Seminars and Colloquia by Series

Complex Ball Quotients and New Symplectic 4-Manifolds with Nonnegative Signatures

Series
Geometry Topology Seminar
Time
Tuesday, March 29, 2022 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Sumeyra Sakalli – University of Arkansas

Please Note: Note this talk is at a different time and day

We first construct a complex surface with positive signature, which is a ball quotient. We obtain it as an abelian Galois cover of CP^2 branched over the Hesse arrangement. Then we analyze its fibration structure, and by using it we build new symplectic and also non-symplectic exotic 4-manifolds with positive signatures.

 

In the second part of the talk, we discuss Cartwright-Steger surfaces, which are also ball quotients. Next, we present our constructions of new symplectic and non-symplectic exotic 4-manifolds with non-negative signatures that have the smallest Euler characteristics in the so-called ‘arctic region’ on the geography chart.

 

More precisely, we prove that there exist infinite families of irreducible symplectic and infinite families of irreducible non-symplectic, exotic 4-manifolds that have the smallest Euler characteristics among the all known simply connected 4-manifolds with nonnegative signatures and with more than one smooth structures. This is a joint work with A. Akhmedov and S.-K. Yeung.

Image formation ideals

Series
Algebra Seminar
Time
Tuesday, March 29, 2022 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Tim Duff – University of Washington –

Projective space, rational maps, and other notions from algebraic geometry appear naturally in the study of image formation and various camera models in computer vision. Considerable attention has been paid to multiview ideals, which collect all polynomial constraints on images that must be satisfied by a given camera arrangement. We extend past work on multiview ideals to settings where the camera arrangement is unknown. We characterize various "image formation ideals", which are interesting objects in their own right. Some nice previous results about multiview ideals also fall out from our framework. We give a new proof of a result by Aholt, Sturmfels, and Thomas that the multiview ideal has a universal Groebner basis consisting of k-focals (also known as k-linearities in the vision literature) for k in {2,3,4}. (Preliminary report based on ongoing joint projects with Sameer Agarwal, Max Lieblich, Jessie Loucks Tavitas, and Rekha Thomas.)

Quasi-morphisms on Surface Diffeomorphism groups

Series
Geometry Topology Seminar
Time
Monday, March 28, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
Speaker
Jonathan Bowden –

We discuss the problem of constructing quasi-morphisms on the group of diffeomorphisms of a surface that are isotopic to the identity, thereby resolving a problem of Burago-Ivanov-Polterovich from the mid 2000’s. This is achieved by considering a new kind of curve graph, in analogy to the classical curve graph first studied by Harvey in the 70’s, on which the full diffeomorphism group acts isometrically. Joint work with S. Hensel and R. Webb. 

How Differential Equations Insight Benefit Deep Learning

Series
Applied and Computational Mathematics Seminar
Time
Monday, March 28, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
https://gatech.zoom.us/j/96551543941 (note: Zoom, not Bluejeans)
Speaker
Prof. Bao Wang – University of Utah –

We will present a new class of continuous-depth deep neural networks that were motivated by the ODE limit of the classical momentum method, named heavy-ball neural ODEs (HBNODEs). HBNODEs enjoy two properties that imply practical advantages over NODEs: (i) The adjoint state of an HBNODE also satisfies an HBNODE, accelerating both forward and backward ODE solvers, thus significantly accelerate learning and improve the utility of the trained models. (ii) The spectrum of HBNODEs is well structured, enabling effective learning of long-term dependencies from complex sequential data.

Second, we will extend HBNODE to graph learning leveraging diffusion on graphs, resulting in new algorithms for deep graph learning. The new algorithms are more accurate than existing deep graph learning algorithms and more scalable to deep architectures, and also suitable for learning at low labeling rate regimes. Moreover, we will present a fast multipole method-based efficient attention mechanism for modeling graph nodes interactions.

Third, if time permits, we will discuss proximal algorithms for accelerating learning continuous-depth neural networks.

Application of optimal transport theory on numerical computation, analysis, and dynamical systems on graph

Series
Dissertation Defense
Time
Wednesday, March 23, 2022 - 14:00 for
Location
ONLINE
Speaker
Shu Liu – Georgia Institute of Technology –

Abstract: 

In this talk, we mainly focus on the applications of optimal transport theory from the following two aspects:

(1)Based on the theory of Wasserstein gradient flows, we develop and analyze a numerical method proposed for solving high-dimensional Fokker-Planck equations (FPE). The gradient flow structure of FPE allows us to derive a finite-dimensional ODE by projecting the dynamics of FPE onto a finite-dimensional parameter space whose parameters are inherited from certain generative model such as normalizing flow. We design a bi-level minimization scheme for time discretization of the proposed ODE. Such algorithm is sampling-based, which can readily handle computations in high-dimensional space. Moreover, we establish theoretical bounds for the asymptotic convergence analysis as well as the error analysis for our proposed method.

(2)Inspired by the theory of Wasserstein Hamiltonian flow, we present a novel definition of stochastic Hamiltonian process on graphs as certain kinds of inhomogeneous Markov process. Such definition is motivated by lifting to the probability space of the graph and considering the Hamiltonian dynamics on this probability space. We demonstrate some examples of the stochastic Hamiltonian process in classical discrete problems, such as the optimal transport problems and Schrödinger bridge problems (SBP).

The Bluejeans link: https://bluejeans.com/982835213/2740

Definable combinatorics in hyperfinite graphs

Series
Combinatorics Seminar
Time
Friday, March 18, 2022 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Matthew Bowen – McGill University –

We discuss a few new results concerning the descriptive combinatorics of bounded degree hyperfinite Borel graphs. In particular, we show that the Baire measurable edge chromatic number of $G$ is at most $\lceil\frac{3}{2}\Delta(G)\rceil+6$ when G is a multigraph, and for bipartite graphs we improve this bound to $\Delta(G)+1$ and show that degree regular one-ended bipartite graphs have Borel perfect matchings generically. Similar results hold in the measure setting assuming some hyperfiniteness conditions. This talk is based on joint work with Kun and Sabok, Weilacher, and upcoming work with Poulin and Zomback.

On mix-norms and the rate of decay of correlations

Series
CDSNS Colloquium
Time
Friday, March 18, 2022 - 13:00 for 1 hour (actually 50 minutes)
Location
Online via Zoom
Speaker
Bryan Oakley – University of Wisconsin-Madison –

Please Note: Zoom link: https://us06web.zoom.us/j/83392531099?pwd=UHh2MDFMcGErbzFtMHBZTmNZQXM0dz09

Two quantitative notions of mixing are the decay of correlations and the decay of a mix-norm --- a negative Sobolev norm --- and the intensity of mixing can be measured by the rates of decay of these quantities. From duality, correlations are uniformly dominated by a mix-norm; but can they decay asymptotically faster than the mix-norm? We answer this question by constructing an observable with correlation that comes arbitrarily close to achieving the decay rate of the mix-norm. Therefore the mix-norm is the sharpest rate of decay of correlations in both the uniform sense and the asymptotic sense. Moreover, there exists an observable with correlation that decays at the same rate as the mix-norm if and only if the rate of decay of the mix-norm is achieved by its projection onto low-frequency Fourier modes. In this case, the function being mixed is called q-recurrent; otherwise it is q-transient. We use this classification to study several examples and raise questions for future investigations. 

Hilbert's Tenth Problem and Generalizations

Series
Algebra Student Seminar
Time
Friday, March 18, 2022 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006 and Teams
Speaker
Ian Lewis – Georgia Tech
Hilbert's Tenth Problem asks whether there exists an algorithm to determine whether an arbitrary polynomial with integer coefficients has a solution or not. This problem was resolved by Matiyasevich building of the work in of Robinson, Davis, and Putnam in the 70s. We will give an overview of how this problem was resolved and the current status of various generalizations.
 

Around Bismut-type formulas for symmetric alpha-stable probability measures

Series
Stochastics Seminar
Time
Thursday, March 17, 2022 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Benjamin Arras – Université de Lille –
In this talk, I will speak about recent results regarding Bismut-type formulas for non-degenerate symmetric alpha-stable probability measures. In particular, I will present its applications to continuity properties of certain singular operators as well as to certain functional inequalities. These recent results are based on joint works with Christian Houdré.

Symmetric generating functions and permanents of totally nonnegative matrices

Series
Algebra Seminar
Time
Thursday, March 17, 2022 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Mark Skandera – Lehigh University –

For each element $z$ of the symmetric group algebra we define a symmetric generating function

$Y(z) = \sum_\lambda \epsilon^\lambda(z) m_\lambda$, where $\epsilon^\lambda$ is the induced sign

character indexed by $\lambda$. Expanding $Y(z)$ in other symmetric function bases, we obtain

other trace evaluations as coefficients. We show that we show that all symmetric functions in

$\span_Z \{m_\lambda \}$ are $Y(z)$ for some $z$ in $Q[S_n]$. Using this fact and chromatic symmetric functions, we give new interpretations of permanents of totally nonnegative matrices.

For the full paper, see https://arxiv.org/abs/2010.00458v2.

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