Seminars and Colloquia by Series

On maximal perimeters of convex sets with respect to measures

Series
High Dimensional Seminar
Time
Wednesday, April 17, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Galyna Livshyts – Georgia Tech –

We discuss the asymptotic value of the maximal perimeter of a convex set in an n-dimensional space with respect to certain classes of measures. Firstly, we derive a lower bound for this quantity for a large class of probability distributions; the lower bound depends on the moments only. This lower bound is sharp in the case of the Gaussian measure (as was shown by Nazarov in 2001), and, more generally, in the case of rotation invariant log-concave measures (as was shown by myself in 2014). We discuss another class of measures for which this bound is sharp. For isotropic log-concave measures, the value of the lower bound is at least n^{1/8}.

In addition, we show a uniform upper bound of Cn||f||^{1/n}_{\infty} for all log-concave measures in a special position, which is attained for the uniform distribution on the cube. We further estimate the maximal perimeter of isotropic log-concave measures by n^2. 

Swindles in Mathematics

Series
Geometry Topology Student Seminar
Time
Wednesday, April 17, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Sudipta Kolay – Georgia Tech

We will see some instances of swindles in mathematics, primarily focusing on some in geometric topology due to Barry Mazur.

Discrete Optimal Transport With Applications in Path Planning and Data Clustering

Series
Dissertation Defense
Time
Wednesday, April 17, 2019 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Haoyan Zhai – Georgia Tech –

Optimal transport is a thoroughly studied field in mathematics and introduces the concept of Wasserstein distance, which has been widely used in various applications in computational mathematics, machine learning as well as many areas in engineering. Meanwhile, control theory and path planning is an active branch in mathematics and robotics, focusing on algorithms that calculates feasible or optimal paths for robotic systems. In this defense, we use the properties of the gradient flows in Wasserstein metric to design algorithms to handle different types of path planning and control problems as well as the K-means problems defined on graphs.

Analysis on Keller-Segel Models in Chemotaxis.

Series
PDE Seminar
Time
Tuesday, April 16, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Li Chen – University of Mannheim –

I this talk I will summerize some of our contributions in the analysis of parabolic elliptic Keller-Segel system, a typical model in chemotaxis. For the case of linear diffusion, after introducing the critical mass in two dimension, I will show our result for blow-up conditions for higher dimension. The second part of the talk is concentrated in the critical exponent for Keller-Segel system with porus media type diffusion. In the end, motivated from the result on nonlocal Fisher-KPP equation, we show that the nonlocal reaction will also help in preventing the blow-up of the solutions.  

Projective geometry of Wachspress coordinates

Series
Algebra Seminar
Time
Tuesday, April 16, 2019 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Kathlén Kohn – ICERM and University of Oslo –
Wachspress defined barycentric coordinates on polygons in 1975. Warren generalized his construction to higher dimensional polytopes in 1996. In contrast to the classical case of simplices, barycentric coordinates on other polytopes are not unique. So the coordinates defined by Warren are now commonly known as Wachspress coordinates. They are used in a variety of applications, such as geometric modeling.
We connect the constructions by Warren and Wachspress by proving the conjecture that there is a unique polynomial of minimal degree which vanishes on the non-faces of a simple polytope. This is the adjoint polynomial introduced by Warren. Our formulation is the natural generalization of Wachspress' original idea.
The algebraic geometry of the map defined by the Wachspress coordinates was studied in the case of polygons by Irving and Schenk in 2014. We extend their results to higher dimensional polytopes. In particular, we show that the image of this Wachspress map is the projection from the image of the adjoint. For three-dimensional polytopes, we show that their adjoints are adjoints of K3- or elliptic surfaces. This talk is based on joint works with Kristian Ranestad, Boris Shapiro and Bernd Sturmfels.

Doubly slice knots and L^2 signatures by Patrick Orson

Series
Geometry Topology Seminar
Time
Monday, April 15, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Patrick Orson – Boston College –

The question of which high-dimensional knots are slice was entirely solved by Kervaire and Levine. Compared to this, the question of which knots are doubly slice in high-dimensions is still a largely open problem. Ruberman proved that in every dimension, some version of the Casson-Gordon invariants can be applied to obtain algebraically doubly slice knots that are not doubly slice. I will show how to use L^2 signatures to recover the result of Ruberman for (4k-3)-dimensional knots, and discuss how the derived series of the knot group might be used to organise the high-dimensional doubly slice problem.

Solving Inverse Problems on Networks: Graph Cuts, Optimization Landscape, Synchronization

Series
Applied and Computational Mathematics Seminar
Time
Monday, April 15, 2019 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Shuyang Ling – New York University –
Information retrieval from graphs plays an increasingly important role in data science and machine learning. This talk focuses on two such examples. The first one concerns the graph cuts problem: how to find the optimal k-way graph cuts given an adjacency matrix. We present a convex relaxation of ratio cut and normalized cut, which gives rise to a rigorous theoretical analysis of graph cuts. We derive deterministic bounds of finding the optimal graph cuts via a spectral proximity condition which naturally depends on the intra-cluster and inter-cluster connectivity. Moreover, our theory provides theoretic guarantees for spectral clustering and community detection under stochastic block model. The second example is about the landscape of a nonconvex cost function arising from group synchronization and matrix completion. This function also appears as the energy function of coupled oscillators on networks. We study how the landscape of this function is related to the underlying network topologies. We prove that the optimization landscape has no spurious local minima if the underlying network is a deterministic dense graph or an Erdos-Renyi random graph. The results find applications in signal processing and dynamical systems on networks.

Prime tropical ideals

Series
Algebra Seminar
Time
Monday, April 15, 2019 - 12:50 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Kalina Mincheva – Yale University –

Tropical geometry provides a new set of purely combinatorial tools, which has been used to approach classical problems. In tropical geometry most algebraic computations are done on the classical side - using the algebra of the original variety. The theory developed so far has explored the geometric aspect of tropical varieties as opposed to the underlying (semiring) algebra and there are still many commutative algebra tools and notions without a tropical analogue. In the recent years, there has been a lot of effort dedicated to developing the necessary tools for commutative algebra using different frameworks, among which prime congruences, tropical ideals, tropical schemes. These approaches allows for the exploration of the  properties of tropicalized spaces without tying them up to the original varieties and working with geometric structures inherently defined in characteristic one (that is, additively idempotent) semifields. In this talk we explore the relationship between tropical ideals and congruences to conclude that the variety of a prime (tropical) ideal is either empty or consists of a single point. This is joint work with D. Joó.

High-dimensional knots, and rho-invariants by Patrick Orson

Series
Geometry Topology Seminar Pre-talk
Time
Monday, April 15, 2019 - 12:45 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Patrick Orson – Boston College –

I will give a brief survey of concordance in high-dimensional knot theory and how slice results have classically been obtained in this setting with the aid of surgery theory. Time permitting, I will then discuss an example of how some non-abelian slice obstructions come into the picture for 1-knots, as intuition for the seminar talk about L^2 invariants.

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