Title: Sums of quadratic functions with two discriminants and Farkas’ identities with quartic characters
Abstract:
In a 1999 paper, Zagier discusses a construction of a function $F_{k,D}(x)$ defined for an even integer $k ge 2$, and a positive discriminant $D$. This construction is intimately related to half-integral weight modular forms. In particular, the average value of this function is a constant multiple of the $D$-th Fourier coefficient of weight $k+1/2$ Eisenstein series constructed by H. Cohen in cite{Cohen}.
In this dissertation, we consider a construction which works both for even and odd positive integers $k$. Our function $F_{k,D,d}(x)$ depends on two discriminants $d$ and $D$ with signs $sign(d)=sign(D)=(-1)^k$, degenerates to Zagier’s function when $d=1$, namely,
$$
F_{k,D,1}(x)=F_{k,D}(x),
$$
and has very similar properties. In particular, we prove that the average value of $F_{k,D,d}(x)$ is again a Fourier coefficient of H. Cohen’s Eisenstein series of weight $k+1/2$, while now the integer $k ge 2$ is allowed to be both even and odd.
In a 2004 paper, Farkas introduces a new arithmetic function and proves an identity involving this function. Guerzhoy and Raji cite{Guerzhoy} generalize this function for primes that are congruent to 3 modulo 4 by introducing a quadratic Dirichlet character and find another identity of the same type. We look at the case when $p equiv 5 Mod 8$ by introducing quartic Dirichlet characters and prove an analogy of their generalization.
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Title: Scattering Theory
Abstract:
Scattering theory is the mathematical formalism of interactions in quantum mechanics. Quantum mechanics being the physical theory that appeared at the turn of the 20th century when it became apparent that not only did light behave as a wave, but matter as well. Further, the waves describing matter were quantized. A classic example of this is the photoelectric effect, where light can only interact with electrons when the light has the same quantized energy as the electron. In this work I will derive mathematically properties of the Schroedinger operator, an unbounded operator on Hilbert space and how its spectra can include both continuous and discrete components. The solutions to the Schroedinger equation are either normalizable or not depending on whether they have eigenvalues in the continuous or discrete spectra respectively. However, the solutions/eigenfunctions for the continuous or scattered spectra are only approximate eigenfunctions, in a sense we will make explicit. Their existence is shown through a rigorous treatment of the Schroedinger operator with no potential – the free state. Next scattering processes for some simple one dimensional cases will be shown, in which the time-independent solutions are related to time-dependent solutions. This will be followed by two dimensional scattering with localized and long range potentials.
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Speaker: Nayantara Bhatnagar (U. Delaware)
Title: Subsequence Statistics in Random Mallows Permutations
Abstract: The longest increasing subsequence (LIS) of a uniformly random permutation is a well studied problem. Vershik-Kerov and Logan-Shepp first showed that asymptotically the typical length of the LIS is 2sqrt(n). This line of research culminated in the work of Baik-Deift-Johansson who related this length to the GUE Tracy-Widom distribution.
We study the length of the LIS of random permutations drawn from the Mallows measure, introduced by Mallows in connection with ranking problems in statistics. We prove limit theorems for the LIS for different regimes of the parameter of the distribution. I will also describe some recent results on the longest common subsequence of independent Mallows permutations.
Relevant background for the talk will be introduced as needed.
This semester, the Analysis Seminar will meet on Tuesdays 3:30 – 4:20 pm in Keller 402.
Title :
What are the possible shapes of polynomial Julia sets?
Abstract :
Ever since the digital revolution and the emergence of computers, mathematicians have been fascinated by fractals, those geometric figures showing self-similar patterns and irregular structures. A well-known family of fractals introduced by Gaston Julia and Pierre Fatou in the early 20th century are the so-called Julia sets, obtained from the iteration of a polynomial of one complex variable. It has been known for a long time how rich and diverse the geometry of these Julia sets are, from Cantor sets to smooth curves as well as highly irregular figures.
But what exactly are the possible shapes of polynomial Julia sets? This question, raised by Bill Thurston shortly before he passed away, has a rather surprising answer : they can have any shape, except some trivial topological obstructions. In this talk, I will present the ideas underlying the proof of this result, which gives an explicit construction. In particular, we will see how potential theory comes into play. I will also discuss some related computational aspects.