From prime distribution in arithmetic progression to modern number theory

Speaker: 

Liang Xiao

Institution: 

UC Irvine

Time: 

Friday, May 17, 2013 - 4:00pm

Location: 

MSTB 120

To start, I will discuss Dirichlet's proof of infinitude of primes in an arithmetic progression.  This leads up to the study of special values of L-functions and their arithmetic properties.  If time permits, I will try to explain some conjectures and philosophy in this direction.

Dissipation in Composities with High-Loss and Lossles Components

Speaker: 

Alex Figotin

Institution: 

UC Irvine

Time: 

Friday, May 3, 2013 - 4:00pm

Location: 

MSTB 120

We study the energy dissipation features of systems comprised of two components one of which is highly lossy and the other lossless. One of the principal results is that all the eigenmodes of any such system split into two distinct classes, high-loss and low-loss,according to their dissipative properties. Interestingly, this splitting is more pronounced the higher the loss of the lossy component. In addition to that, the real frequencies of the high-loss eigenmodes can become very small and even can vanish entirely, which is the case of overdamping. An exhaustive analytical study of the energy, dissipated power, and quality factor for such composite systems is given.

Quantization and Algebra

Speaker: 

Vladimir Baranovsky

Institution: 

UC Irvine

Time: 

Friday, April 26, 2013 - 4:00pm

Location: 

MSTB 120

Transition from Classical Mechanics to Quantum Mechanics involves replacing a certain commutative ring by its "non commutative deformation". We will discuss some algebraic aspects of this transition, and recent developments in algebra and algebraic geometry which are motivated by applications to quantum field theory

Structured Sparse Representations with Coherent Dictionaries Based on Ratio and Difference of L1 and L2 Norms

Speaker: 

Jack Xin

Institution: 

UC Irvine

Time: 

Friday, April 19, 2013 - 4:00pm

Location: 

MSTB 120

De-mixing problems in spectroscopic imaging often require finding sparse non-negative linear combinations of library functions to match observed data. Due to misalignment and uncertainty in data measurement, the known library functions may not represent the data as well as their proper deformations. To improve data adaptivity, we expand the library to one with a group structure and impose a structured sparsity constraint so that the coefficients for each group should be sparse or even 1-sparse. Since the expanded library is a highly coherent (redundant) dictionary, it is difficult to obtain good solutions using convex methods such as non-negative least squares (NNLS) or L1 norm minimization. We study efficient non-convex penalties such as the ratio/difference of L1 and L2 norms, as sparsity penalties to be added to the objective in NNLS-type models. We show an exact recovery theory of the sparsest solution by minimizing the ratio/difference norms under a uniformity condition. For solving the related unconstrained non-convex models, we develop a scaled gradient projection algorithm that requires solving a sequence of strongly convex quadratic programs.

 

How do Computers Use Mathematics to Solve Real World Problems: A Glimpse into Computational Mathematics

Speaker: 

Hongkai Zhao

Institution: 

UC Irvine

Time: 

Friday, April 5, 2013 - 4:00pm

Location: 

MSTB 120

 

I will use concrete examples to argue why mathematics is even more important and powerful when computers become more and more powerful and to show what computational mathematics is about.

How many values a polynomial map misses?

Speaker: 

Daqing Wan

Institution: 

UCI

Time: 

Friday, February 8, 2013 - 4:00pm

Host: 

Location: 

MSTB 120

For a polynomial map f(x) from a field F to itself, we are interested in the size of the values that f misses, that is, the cardinality of F - f(F). For F = C (the complex numbers), if f misses one value, then f is a constant (this is the fundamental theorem of algebra). For F = C, if a holomorphic map f misses two values, then f is again a constant (this is Picard's little theorem). What about when f: F^n -> F^n is a polynomial vector map? When F is a finite field F_q of q elements, this problem becomes very interesting. There are extensive results and open problems available. For example, if a polynomial f of degree d>1 misses one value of F_q, then it must miss at least (q-1)/d values. In this lecture, we give a self-contained exposition of the main results and the open problems on the value set problem, and explain its link to different parts of mathematics.

Ranks of elliptic curves

Speaker: 

Karl Rubin

Institution: 

UC Irvine

Time: 

Friday, March 1, 2013 - 4:00pm

Location: 

MSTB 120

Which natural numbers occur as the area of a right triangle with three rational sides? This is a very old question and is still unsolved, although partial answers are known (for example, five is the smallest such natural number). In this talk we will discuss this problem and recent progress that has come about through its connections with elliptic curves and other important open questions in number theory.

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