There are two broad areas of application of mathematics in economics: (a) economic theory, and (b) econometrics. Let me comment on both.

Regarding the former, there is need to appreciate that over the last few decades a transition of sorts has been underway in the nature of mathematics involved. Economic theory now relies more heavily on real analysis and topology. Understanding text books on modern economic theory, and more recent journal papers, requires real analysis, set theoretic  and abstract mathematical reasoning skills more than the ability to, more or less mechanically, crank out derivatives or integrals of functions.

This requires that economists consider formal maths training starting with mathematical logic/ abstract mathematical reasoning, where they learn to understand the logical anatomy of proposition and the needed proof techniques. The next steps could be to proceed to topology (see Ramrattan's comment above), followed by advanced calculas (set theoretic approach, such as in Rudin). Finally, the real analysis courses should be considered that would involve measure theory.

On the econometrics side, people either want to be able to understand and apply econometric techniques, or be able to develop such techniques themselves. I will only talk about people who are interested in applications, as this constitutes perhaps the largest group. To understand how certain estimators behave as sample size increases, knowledge of probability distributions, asymptotic distributions, sequences and convergence of sequences is needed. Real analysis will cover tbe sequence and convergence issues. Also needed is facility with matrix algebra. Most econometric papers and text books use matrix algebra and vector differentiation. So the reader should get familiar with the type of linear algebra found in Greene econometrics book. This would enable the readers to understand very well the econometric  techniques they want to apply. 

These requirements are quite substantial. They cannot be met simply by taking a course or two. But they can be gradually added to one's academic program. The key is to start early (undergrad level). But these skills can also be picked up at any stage of professional career, as learning is a lifelong process.

Apart from economic theory and econometrics skills, there have been exiting developments usually described as Big Data or Data Science. Although these are not math skills, expertise in Big data / data science would become increasingly important as huge quantities of structured and unstructured data is becoming available. To keep discussion short here, interested readers are referred to data science courses on Coursera, udacity and elsewhere.

Basically, you have three areas of application:

You can also find applications in areas like, e.g., graph theory (social networks), fuzzy sets (management with uncertainty), differential equations (complex systems), optimization (operations research) but the first three are, at least to me, the most common source of knowledge in economics.

Dear Julio and George,

Thanks for your guides.

All the best,

Omid

May I add Nonlinear regression  models with GARCH error term. VAR models in time series analysis.

Information theory, bootstrapping and recently Data Science provides a wide area of economics applications.

Topology should be mention in regard to fixed point solutions for general equilibrium models of the Arrow-Debreu type. Debreu himself has entertained Measure Theory as an alternative way to solve for general equilibrium. Others have use Non-Standard analysis, which is still a budding research program. Here are some references:

1. Gerard Debreu, "Mathematical Economics: Twenty Papers of Gerard Debreu," Cambridge Press. 1989.

2. Salim Rashid, "Economies with Many Agents: An Approach Using Nonstandard Analysis," John Hopkins University Press, 1987.

3. "Handbook of Mathematical Economics," Volumes I and II, Elsevier, N. Holland.

Dear Lall, Jose and Olanrewaju,

Thanks for your answers

Game Theory!

I would add, on top of the aforementioned fields, the field of Input-Output analysis and its non-linear applications.

Long memory stochastic volatility model and application on (option pricing ,intrest rate , portofolio,etc)

There is a growing number of applications for Markov Processes being applied to economic information, and for network analysis in particular. Bamut & Jackson (2010)'s derivation of DeGroot learning is one excellent example, as is the increasing use of MCMC for inferential statistics in complex network systems, both static (as in Expontential Random Graph Models (ERGMs)) and dynamic (Stochastic Actor-Oriented Models or SOAMs).

Game theory would be very interesting. But should I request also chaos theory and nonlinear optimization models!

There are two broad areas of application of mathematics in economics: (a) economic theory, and (b) econometrics. Let me comment on both.

Regarding the former, there is need to appreciate that over the last few decades a transition of sorts has been underway in the nature of mathematics involved. Economic theory now relies more heavily on real analysis and topology. Understanding text books on modern economic theory, and more recent journal papers, requires real analysis, set theoretic  and abstract mathematical reasoning skills more than the ability to, more or less mechanically, crank out derivatives or integrals of functions.

This requires that economists consider formal maths training starting with mathematical logic/ abstract mathematical reasoning, where they learn to understand the logical anatomy of proposition and the needed proof techniques. The next steps could be to proceed to topology (see Ramrattan's comment above), followed by advanced calculas (set theoretic approach, such as in Rudin). Finally, the real analysis courses should be considered that would involve measure theory.

On the econometrics side, people either want to be able to understand and apply econometric techniques, or be able to develop such techniques themselves. I will only talk about people who are interested in applications, as this constitutes perhaps the largest group. To understand how certain estimators behave as sample size increases, knowledge of probability distributions, asymptotic distributions, sequences and convergence of sequences is needed. Real analysis will cover tbe sequence and convergence issues. Also needed is facility with matrix algebra. Most econometric papers and text books use matrix algebra and vector differentiation. So the reader should get familiar with the type of linear algebra found in Greene econometrics book. This would enable the readers to understand very well the econometric  techniques they want to apply. 

These requirements are quite substantial. They cannot be met simply by taking a course or two. But they can be gradually added to one's academic program. The key is to start early (undergrad level). But these skills can also be picked up at any stage of professional career, as learning is a lifelong process.

Apart from economic theory and econometrics skills, there have been exiting developments usually described as Big Data or Data Science. Although these are not math skills, expertise in Big data / data science would become increasingly important as huge quantities of structured and unstructured data is becoming available. To keep discussion short here, interested readers are referred to data science courses on Coursera, udacity and elsewhere.

la teoría de colas aún no "está de moda" pero lo estará cuando los economistas descubran todas sus posibilidades de aplicación a numerosos campos de la economía, y eso será pronto

Stochastic differential equations, also applications of Feynmann-Kac formula to a financial problem

Game Theory is taking pace. This is due to increased global competition and hence calls for a strategic business practice.

Stochastic programming for macroeconomics is an interesting area too..it requires the understanding economic theory and mathematical optimization techniques. Some of the models deals with economic growth with perspective to GDP and unemployment rate

Calculus could be applied in the field of Economics. Specifically, it is very useful in optimization problems such as finding equilibrium points

i would say that  Multtricriteria Decisión Making models have a very imporgtant role, in analyzing economic issues, as for instance the determination of an index to measure country growth, considering economics, social, environmental and otrher aspects

At he Politechnic University Of Valencia we aree working on fthat issue. If you think it could help you will will be glad in sharting it with you

Read the books written by mas-colell (Microeconomic Theory)  and by De la Fuente (Mathematical methods and models for economists). I hope help you.

In the class of "Mathematical economics," there are 36 questions. Majority of questions are statistical or probabilistic:  time series, cointergration, identification, calibration, computation methods, software, etc. There are, it seems, only 10 questions including this one, which are out of above group. How do you think of this state of mathematical economics?

Mathematical economics helped theoretical reasoning of various side of economics. The present state signifies  that theoretical reasoning is loosing its relevance and interest in the economics. Do you think this is a right situation? 

 Ten questions which I counted not related to statistics and probability theory are as follows:

Even if I see these non-statistical questions, really economical questions are few. This may stand for a kind of defeatism among economists. At least, it seems that majority of economists are avoiding to think really theoretical questions.          

Khalid Riaz's classification (popular answer in this question and answers page) tells that there are 2 great divisions: (1) economic theory and (2) econometrics.

In the above post, I talked about the weakness of economic theory investigations and interests. I again examined what is listed as interesting or hopeful field of mathematical economics. Many mentioned econometrics and related fields (I included time series in this group). As for economic theory, there are a few mentions and, even if they are mentioned, the topics are concentrated on game theory. Mosahar Tarimoradi, Mohamed Chakroun and Niti Khandelwal listed only Game Theory as a field of interest. Julio Rojas-Mora cited game theory as one of three fields of mathematical economics and Mohamed Chakroun listed chaos theory and nonlinear optimization models in addition to game theory. As an unique exception, Konstantinos Konstantakis added to the list Input-Output analysis and Lall B. Ramrattan mentioned Arrow-Debreu type models. 

It is true that game theory has been flourishing for these 30 years. Is it sufficient that our investigation in economics is largely restricted to game theory? Game theory contributed enormously in the analysis of origins and stability of institutions. An example is efficient wage hypothesis. Samuel Bowls in his Microeconomics explained many institutions (good or bad) as a classical game or evolutionary game. I do not deny their significance. However, I wonder if it is sufficient to assume Walrassian or Arrow-Debreu type general equilibrium models are valid. 

In the macroeconomics, the field much influenced by microeconomics, there is now a reflection on the validity of research programs for these 30 years, for example P. Krugman's remarked that macroeconomics of these 30 year are "spectacularly useless at best, and positively harmful at worst." Game theory may be a part of economic theory abut cannot answer how the economy as a whole works. We need a new research and mathematics. The first step in reconstruction economics will be to re-examine the price theory (or value theory).

I'm not sure about all fields of economics but in oil and gas analysis, Dynamic Optimization, Optimal Control and its applications are used.