Seminars and Colloquia by Series

Surfaces bounded by knots in the 3-sphere

Series
Research Horizons Seminar
Time
Wednesday, November 10, 2021 - 12:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jonathan SimoneGeorgia Institute of Technology

Given a knot $K$ in the 3-sphere, one can ask: what kinds of surfaces in the 3-sphere are bounded by $K$? One can also ask: what kinds of surfaces in the 4-ball (which is bounded by the 3-sphere) are bounded by $K$? In this talk we will discuss how to construct surfaces in both the 3-sphere and in the 4-ball bounded by a given knot $K$, how to obstruct the existence of such surfaces, and explore what is known and unknown about surfaces bounded by so-called torus knots.

Semidefinite programming, convex relaxations, and low rank structure

Series
Research Horizons Seminar
Time
Wednesday, November 3, 2021 - 12:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Diego CifuentesGeorgia Tech

Semidefinite programming (SDP) is a very well behaved class of convex optimization problems. We will introduce this class of problems, illustrate how it allows to approximate many practical nonconvex optimization problems, and discuss the role of low rank structure.

Unknotting operations

Series
Research Horizons Seminar
Time
Wednesday, October 27, 2021 - 12:30 for 1 hour (actually 50 minutes)
Location
Skiles 006 / https://bluejeans.com/396232086/4264
Speaker
Hannah TurnerGeorgia Tech

Please Note: Talk will be presented live as well as streamed. Questions will be fielded by the organizer.

We'll discuss various operations which can be applied to a knot to "simplify" or "unknot" it. Study of these "unknotting operations" began in the 1800s and continues to be an active area of research in low-dimensional topology. Many of these operations have applications more broadly in topology including to 3- and 4-manifolds and even to DNA topology. I will define some of these operations and highlight a few open problems.

Combinatorics of Neural Codes

Series
Research Horizons Seminar
Time
Wednesday, October 20, 2021 - 12:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Alexander Ruys De PerezGeorgia Tech


Neural codes are inspired by John O'Keefe's discovery of the place cell, a neuron in the mammalian brain which fires if and only if its owner is in a particular region of physical space. Mathematically, a neural code $C$ on n neurons is a collection of subsets of $\{1,...,n\}$, with the subsets called codewords. The implication is that $C$ encodes how the members of some collection $\{U_i\}_{i=1}^n$ of subsets of $\mathbb{R}^d$ intersect one another. 

The principal question driving the study of neural codes is that of convexity. Given just the codewords of $C$, can we determine if there is a collection of open convex subsets $ \{U_i\}_{i=1}^n$ of some $\mathbb{R}^d$ for which $C$ is the code? A convex code is a code for which there is such a realization of open convex sets. While the question of determining which codes are convex remains open, there has been significant progress as many large families of codes can now be ruled as convex or nonconvex. In this talk, I will give an overview of some of the results from this work. In particular, I will focus on a phenomenon called a local obstruction, which if found in a code forbids convexity.    

Structure and computation of data-driven brain networks

Series
Research Horizons Seminar
Time
Wednesday, October 13, 2021 - 12:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Hannah ChoiGeorgia Tech

Please Note: The seminar will also be streamed live at https://bluejeans.com/787128769/7101 . Questions will be fielded by the organizer.

The complex connectivity structure unique to the brain network is believed to underlie its robust and efficient coding capability. One of many unique features of the mammalian brain network is its spatial embedding and hierarchical organization. I will discuss effects of these structural characteristics on network dynamics as well as their computational implications with a focus on the flexibility between modular and global computations and predictive coding.  

Surfaces of Infinite Type

Series
Research Horizons Seminar
Time
Wednesday, October 6, 2021 - 12:30 for 1 hour (actually 50 minutes)
Location
ONLINE https://bluejeans.com/506659049/8073
Speaker
Yvon VerberneGeorgia Tech

The mapping class group of a surface is well understood for surfaces of finite type. In contrast, the study of mapping class groups of infinite type surfaces is a new field with many opportunities to establish new results. In this talk, we will introduce infinite type surfaces and their mapping class groups.

https://bluejeans.com/506659049/8073

q-calculus and Stirling numbers

Series
Research Horizons Seminar
Time
Wednesday, September 29, 2021 - 12:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Orli HerscoviciGeorgia Tech

Different aspects of q-calculus are widely used in number theory, combinatorics, orthogonal polynomials, to name a few. In this talk we introduce q-calculus and consider its applications  to the Stirling numbers.

Geometric equations for matroid varieities

Series
Research Horizons Seminar
Time
Wednesday, September 22, 2021 - 12:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Ashley WheelerGeorgia Institute of Technology

Each point x in Gr(r, n) corresponds to an r × n matrix Ax which gives rise to a matroid Mx on its columns. Gel’fand, Goresky, MacPherson, and Serganova showed that the sets {y ∈ Gr(r, n)|My = Mx} form a stratification of Gr(r, n) with many beautiful properties. However, results of Mnëv and Sturmfels show that these strata can be quite complicated, and in particular may have arbitrary singularities. We study the ideals Ix of matroid varieties, the Zariski closures of these strata. We construct several classes of examples based on theorems from projective geometry and describe how the Grassmann-Cayley algebra may be used to derive non-trivial elements of Ix geometrically when the combinatorics of the matroid is sufficiently rich.

The lattice metric space and its applications

Series
Research Horizons Seminar
Time
Friday, November 13, 2020 - 11:30 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Yuchen HeGeorgia Tech
Lattice patterns are commonly observed in material sciences where microscopic structural nuances induce distinct macroscopic physical or chemical properties. Provided with two lattices of the same dimension, how do we measure their differences in a visually consistent way? Mathematically, any n-D lattice is determined by a set of n independent vectors. Since such basis-representation is non-unique, a direct comparison among basis-representations in Euclidean space is highly ambiguous. In this talk, I will focus on 2-D lattices and introduce the lattice metric space proposed in my earlier work. This geometric space was constructed mainly based on integrating the Modular group theory and the Poincaré metric. In the lattice metric space, each point represents a unique lattice pattern, and the visual difference between two patterns is measured by the shortest path connecting them. Some applications of the lattice metric space will be presented. If time allows, I will briefly discuss potential extensions to 3D-lattices.

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