IMA Webinar

Speaker: Dr. Pratik Misra, North Carolina State University

Abstract: Gaussian graphical models are semi-algebraic subsets of the cone of positive definite covariance matrices. They are widely used throughout natural sciences, computational biology and many other fields. In this talk, I will give a brief idea of some concepts of algebraic geometry and graph theory like vanishing ideals, toric ideals, complete graphs and graph separation. Computing the vanishing ideal of the model gives us an implicit description of the model. I will mainly talk about the conjecture given by Sturmfels and Uhler. In particular, we characterize those graphs for which the vanishing ideal of the Gaussian graphical model is generated in degree 1 and 2. These turn out to be the Gaussian graphical models whose ideals are toric ideals, and the resulting graphs are the 1-clique sums of complete graphs.

Speaker: Pratyush Kumar Mishra, North Dakota State University

Abstract: Group actions on manifolds have attracted people from different fields of mathematics such as group theory, low dimensional topology, dynamical systems, and differential geometry. Among group action on 1-manifolds, in this talk we will be looking at the group of piecewise linear homeomorphisms of the closed interval I= [0,1] (denoted PL_+(I)). We will study this infinite-dimensional group using algebraic and dynamical methods. By the end of the talk, we will see a complete classification of solvable and non-solvable subgroups of PL_+(I) as appeared in the PhD thesis of Collin Bleak in 2005.

Speaker: Prayagdeep Parija, University of Wisconsin-Milwaukee

Abstract: What properties does a "typical" group have? In this talk we will introduce Gromov's notion of Random groups and discuss the foundational density 1/2 theorem.

Speaker: Nilkantha Das, NISER-Bhubaneswar

Abstract: We give a brief introduction to enumerative geometry of curves. We show that the number of cubics in CP2 that have only one node, and that pass through 8 generic points in the plane is 12. We also consider a curve counting problem involving excess intersection theory..

Speaker: Dr. Azizul Hoque, Rangapara College

Abstract: We will discuss some basic results about orders in the ring of integers of quadratic fields and then we will show how (classes of) binary quadratic forms can be associated to (classes of) ideals in such orders and vice-versa. Combining these results, we will give the precise correspondence between classes of binary quadratic forms and narrow Picard groups of orders/classes of ideals in orders in quadratic fields. If time permits, we will look into some of its applications.

Speaker: Dr. Ashani Dasgupta, University of Wisconsin-Milwaukee

Abstract: There is a large class of groups called relatively hyperbolic groups. These groups have boundaries (we will intuitively learn what that means). There is a 30-years old question regarding these boundaries: are they locally connected? We will explore a weird (and fun) journey toward its solution in the most general case.

[JOINT WORK WITH PROFESSOR CHRIS HRUSKA, UNIVERSITY OF WISCONSIN-MILWAUKEE. THIS WORK IS BASED ON AN UNPUBLISHED RESULT THAT IS ACCEPTED BY THE DOCTORAL COMMITTEE]

Speakers:

1. Dr. Tattwamasi Amrutam, PDF, Ben-Gurion University of theNegev, ISRAEL

2. Dr. Pratik Misra, PDF, KTH, Royal Institute of Technology, Stockholm, Sweden

3. Prayagdeep Parija, Graduate Student, University of Wisconsin-Milwaukee, USA

4. Pratyush Kumar Mishra, Graduate Student, North Dakota StateUniversity, USA

Seminar PPT slides AND Recorded Video:

https://drive.google.com/file/d/1ft9Z3Mzm_MIXSMfInFQNb4YO-Xj6kAaN/view?usp=sharing and https://www.youtube.com/watch?v=xDtghjPY3Yo

Speaker: Dr. Deepak Kumar Pradhan, ISI, Bangalore, India

Abstract: A little more than a hundred years ago Georg A. Pick and Rolf Nevanlinna discovered an interpolation result on the unit disc. We will discuss the Pick-Nevanlinna interpolation and the far reaching consequences the interpolation problem has on contemporary operator theory.

Speaker: Dr. Manjil Saikia

School of Mathematics, Cardiff University, United Kingdom

Abstract: A very interesting combinatorial phenomena is that of combinatorial reciprocity. In brief, a polynomial whose values at positive integers count certain combinatorial objects, sometimes gives the number of combinatorial objects of a different kind when the polynomial is evaluated at negative integers (with normalization). We will work out some examples of this phenomenon, involving graphs, posets and other combinatorial objects. No background knowledge will be assumed, apart from a certain level of mathematical maturity.