendricks and C. Manolescu, which additionally considers a chain map on CFK^\infty(K) called \iota_K, and extracts from this data two new numerical invariants of knot concordance. These new invariants are interesting, because, unlike other concordance invariants from Heegaard Floer homology, they do not necessarily vanish on knots of finite order in the group of concordance classes of knots. The map \iota_K is in general difficult to compute, and computations have been carried out for relatively few knots. We discuss computations of iota_K for 10 and 11-crossing knots satisfying a certain simplicity condition, called the (1,1)-knots. Our methods are principally focused in homological algebra.

### 12/2, 12 pm ~ 1 pm - Haoyu

Title: Consensus on Dynamic Stochastic Block Models

Abstract: We introduce two models of consensus following a majority rule on time-evolving stochastic block models (SBM), in which the network evolution is Markovian or non-Markovian. Under the majority rule, in each round, each agent simultaneously updates his/her opinion according to the majority of his/her neighbors. Our network has a community structure and randomly evolves with time. In contrast to the classic setting, the dynamics is not purely deterministic, and reflects the structure of SBM by resampling the connections at each step, making agents with the same opinion more likely to connect than those with different opinions. In the Markovian model, connections between agents are resampled at each step according to the SBM law and each agent updates his/her opinion via the majority rule. In the non-Markovian model, a connection between two agents is resampled according to the SBM law only when some of the two changes opinion and is otherwise kept the same. We study the phase transition between the fast convergence to the consensus and a halt of the dynamics. Moreover, we establish thresholds of the initial lead for various convergence speeds. This is based on joint work with J. Wei and Z. Zhang.

### 12/9 - Linh

Title: Recent progresses in the unanimity problem in Majority Dynamics on random graphs

Abstract: Majority Dynamics is a process on a graph, where each vertex starts out with a Red or Blue color, then on each day changes its color to the majority color among its neighbors the previous day. If at some point one color covers every vertex, that color is said to win and such state is called unanimity. Research on unanimity has traditionally focused on the model where initial colors are independently chosen with 1/2 chance, and the graph is generated on them from the G(n, p) model. In this talk, we discuss how recent studies by many authors point out that, fixing the initial colors instead can help answer some of the questions in the traditional model, and pose new interesting conjectures.