Time & Location: All talks are on Thursdays in TBA at 3:30 pm unless otherwise noted. Refreshments in Gibson 426 after the talk.
Organizer: Tommaso Buvoli and Samuel Punshon-Smith
Thursday, January 16
Title: On the flow of zeros of derivatives of polynomials
Andrei Martinez-Finkelshtein - Baylor University (Host: Ken)
Abstract: Assume we have a sequence of polynomials whose asymptotic zero distribution is known. What can be said about the zeros of their derivatives? Especially if we differentiate each polynomial several times, proportional to its degree? This simple-to-formulate problem has recently attracted the attention of researchers. Both the problem and the methods of its solution have exciting connections with free probability, random matrices, and approximation theory on the complex plane. In this talk, I will explain some known results in this direction and our approach to the problem, which uses only some elementary complex analysis. This is a joint work with E. Rakhmanov from the University of South Florida.
Location: Gibson Hall 126
Time: 3:30
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Special Friday, January 31
Title: Partitions Detect Primes
Speaker: Ken Ono - University of Virginia (Host: Olivia)
Abstract: This talk presents “partition theoretic” analogs of the classical work of Matiyasevich that resolved Hilbert’s Tenth Problem in the negative. The Diophantine equations we consider involve equations of MacMahon’s partition functions and their natural generalizations. Here we explicitly construct infinitely many Diophantine equations in partition functions whose solutions are precisely the prime numbers. To this end, we produce explicit additive bases of all graded weights of quasimodular forms, which is of independent interest with many further applications. This is joint work with Will Craig and Jan-Willem van Ittersum.
Location: Gibson Hall 126 Subject to change
Time: 4:00
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Tuesday, February 25
Title: Dynamical Systems Insights into Map Enumeration
Joceline Lega - University of Arizona - Host: (Ken McLaughlin)
Abstract: This talk will highlight techniques from discrete dynamical systems theory that have led to significant advances in map enumeration. We will start with a brief overview of generating functions for map counts and their relation to solutions of the discrete Painlevé I equation. We will then present computational and analytical results that lead to specific generating functions and, in the case of 4-valent maps, derive explicit expressions for map counts as functions of the number of vertices and the genus of the surface on which the map is embedded. We will conclude with open as well as recently solved questions associated with this research program. This is joint work with Nick Ercolani and Brandon Tippings..
Location: TBA
Time: 4:30
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Thursday, March 13
Title: The Feynman-Lagerstrom criterion for boundary layers
Theo Drivas - Affiliation: SUNY Stony Brook (Host: Sam)
Abstract: We study the boundary layer theory for slightly viscous stationary flows forced by an imposed slip velocity at the boundary. According to the theory of Prandtl (1904) and Batchelor (1956), any Euler solution arising in this limit and consisting of a single ``eddy" must have constant vorticity. Feynman and Lagerstrom (1956) gave a procedure to select the value of this vorticity by demanding a necessary condition for the existence of a periodic Prandtl boundary layer description. We will show that this condition is also sufficient for the existence of a periodic boundary layer described by the Prandtl equations. This is joint work with S. Iyer and T. Nguyen.
Location: Gibson Hall 126
Time: 3:30
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Thursday, March 27
Title: INTEGRABLE COMBINATORICS
Philippe Di Francesco - University of Illinois Urbana-Champaign Host: (Ken McLaughlin)
Abstract: Combinatorics has constantly evolved from the mere counting of classes of objects to the study of their underlying algebraic or analytic properties, such as symmetries or deformations. This was fostered by interactions with in particular statistical physics, where the objects in the class form a statistical ensemble, where each element comes with some probability. Integrable systems form a special subclass: that of systems with sufficiently many symmetries to be amenable to exact solutions.
In this talk, we explore various basic combinatorial problems involving discrete surfaces, dimer models of cluster algebra, or two-dimensional vertex models, whose (discrete or continuum) integrability manifests itself in different manners: commuting operators, conservation laws, flat connections, quantum Yang-Baxter equation, etc. All lead to often simple and beautiful exact solutions.
Location: Gibson Hall 126
Time: 3:30
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Thursday, April 3
Title: Matrices, Polynomials, and Linear Recurrences over Finite Fields
Sudhir Ghorpade - IIT Bombay Host: (Can)
Abstract: Let F be a finite field. We will discuss a variety of questions concerning the enumeration of certain classes of matrices with entries F, polynomials with coefficients in F, and homogeneous linear recurrence sequences of elements of F. Here is a sample of the questions, in their simplest form, that we will consider.
1. What is the number of nonsingular matrices of a given size whose order is maximum in the corresponding general linear group?
2 What is the probability that two polynomials of positive degrees are relatively prime?
3. What is the number of Hankel matrices of a given rank?
4. What is the number of primitive linear recurrence sequences of a given order?
We will attempt to outline some results and conjectures concerning these questions and their extensions and generalisations. Connection of these with combinatorics, cryptography, and algebra may also be indicated.
Parts of this talk are a joint work with Sartaj Ul Hasan and Meena Kumari, with Samrith Ram, and with Mario Garcia Armas and Samrith Ram.
Location: Gibson Hall 126
Time: 3:30
Special Tuesday, April 8
Title: The inviscid limit from Navier-Stokes to compressible Euler equations involving shock waves.
Geng Chen - University of Kansas Host: (Scott McKinley)
Abstract: The solutions of compressible Euler equations often form discontinuities, i.e. shock waves, in finite time, notably observed behind supersonic planes. A very natural way to justify these singularities involves studying solutions from inviscid limits of Navier-Stokes solutions. The mathematical study of this problem is however very difficult because of the destabilization effect of the viscosities.
Bianchini and Bressan proved the inviscid limit to small BV solutions in one space dimension using the so-called artificial viscosities in 2004. However, until very recently, achieving this limit with physical viscosities remained an open question. In this presentation, the recent advances on the L2 theory of compressible fluid mechanics will be introduced. This method is employed to describe the physical inviscid limit in the context of the barotropic Euler equations, and to solve the Bianchini and Bressan's conjecture. This is a joint work with Moon-Jin Kang and Alexis Vasseur.
Our current research program on the uniqueness and stability of shock wave in mutliple space dimension will be introduced in the end of the presentation.
A Colloquium Reception will be held in the common room at 3 pm BEFORE the colloquium.
Location: Tilton Hall 305
Time: 3:30 PM
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Special Tuesday, April 22
Title: Resonances
Toan T Nguyen - Penn State University
Abstract: Of great interest is to establish the large time dynamical behavior of a fluid or plasma in a non-equilibrium state, whether transitioning to turbulence or relaxing to neutrality, a resolution of which requires a deep understanding of resonances (mainly between particles and waves). This talk aims to provide a roadmap towards resolving the problem, plus the current state of the art.
Location: Tilton Hall, Room 305
Time: 3:30 PM
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