What are the branches of maths where we can see undoubtful connections with economics? Where can we use mathematical methods or models and apply them to analyze economic concepts?
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Lots of branches of maths are used in lots of areas of economics, so it would be hard to give a concise answer. Have a look at Essential Mathematics for Economic Analysis by Sydsaeter & Hammond, and the follow-up Further Mathematics for Economic Analysis.
Very late answer, but I figured I give it a shot, from the perspective of an economics graduate student.
There are clear connections between econ and math but the statement needs some qualification: Economics is about solving economics problem in the economics paradigm. How much math is involved in the solution depends on the subfield and the economist in question. The mathematical sophistication across the econ community varies greatly. Some economists are mathematicians in their own right, while others (probably most) don't know much math at all. So, yes, at its most rigorous level, economics is highly mathematical but that's small subcommunity in econ.
Having given that caveat, mathematics used, when they are, in the three main econ subfields, along with some references:
Microeconomics: Consumer theory is about how an individual consumer makes his decision. The mathematical context is convex analysis. See Theory of Value by Debreu (that level of rigor, standard in math, will never been seen in an econ course.) Game theory---used in analysing conflict. See Game Theory by Fudenberg and Tirole.
Macroeconomics: Real and (basic) functional analysis. The main mathematical tool used in macro is dynamic programming, whose rigorous formulation requires analysis. See Recursive Methods in Economics by Stokey and Lucas, written at the level of Rudin.
Econometrics: This is basically statistics applied to econ. I don't know a PhD level text that would be considered rigorous by a mathematician. I'd refer to journal papers published in Econometrica.
Hope this helps.