# Seminars

## TODAY

On the Moduli Spaces of Metrics with Nonnegative Sectional Curvature

**Speacker:**

*Jackson Goodman*- UPENN

**Differential Geometry**

**Abstract: **The Kreck-Stolz s invariant is used to distinguish connected components of the moduli space of positive scalar curvature metrics. We use a formula of Kreck and Stolz to calculate the s invariant for metrics on Sn bundles with nonnegative sectional curvature. We then apply it to show that the moduli spaces of metrics with nonnegative sectional curvature on certain 7-manifolds have infinitely many path components. These include the first non-homogeneous examples of this type and certain positively curved Eschenburg and Aloff-Wallach spaces.

## UPCOMING

Canonical singularities, multiplier ideals and Kawamata-Vieweg vanishing theorem

**Speacker:**

*Misha Verbitsky*- IMPA

**Algebra**

**Abstract: **Let $g$ be a singular metric on a trivial line bundle $L$ on a complex manifold. The multiplier ideal of $g$ is the ideal of $L^2$-integrable holomorphic sections of $L$. When $L$ is a nontrivial line bundle, its multiplier ideal is the ideal sheaf $I$ where $L\otimes I$ is the sheaf of $L^2$-integrable holomorphic sections of $L$. This complex-analytic notion can be used to define several classical concepts of algebraic geometry in more geometric fashion; also it allows to give a one-line proof of Kawamata-Viehweg vanishing theorem (called Kawamata-Viehweg-Nadel theorem in this version). I will define "singular metric" and "multiplier ideals" formally and state Nadel's theorems. Then I will define the canonical singularities via multiplier ideal sheaves and explain why all canonically embedded varieties have canonical singularities. The talk is supposed to be as elementary as possible; only basic knowledge of complex algebraic geometry (Kahler metrics, connection, curvature) is assumed.

Volumes of open surfaces

**Speacker:**

*Valery Alexeev*- University of Georgia

**Algebra**

**Abstract: **Volumes of smooth surfaces measure the rate of growth of pluri canonical forms with simple poles at infinity. There are many parallels between these and volumes of hyperbolic 3-folds. I willexplain recent results, joint with Wenfei Liu, about the set ofvolumes and its accumulation points.

Embedded contact homology and the volume property I

**Speacker:**

*Michael Hutchings*- UC Berkeley

**Mathematical Physics**

**Abstract: **We will explain the definition of the embedded contact homology (ECH) of a contact three-manifold. This is an invariant which is built out of periodic orbits of the Reeb vector field and holomorphic curves between them. We are also interested in spectral invariants in ECH, which measure the total length of certain collections of periodic orbits that arise in ECH. The goal is to explain how to compute examples of spectral invariants for the three-sphere and relate them to contact volume. The relation between spectral invariants and volume has applications to dynamics which will be explained in a talk in the Geometry Day.

Embedded contact homology and the volume property II

**Speacker:**

*Michael Hutchings*- UC Berkeley

**Mathematical Physics**

**Abstract: **We will explain the definition of the embedded contact homology (ECH) of a contact three-manifold. This is an invariant which is built out of periodic orbits of the Reeb vector field and holomorphic curves between them. We are also interested in spectral invariants in ECH, which measure the total length of certain collections of periodic orbits that arise in ECH. The goal is to explain how to compute examples of spectral invariants for the three-sphere and relate them to contact volume. The relation between spectral invariants and volume has applications to dynamics which will be explained in a talk in the Geometry Day.

Optimization approach for computing shortest constrained paths

**Speacker:**

*Phan Thanh An*- Institute of Mathematics, Hanoi, Vietnam

**Computer Graphics**

**Abstract: **The method of orienting curves and the method of multiple shooting for computing shortest constrained paths joining two points on a polytope are presented. These methods are originally inspired from optimization. Here, constrained paths may be descending paths, gentle paths.

Fibrewise bundles over the Gromov-Hausdorff space

**Speacker:**

*C. A. Morales*- UFRJ/Chungnam National University, Daejeon, South Korea.

**Dynamical Systems**

**Abstract: **The Gromov-Hausdorff space is the set *M*of compact metric spaces up to isometries, equippedwith the Gromov-Hausdorff metric.In this lecture we will study*Act(G), *the spaceof*G-*actions over compact metric spacesup to isometric conjugacy. We present a metricin*Act(G)*extending [Arbieto-Morales, DCDS2017] yieldinga fibrewise space structure over*M.*Some properties of *Act(G)*asfibrewise space over *M*will be discused.

Discrete Exterior Calculus on General Polygonal Meshes

**Speacker:**

*Lenka Ptackova*- IMPA

**Computer Graphics**

**Abstract: **After a brief introduction on the discrete exterior calculus, novel discretization of several differential operators on general polygonal meshes is presented. Several applications of these methods in Computer Graphics are also shown.

On the Visual Integration of Training and Unseen Data in Classification

**Speacker:**

*Bruno Schneider*- University of Konstanz

**Computer Graphics**

**Abstract: **There is a growing interest in the field of machine learning, which enables a computer to learn from data instead of explicitly programming it to execute a particular task. Classification is one of the problems in machine learning. Examples go from automatically recognizing images after training a model with known instances, supporting the diagnosis of diseases using medical records, or categorizing our e-mail messages as spam or not spam.

One of the problems in Classification is that we never know if the performance obtained during the construction of a model will be the same with new and unseen data. To better understand the reasons for poor model generalization, I propose the visual integration of training and unseen data. In this work, we want to explore, understand and explain how the lack of similar learning examples affects the classification outputs with unseen data at the time of training.

I will also show my previous work on the visual integration of data and models in Ensemble Learning. The goal, in this case, is to give direct access to models and data spaces in classification, thus enabling the user to explore the relationships between these spaces and seek for classification patterns that are not visible through aggregated model performancemetrics.