Time & Location: All talks are on Thursdays in Norman Mayer 200B at 3:30 pm unless otherwise noted. Refreshments in Gibson 426 after the talk.
Organizer: Tommaso Buvoli and Samuel Punshon-Smith
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***Tuesday, January 20,***
Special Colloquium
Title: Evolution equations in physical and biological systems
Speaker: Selim Sukhtaiev - Auburn University
Abstract: Disorder and pattern formation are central themes in modern science, and both play a fundamental role in the behavior of complex physical and biological systems. In this talk, we will discuss two mathematical models that illustrate these phenomena: the Anderson model of electronic transport in random media and the Keller–Segel model of chemotaxis.
We will first turn to a mathematical treatment of the Anderson model. We will discuss several natural Hamiltonians on metric trees with random branching numbers and show that their transport properties are suppressed by disorder. This phenomenon, known as Anderson localization, is a hallmark of the spectral theory of Schrodinger operators.
We will then consider the Keller–Segel system, a coupled pair of reaction–advection–diffusion equations describing the collective motion of cells in response to chemical signals. We will focus on well-posedness of this system on arbitrary compact networks, as well as the asymptotic stability, instability, and bifurcation of steady states in both the parabolic–parabolic and parabolic–elliptic realizations of the Keller–Segel model.
Location: Dinwiddie 108
Time: 3:30PM
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***Thursday, January 22,***
Special Colloquium
Title: Large Effects in Perturbed Hamiltonian Systems
Speaker: Marian Gidea - Yeshiva University
Abstract: One of the fundamental laws of physics is the conservation of energy, which states that the total energy of an isolated system remains constant.
Hamiltonian dynamics provides a natural framework for modeling this law. However, real-life systems are rarely isolated and are subject to external perturbations of various types, such as periodic / quasi-periodic forcing, random perturbations, or dissipation. In this lecture, we will consider several models from celestial mechanics, engineering, and biology, and study the effects of perturbations on these systems. The upshot is that even small perturbations can accumulate over time, giving rise to large effects, such as significant energy growth, and trajectories that wander far from their initial point. In particular, we will address conjectures proposed by Arnold (1964) and Chirikov (1979).
Location: Dinwiddie 108
Time: 3:30PM
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***Thursday, January 29,***
Special Colloquium
Title: Tractability of chaotic dynamics in noisy systems
Speaker: Alex Blumenthal - Georgia Tech
Abstract: Many real-world systems exhibit dynamical chaos, characterized by sensitive dependence on initial conditions and intricate, seemingly disordered behavior. While existing abstract tools from smooth ergodic theory provide a rich framework for understanding chaotic dynamics, verifying this framework in concrete systems remains a notoriously difficult problem. Even in low-dimensional toy models, rigorous proofs often lag significantly behind compelling numerical evidence. Remarkably, this problem becomes far more tractable when systems are subjected to external, time-dependent stochastic forcing. In such settings, the scope of systems for which chaotic hallmarks can be rigorously established expands dramatically, offering substantive progress toward the original promise of chaos theory: to explain and quantify dynamical disorder in nature. I will present several applications of these ideas, including towards disordered dynamical behavior exhibited in systems from fluid mechanics. This talk will include joint work with many collaborators, including Lai-Sang Young, Jinxin Xue, Jacob Bedrossian, and Sam Punshon-Smith.
Location: Dinwiddie 108
Time: 3:30PM
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Thursday, February 05,
Colloquium
Title: The prime number theorem in short intervals
Speaker: Ayla Gafni - Ole Miss (Host: Olivia)
Abstract: One form of the prime number theorem asserts that
$$\sum_{n\le x} \Lambda(n) = x(1 + o(1)),$$
where $\Lambda(n)$ is the von Mangoldt function. By the triangle inequality, this also gives
$$\sum_{x < n\le x+y} \Lambda(n) = y(1 + o(1)),$$
in the ``long interval'' setting $y\sim x$. It is expected that the prime number theorem holds for much shorter intervals, namely for $y\sim x^{\theta}$ for any fixed $\theta\in (0,1]$. From the recent zero density estimates of Guth and Maynard, this result is known for all $x$ when $\theta > \frac{17}{30} $ and for almost all $x$ when $\theta > \frac{2}{15}$. In this talk, we will discuss the connections between zero density estimates, the prime number theorem in short intervals, and the distribution of prime numbers. Further, we will present some quantitative upper bounds on the size of the exceptional set where the prime number theorem in short intervals fails. We give an explicit relation between zero density estimates and exceptional set bounds, allowing for the most recent zero density estimates to be directly applied to give upper bounds on the exceptional set via a small amount of computer assistance. This talk is based on joint work with Terence Tao.
Location: Norman Mayer 200B
Time: 3:30
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Thursday, February 26,
Colloquium
Title: TBA
Speaker: Nancy Neudauer - Pacific University (Host: Mahir)
Abstract: TBA
Location: Norman Mayer 200B
Time: 3:30
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Thursday, March 05,
Colloquium
Title: TBA
Speaker: Sergio R. López-Permouth - Ohio University (Host: Mahir)
Abstract: TBA
Location: Norman Mayer 200B
Time: 3:30
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Thursday, March 12,
Colloquium
Title: TBA
Speaker: Bernhard Heim - Universitat Koln (Host: Olivia)
Abstract: TBA
Location: Norman Mayer 200B
Time: 3:30
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Thursday, March 19,
Colloquium
Title: TBA
Speaker: Ivan Corwin - Columbia University (Host: Guido)
Abstract: TBA
Location: Norman Mayer 200B
Time: 3:30
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Thursday, April 09,
Colloquium
Title: Linear Flows on Translation Prisms
Speaker: Jayadev S. Athreya - University of Washington (Host: Kalina & Edna)
Abstract: We will share a story that brings together geometry, dynamics, and number theory in interesting and novel ways, and also creates some very compelling imagery- the talk will have lots of pictures, and all relevant background notions will be introduced and explained. Motivated by the study of billiards in polyhedra, we study linear flows in a family of singular flat 3-manifolds which we call translation prisms. Using ideas of Furstenberg, and Veech, we connect results about weak mixing properties of flows on translation surfaces to ergodic properties of linear flows on translation prisms, and use this to obtain several results about unique ergodicity of these prism flows and related billiard flows. Furthermore, we construct explicit eigenfunctions for translation flows in pseudo-Anosov directions with Pisot-Vijayraghavan expansion factors, and use this construction to build explicit examples of non-ergodic prism flows, and non-ergodic billiard flows in a right prism over a regular n-gon for n = 7, 9, 14, 16, 18, 20, 24, 30. This is joint work with Nicolas Bedaride, Pat Hooper, and Pascal Hubert.
Location: Norman Mayer 200B
Time: 3:30
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Thursday, April 16,
Colloquium
Title: TBA
Speaker: Henry Adams - University of Florida (Host: Rafal)
Abstract: TBA
Location: Norman Mayer 200B
Time: 3:30
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Thursday, April 23,
Colloquium
Title: TBA
Speaker: Nick Andersen - BYU (Host: Olivia)
Abstract: TBA
Location: Norman Mayer 200B
Time: 3:30
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