All Seminars
| Title: Designing lenses with help from geometry and optimal transport |
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| Seminar: Analysis and Differential Geometry |
| Speaker: Vladimir Oliker of Emory University |
| Contact: Vladimir Oliker, oliker@mathcs.emory.edu |
| Date: 2010-02-09 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: This is a continuation (and, hopefully, conclusion) of the report on the topic discussed in previous two seminars. |
| Title: Generalized Borcherds Products and Two number theoretic applications |
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| Colloquium: Algebra |
| Speaker: Ken Ono of University of Wisconsin at Madison |
| Contact: Susan Guppy, sguppy@emory.edu |
| Date: 2010-02-08 at 3:00PM |
| Venue: MSC W201 |
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| Abstract: In his 1994 ICM lecture, Borcherds famously introduced an entirely new concept in the theory of modular forms. He established that modular forms with very special divisors can be explicitly constructed as infinite products. Motivated by problems in geometry, number theorists recognized a need for an extension of this theory to include a richer class of automorphic form. In joint work with Bruinier, the speaker has generalized Borcherds's construction to include modular forms whose divisors are the twisted Heegner divisors introduced in the 1980s by Gross and Zagier in their celebrated work on the Birch and Swinnerton-Dyer Conjecture. This generalization, which depends on the new theory of harmonic Maass forms, has many applications. The speaker will illustrate the utility of these products by resolving open problems on the following topics: 1) Parity of the partition function 2) Birch and Swinnerton-Dyer Conjecture and ranks of elliptic curves. |
| Title: Mathematical Approaches to Two Problems Related to Intravascular Blood |
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| Seminar: Numerical Analysis and Scientific Computing |
| Speaker: Aaron Fogelson of University of Utah |
| Contact: Alessandro Veneziani, ale@mathcs.emory.edu |
| Date: 2010-02-04 at 2:30PM |
| Venue: MSC W301 |
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| Abstract: Damage to the lining of a blood vessel triggers the intertwined processes of platelet aggregation and coagulation that result in the formation of a thrombus (clot) at the injury site. The thrombus itself is made up of platelets adherent to the vessel and to one another, and of a fibrin protein gel surrounding and between the platelets. An enzyme thrombin is critical to both platelet deposition and to fibrin gelation and is produced by a complex network of reactions on the vascular surface, in the blood plasma, and on the surfaces of platelets. This process happens under flow and, in turn, can strongly influence the flow. I will present work addressing two problems related to these processes: 1) How do platelet deposition and coagulation up through thrombin production interact under flow? 2) How can the rate at which thrombin produces fibrin momoners affect the ultimate branching structure of the fibrin gel? |
| Title: Mock theta functions, q-series, combinatorial probability, and percolation models |
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| Seminar: Algebra |
| Speaker: Karl Mahlburg of Princeton University |
| Contact: Susan Guppy, sguppy@emory.edu |
| Date: 2010-02-04 at 4:00PM |
| Venue: MSC W201 |
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| Abstract: In probabilistic percolation models (such as the Ising model for ferromagnetism), the sites of a lattice are randomly set to be active or inactive through an independent Bernoulli process. The system then evolves through simple deterministic growth rules that allow active sites to ``flow'' to inactive neighbors; for example, an inactive site may become active if at least two of its neighbors are active. These systems are very interesting due to the fact that they possess critical behavior, which means that there is a sharp density cutoff below which the system experiences very little growth, and above which almost every site is likely to eventually become active. Furthermore, as the size of the lattice grows to infinity, the critical probability approaches zero, and the critical window follows a finite-size (threshold) scaling relationship in this limit. Amazingly, this limiting process also leads to the surprise appearance of functions of combinatorial and number theoretic interest, including Ramanujan's famous mock theta functions as well as the generating functions for partitions without sequences. The interplay between combinatorial probability, hypergeometric q-series, and automorphic forms leads to new proofs of conjectures posed by both probabilists and number theorists, involving slow convergence results for metastability thresholds and improved asymptotics for the generating functions of partitions without k-sequences. |
| Title: Chow groups of quadrics II |
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| Seminar: Algebra |
| Speaker: Asher Auel of Emory University |
| Contact: R. Parimala, parimala@mathcs.emory.edu |
| Date: 2010-02-02 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: |
| Title: Designing lenses with help from geometry and optimal transport |
|---|
| Seminar: Analysis and Differential Geometry |
| Speaker: Vladimir Oliker of Emory University |
| Contact: Vladimir Oliker, oliker@mathcs.emory.edu |
| Date: 2010-02-02 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: This is a continuation of the presentation started at the previous seminar |
| Title: Galois group actions over the integers |
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| Colloquium: Number theory |
| Speaker: George Pappas of Michigan State University |
| Contact: Skip Garibaldi, skip@mathcs.emory.edu |
| Date: 2010-01-26 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: According to the ``normal basis theorem", if $L/K$ is a Galois extension of fields with finite Galois group $G$, then there is an element $x$ in $L$, such that the collection of all its conjugates, $g(x)$, for $g$ in $G$, forms a basis of $L$ as a vector space over $K$. This talk will describe a theme of ``integral" extensions of this classical fact to situations where a finite group acts on a system of polynomial equations with integer coefficients, i.e., when a finite group acts on a ``scheme over $\mathbf{Z}$". |
| Title: Designing lenses with help from geometry and optimal transport |
|---|
| Seminar: Analysis and Differential Geometry |
| Speaker: Vladimir Oliker of Emory University |
| Contact: Vladimir Oliker, oliker@mathcs.emory.edu |
| Date: 2010-01-26 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: Mirror and lens devices converting an incident plane wave of a given cross section and intensity distribution into an output plane wave irradiating at a given target set with prescribed intensity are required in many applications. Most of the known designs are restricted to rotationally symmetric mirrors/lenses. In this talk I will discuss designs with freeform lenses, that is, without a priori assumption of rotational symmetry. Assuming the geometrical optics approximation, it can be shown that the functions describing such freeform lenses satisfy Monge-Amp\`{e}re type partial differential equations. Because of strong nonlinearities analysis of these PDE's is difficult. Fortunately, many such problems can also be formulated geometrically and lead to problems in calculus of variations in which instead of solving the nonlinear PDE's one needs to find extrema of some Fermat-like functionals. Furthermore, discrete versions of such problems can be formulated and are useful for numerics. In this talk I will describe some of these results in the case of the lens design problem. This is work "in progress", so the presentation may be technical and require more than one seminar. |
| Title: Chow groups of quadrics |
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| Seminar: Algebra |
| Speaker: Asher Auel of Emory University |
| Contact: R. Parimala, parimala@mathcs.emory.edu |
| Date: 2010-01-19 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: |
| Title: Unitary descent properties |
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| Seminar: Algebra |
| Speaker: Eva Bayer of Swiss Federal Institute of Technology, Lausanne (EPFL) |
| Contact: R. Parimala, parimala@mathcs.emory.edu |
| Date: 2009-12-07 at 4:00PM |
| Venue: W306 |
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| Abstract: Let $k$ be a field of characteristic $\not = 2$, let $L/k$ be an odd degree extension and let $U$ be a unitary group defined over $k$. It is well--known that the natural map $H^1(k,U) \to H^1(L,U)$ is injective. Suppose that $L/k$ is Galois with group $G$. Is then $H^1(k,U) \to H^1(L,U)^G$ a bijection? This is true for orthogonal groups, and one of the main ingredients in the proof is a result of Rosenberg and Ware concerning a descent property for Witt rings of quadratic forms, namely that $W(L)^G \simeq W(k)$. This talk will present a generalization of the Rosenberg--Ware theorem to Witt groups of hermitian forms, as well as some applications of this result, in particular to the above mentioned Galois cohomology descent question. |