The holonomy group of a Riemannian manifold exhibits various geometric structures compatible with the metric. In 1955, M.Berger classified all possible Riemannian holonomy groups. Studying all these are more than one semester subject. So, in this talk after a brief introduction we overview very basics of these holonomy groups.

Title: Arithmetic intersection theory and Arakelov's Hodge Index Theorem

Date: 11/04/2019

Time: 4:30 PM - 5:30 PM

Place: C304 Wells Hall

The famous Mordell-Weil conjecture was first proved by Faltings in a classical way, then Vojta gave an alternative proof using arithmetic Arakelov geometry, which is one big motivation for developing Arakelov theory into a mature tool. In this talk I will introduce Neron functions and divisors, which is an arithmetic approach to define divisors rather than classical algebraic geometry. We shall also cover arithmetic chow groups and the arithmetic intersection number. In the end I will present Neron symbols and use it to give a sketch proof of Arakelov’s Hodge Index Theorem.

Speaker: Carl Wang-Erickson, University of Pittsburg

Title: Bi-ordinary modular forms

Date: 11/05/2019

Time: 3:00 PM - 4:00 PM

Place: C304 Wells Hall

It is known that p-ordinary cuspidal Hecke eigenforms give rise to 2-dimensional global Galois representations which become reducible after restriction to a decomposition group at p. For which such forms is this restriction not only reducible but also splittable? Complex multiplication (CM) forms satisfy this p-local property, but is such a restrictive global property as CM necessary? In classical weights at least 2, it is expected that this is the case. We present a construction of "bi-ordinary" p-adic modular forms, which can measure exceptions to this expectation. We also give evidence that there are non-CM but p-locally splittable forms in p-adic weights. This is joint work with Francesc Castella.

Normal rulings are combinatorial structures associated to the front diagrams of 1-dimensional Legendrian knots in R^3. They were introduced independently by Fuchs and Chekanov-Pushkar in the context of augmentations of the Legendrian DG-algebra and generating families. In this talk I will present joint work with B. Henry in which we construct a decomposition of the augmentation variety into disjoint pieces indexed by normal rulings. The pieces of the decomposition are products of algebraic tori and affine spaces with dimensions determined by the combinatorics of the ruling. As a consequence, the ruling polynomial invariants of Chekanov-Pushkar are seen to be equivalent to augmentation number invariants defined by counting augmentations to finite fields. The construction of the decomposition is based on considering Morse complex sequences which are combinatorial analogs of generating families.

Title: Variations of cops and robbers on infinite graphs

Date: 11/06/2019

Time: 3:00 PM - 3:50 PM

Place: C304 Wells Hall

The game of cops and robbers is a two player pursuit and evasion game played on a discrete graph G. We study a variation of the classical rules which leads to a different invariant when G is an infinite graph. In this variation, called "weak cops and robbers," the cops win by preventing the robber from visiting any vertex infinitely often. In the classical game, if G is connected and planar, then the cops can always win if there are at least three cops. We prove that this is true in the weak game if G is a locally finite plane graph with no vertex accumulation points.

Speaker: Ioakeim Ampatzoglou, University of Texas, Austin

Title: Derivation of a ternary Boltzmann system

Date: 11/06/2019

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

In this talk work we present a rigorous derivation of a new kinetic equation describing the limiting behavior of a classical system of particles with three particle instantaneous interactions, which are modeled using
a non-symmetric version of a ternary distance. The equation, which we call ternary Boltzmann equation, can be understood as a step towards modeling a dense gas in non-equilibrium. This is a joint work with Natasa Pavlović.

Speaker: Marios Velivasakis, University of Western Ontario

Title: Schubert Varieties in Partial Flag Manifolds and Generalized Severi-Brauer Varieties

Date: 11/07/2019

Time: 11:00 AM - 12:00 PM

Place: C329 Wells Hall

Schubert varieties form one of the most important classes of singular algebraic varieties. They are also a kind of moduli spaces. One problem is that these varieties are not easy to understand and manipulate using only their geometric nature. In this talk, we will discuss about Schubert varieties and present a way to characterize them combinatorially. In addition, we will discuss how they relate to Severi-Brauer varieties SB(d,A) and how we can use their combinatorial description to answer questions about subvarieties of SB(d,A)

Title: Localization for the Anderson--Bernoulli model on the integer lattice

Date: 11/07/2019

Time: 11:30 AM - 12:30 PM

Place: C304 Wells Hall

Abstract: I will give a brief mathematical introduction to Anderson localization followed by a discussion of my recent work with Jian Ding. In our work we establish localization near the edge for the Anderson Bernoulli model on the two dimensional lattice. Our proof follows the program of Bourgain--Kenig and uses a new unique continuation result inspired by Buhovsky--Logunov--Malinnikova--Sodin. I will also discuss recent work of by Li and Zhang on the three dimensional case.

Title: Sketching and Clustering Metric Measure Spaces

Date: 11/08/2019

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Two important optimization problems in the analysis of geometric data sets are clustering and simplification (sketching) of data. Clustering refers to partitioning a dataset, according to some rule, into sets of smaller size with the aim of extracting important information from the data. Sketching, or simplification of data, refers to approximating the input data with another dataset of much smaller size in such a way that properties of the input dataset are retained by the smaller dataset. In this sense, sketching facilitates understanding of the data.
There are many clustering methods for metric spaces (mm spaces) already present in literature, such as k-center clustering, k-median clustering, k-means clustering, etc. A natural method for obtaining a k-sketch of a metric space (mm space) is by viewing the space of all metric spaces (mm space) as a metric under Gromov-Hausdorff (Gromov-Wasserstein) distance, and then determining, under this distance, the k point metric space (mm space) closest to the input metric space (mm space).
These two problems of sketching and clustering, a priori, look completely unrelated. However, we establish a duality i.e. an equivalence between these notions of sketching and clustering. For metric spaces, we consider the case where the clustering objective is minimizing the maximum cluster diameter. We show that the ratio between the sketching and clustering objectives is constant over compact metric spaces.
We extend these results to the setting of metric measure spaces where we prove that the ratio of sketching to clustering objectives is bounded both above and below by some universal constants. In this setting, the clustering objective involves minimizing various notions of the $\ell_p$-diameters of the clusters.
We also identify procedures/maps that transform a solution of the sketching problem to a solution of the clustering problem, and vice-versa. These maps give rise to algorithms for performing these transformations and, by virtue of these algorithms, we are able to obtain an approximation to the k-sketch of a metric measure space (metric space) using known approximation algorithms for the respective clustering objectives. This is joint work with Facundo Memoli and Anastasios Sidiropoulos, and is available online at https://arxiv.org/abs/1801.00551.