Math Colloquia

"Variations and Applications of Carleson's theorem" Abstract: Carleson's theorem of the 1960s states that a Fourier series with square summable coefficients converges almost everywhere. In the past ten years this theorem has been much studied and generalized, and its variants have applications to singular integrals along vector fields and to theorems in ergodic theory. We will present the theorem and elaborate on its variants and applications.

The study of computational complexity presents challenging mathematical problems. In Complexity Theory computational problems are classified into complexity classes, the best known include P, NP and Valiant's class #P for counting problems. A central problem in Valiant's theory is the permanent vs. determinant problem. We will report some latest progress on this problem. Graph homomorphism was introduced by Lovasz over 40 years ago, and it is also called the partition functions in Statistical Physics, and can encode a rich class of counting problems: Given an $m \times m$ symmetric matrix $A$ over the complex numbers, compute the function $Z_A(\cdot)$, where for an arbitrary input graph $G$, \[ Z_A(G) = \sum_{\xi:V(G)\rightarrow [m]} \prod_{(u,v)\in E(G)} A_{\xi(u),\xi(v)}.\] Our foucs is the computational complexity of $Z_A(\cdot)$. With Xi Chen and Pinyan Lu, we have achieved a complete classification theorem for the complexity of $Z_A(\cdot)$. The classification proof is too complicated to present, but we will present the proof of a lemma. It states that in order to be computable in polynomial time, the matrix $A$ must possess a group structure. Another component of the proof uses Gauss sums. (In a subsequent Number Theory Seminar I will present some related work.) No prior knowledge of complexity theory is assumed.

Many classical fractals, such as the Sierpinski gasket, and Julia sets of polynomials can be described through groups generated by finite automata. Automaton groups also provide a rich source of examples (such as Grigorchuk group of intermediate growth) and play an important role in geometric group theory. I will talk about how random walks on fractals can be used to understand their structure.

The cohesive movement of a biological population is a commonly observed natural phenomenon. With the advent of platforms of unmanned vehicles, such phenomena has attracted a renewed interest from the engineering community. This talk will cover a survey of the speaker's research and related work in this area ranging from aggregation models in nonlinear PDE to control algorithms and robotic testbed experiments. We conclude with a discussion of some interesting problems for the applied mathematics community.

Mirror symmetry is a phenomenon first discovered by string theorists in 1990. This was a surprising conjectural relationship between different Calabi-Yau manifolds. These are Ricci-flat, three-dimensional complex manifolds. String theory predicted that certain very difficult calculations on one of these manifolds (counting rational curves) was transformed into a manageable problem on the other. Since then, a great deal of effort has been devoted to understanding this phenomenon. In general, however, it is quite a complicated story. I will explain a surprisingly elementary theorem, due to myself, Pandharipande and Siebert, which provides a self-contained example of mirror symmetry. One side of the mirror symmetry picture involves certain commutators in a group of automorphisms of the algebraic torus, while the other side involves counting ratinoal curves on algebraic surfaces.

Effective randomness, also known as algorithmic randomness deals with the infinite binary strings which are intuitively ''random''. I will talk about the different ways in which one can take to calibrate randomness, and discuss some of the recent development in this area. In particular, I will focus on the interactions with computability and logic. This talk is aimed at a general audience.

Niccolo Tartaglia's (1449-1557) solution to solving cubic equations, which renowned mathematician and physician Girolamo Cardano wanted but Tartaglia resisted, led to one of the first intellectual property cases in Western history. Eventually, Tartaglia agreed to give Cardano what he so desired, but only if the latter promised he would not publish it. Cardano promised, and Tartaglia sent him the solution. Wasting little time, however, Cardano published the solution (along with a 'general' solution that he himself developed). Tartaglia was, not surprisingly, furious and began a vicious battle with Cardano's assistant, Ludovico Ferrari (Cardano refused to engage Tartaglia directly). But vitriolic polemics aside, there is something else rather curious about this ordeal: the solution Tartaglia gave Cardano was encrypted in a poem. This talk looks at the motives behind his "poetic solution" and what it says about the close relationship between 'poeisis' and 'mathesis' in this period of mathematics' history.

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Start with n particles at each of k points in the d-dimensional lattice, and let each particle perform simple random walk until it reaches an unoccupied site. The law of the resulting random set of occupied sites does not depend on the order in which the walks are performed, as shown by Diaconis and Fulton. We prove that if the distances between the starting points are suitably scaled, then the set of occupied sites has a deterministic scaling limit. In two dimensions, the boundary of the limiting shape is an algebraic curve of degree 2k. (For k = 1 it is a circle, as proved in 1992 by Lawler, Bramson and Griffeath.) The limiting shape can also be described in terms of a free-boundary problem for the Laplacian and quadrature identities for harmonic functions. I will describe applications to the abelian sandpile, and to Propp's rotor-router model, and show simulations that suggest intriguing (yet unproved) connections with conformal mapping. Joint work with Lionel Levine.

Sometimes the way that magic tricks work is even more amazing than the tricks themselves. I will illustrate with tricks that fool magicians (demonstrations provided). The tricks depend on hidden mathematics; combinatorics and group theory (don't worry, the talk is aimed at a general audience). The Math behind the tricks has applications to secret codes, decoding dna, robot vision and much else. Changing the tricks leads to math problems beyond our current understanding.

When several integers are added, carries occur along the way. This carries process has an amazing transition matrix (Holte). The behavior of the carries is intimately related to the mathematics of ordinary riffle shuffles. I will explain the mathematics of carries, shuffling and their connection. This is joint work with Jason Fulman

A broad overview of recent work on affine geometric properties of convex bodies will be presented. In particular, the shadows of a convex body can be used to construct new convex bodies called projection bodies. We discuss how this construction leads to new affine isoperimetric inequalities that are stronger than the classical Euclidean isoperimetric inequality. Connections to Sobolev inequalities, moment-entropy inequalities, and information theory will also be mentioned.

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The structure of rational solutions to a polynomial equation depends on the structure of corresponding algebraic variety. In case of a curve of genus zero, the problem of finding all solutions can be completely solved using Hasse-Minkowski principle. In case of genus one, the obstruction to the Hasse-Minkowski principle is conjectured to be finite; and the set of rational points is a finitely generated group by the Mordell-Weil theorem if it not empty. In case of genus two or bigger, the set of solutions is finite by Faltings theorem. A major unsolved problem today is the effectivity of solutions for curves of genus one or bigger. For elliptic curves, one has the Birch and Swinnerton-Dyer (BSD) conjecture which relates the Mordell-Weil group and the central values of L-series arising from counting rational points over finite fields. For curves of genus two or bigger, one has the ABC conjecture and its refinements providing some effective bounds for curves. In function field case, these conjectures are consequences of Bogomolov-Miyaoka-Yau.

Gross and Schoen have constructed a cohomologically trivial 1-cycle on the triple product of a curve by a modification of the diagonal cycle. In lecture 2, he will explain his recent formula for the height of this cycle in term of relative dualising sheaf. He will also explain the applications of this formula to ABC conjecture, Bogomolov Conjecture, and Tate's conjecture for variety over finite fields.

Gross and Kudla have conjectured a formula to related the height of Gross--Schoen cycles on Shimura curves and the central derivatives of the triple product L-series for triples of newforms of weight two. This conjecture is proved in his joint work with Xinyi Yuan and Wei Zhang with a great generality. In Lecture 3, Shouwu will explain this formula and its applications to rational points to elliptic curves.

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