Seminars and Colloquia by Series

Higher dimensional fractal uncertainty

Series
Analysis Seminar
Time
Wednesday, November 1, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Alex CohenMIT

The fractal uncertainty principle (FUP) roughly says that a function and its Fourier transform cannot both be concentrated on a fractal set. These were introduced to harmonic analysis in order to prove new results in quantum chaos: if eigenfunctions on hyperbolic manifolds concentrated in unexpected ways, that would contradict the FUP. Bourgain and Dyatlov proved FUP over the real numbers, and in this talk I will discuss an extension to higher dimensions. The bulk of the work is constructing certain plurisubharmonic functions on C^n. 

The Erdős-Szekeres problem

Series
Graph Theory Seminar
Time
Tuesday, October 31, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Cosmin PohoataEmory University

For every natural number n, if we start with sufficiently many points in R^d in general position there will always exist n points in convex position. The problem of determining quantitative bounds for this statement is known as the Erdős-Szekeres problem, and is one of the oldest problems in Ramsey theory. We will discuss some of its history, along with the recent developments in the plane and in higher dimensions.

Long time behavior in cosmological Einstein-Belinski-Zakharov spacetimes

Series
PDE Seminar
Time
Tuesday, October 31, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Online: https://gatech.zoom.us/j/95574359880?pwd=cGpCa3J1MFRkY0RUeU1xVFJRV0x3dz09
Speaker
Claudio MuñozUniversidad de Chile

In this talk I will present some recent results in collaboration with Jessica Trespalacios where we consider Einstein-Belinski-Zakharov spacetimes and prove local and global existence, long time behavior of possibly large solutions and some applications to gravisolitons of Kasner type.

The Burau representation and shapes of polyhedra by Ethan Dlugie

Series
Geometry Topology Seminar
Time
Monday, October 30, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Speaker

The Burau representation is a kind of homological representation of braid groups that has been around for around a century. It remains mysterious in many ways and is of particular interest because of its relation to quantum invariants of knots and links such as the Jones polynomial. In recent work, I came across a relationship between this representation and a moduli space of Euclidean cone metrics on spheres (think e.g. convex polyhedra) first examined by Thurston. After introducing the relevant definitions, I'll explain a bit about this connection and how I used the geometric structure on this moduli space to exactly identify the kernel of the Burau representation after evaluating its formal parameter at complex roots of unity. There will be many pictures!

Lie algebra representations, flag manifolds, and combinatorics. An old story with new twists

Series
Algebra Seminar
Time
Monday, October 30, 2023 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Cristian LenartSUNY Albany

Please Note: There will be a pre-seminar from 11am to 11:30am (aimed toward grad students and postdocs) in Skiles 006.

The connections between representations of complex semisimple Lie algebras and the geometry of the corresponding flag manifolds have a long history. Moreover, combinatorics plays an important role in the related computations. My talk is devoted to new aspects of this story. On the Lie algebra side, I consider certain modules for quantum affine algebras. I discuss their relationship with Macdonald polynomials, which generalize the irreducible characters of simple Lie algebras. On the geometric side, I consider the quantum K-theory of flag manifolds, which is a K-theoretic generalization of quantum cohomology. A new combinatorial model, known as the quantum alcove model, is also presented. The talk is based on joint work with S. Naito, D. Sagaki, A. Schilling, and M. Shimozono.

Arnold Tongues in Standard Maps with Drift

Series
CDSNS Colloquium
Time
Friday, October 27, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 249
Speaker
Jing ZhouGreat Bay University

In the early 60’s J. B. Keller and D. Levy discovered a fundamental property: the instability tongues in Mathieu-type equations lose sharpness with the addition of higher-frequency harmonics in the Mathieu potentials. Twenty years later, V. Arnold discovered a similar phenomenon on the sharpness of Arnold tongues in circle maps (and rediscovered the result of Keller and Levy). In this paper we find a third class of object where a similar type of behavior takes place: area-preserving maps of the cylinder. loosely speaking, we show that periodic orbits of standard maps are extra fragile with respect to added drift (i.e. non-exactness) if the potential of the map is a trigonometric polynomial. That is, higher-frequency harmonics make periodic orbits more robust with respect to “drift". This observation was motivated by the study of traveling waves in the discretized sine-Gordon equation which in turn models a wide variety of physical systems. This is a joint work with Mark Levi.

The asymptotics of $r(4,t)$

Series
Combinatorics Seminar
Time
Friday, October 27, 2023 - 15:15 for 1 hour (actually 50 minutes)
Location
Skiles 308
Speaker
Sam MattheusVrije Universiteit Brussel

I will give an overview of recent work, joint with Jacques Verstraete, where we gave an improved lower bound for the off-diagonal Ramsey number $r(4,t)$, solving a long-standing conjecture of Erd\H{o}s. Our proof has a strong non-probabilistic component, in contrast to previous work. This approach was generalized in further work with David Conlon, Dhruv Mubayi and Jacques Verstraete to off-diagonal Ramsey numbers $r(H,t)$ for any fixed graph $H$. We will go over of the main ideas of these proofs and indicate some open problems.

Derivation and analysis of discrete population models with delayed growth

Series
Mathematical Biology Seminar
Time
Friday, October 27, 2023 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Sabrina StreipertUniversity of Pittsburgh, Department of Mathematics

Please Note: The hybrid version of this talk will be available at: https://gatech.zoom.us/j/92357952326

Discrete delay population models are often considered as a compromise between single-species models and more advanced age-structured population models, C.W. Clark, J. Math. Bio. 1976. This talk is based on a recent work (S. Streipert and G.S.K. Wolkowicz, 2023), where we provide a procedure for deriving discrete population models for the size of the adult population at the beginning of each breeding cycle and assume only adult individuals reproduce. This derivation technique includes delay to account for the number of breeding cycles a newborn individual remains immature and does not contribute to reproduction. These models include a survival probability (during the delay period) for the immature individuals, since these individuals have to survive to reach maturity and become members of what we consider the adult population. We discuss properties of this class of discrete delay population models and show that there is a critical delay threshold. The population goes extinct if the delay exceeds this threshold. We apply this derivation procedure to two well-known population models, the Beverton–Holt and the Ricker population model. We analyze their dynamics and compare it to existing delay models.

The clique chromatic number of sparse random graphs

Series
Stochastics Seminar
Time
Thursday, October 26, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Manuel FernandezGeorgia Tech

The clique chromatic number of a graph is the smallest number of colors in a vertex coloring so that no inclusion-maximal clique is monochromatic (ignoring isolated vertices). For the binomial random graph G_{n,p} the clique chromatic number has been studied in a number of works since 2016, but for sparse edge-probabilities in the range n^{-2/5} \ll p \ll 1 even the order of magnitude remained a technical challenge.

Resolving open problems of Alon and Krivelevich as well as Lichev, Mitsche and Warnke, we determine the clique chromatic number of the binomial random graph G_{n,p} in most of the missing regime: we show that it is of order (\log n)/p for edge-probabilities n^{-2/5+\eps} \ll p \ll n^{-1/3} and n^{-1/3+\eps} \ll p \ll 1, for any constant \eps > 0. Perhaps surprisingly for a result about random graphs, a key ingredient in the proof is an application of the probabilistic method (that hinges on careful counting and density arguments).

This talk is based on joint work with Lutz Warnke.

Meet My Muse: the MMM classes

Series
Geometry Topology Student Seminar
Time
Wednesday, October 25, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jaden AiGeorgia Tech

Mapping class groups of surfaces in general have cohomology that is hard to compute. Meanwhile, within something called the cohomologically-stable range, a family of characteristic classes called the MMM classes (of surface bundles) is enough to generate this cohomology and thus plays an important role for understanding both the mapping class group and surface bundles. Moreover, constructing the so-called Atiyah-Kodaira manifold provides the setting to prove that these MMM classes are non-trivial. Most of this beginner-friendly talk will be dedicated to proving the non-triviality of the first MMM class. Maybe as a side quest, we will also give a crash course on the geometric viewpoint of (co)homology and then apply this viewpoint to understand the constructions and the proofs.

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