The Kerala School of Mathematics (14th to 19th century CE...
...led by Madhava, was an intellectual hub that blazed new trails in formulating the power series – the trignometrical sine, cosine, and inverse tangent as well as pi series, which antedated similar mathematical formulations in Europe by a couple of centuries. Geometrical algebra was also developed, as evidenced by the sixteenth-century work, the Kriyakramakari. 
(Reference: ‘Science in India: A historical perspective’ by B. V. Subbarayappa. Page 26)

Calculus in India before Newton
The clearest exposition of the calculus in India, in the Yuktidipika, was written by Sankara Variyar (1500-1560)…
(Reference: http://ckraju.net/papers/calculus_abstract_2.pdf. Page 2)

The place value system…
…that leads to efficient arithmetic, as in the methods of addition, subtraction, multiplication and division, was invented in India.
(Reference: Brahmasphutasiddhanta of Brahmagupta of 7th c. From Professor CK Raju’s
talk at MIT, https://www.youtube.com/watch?v=IaodCGDjqzs)

Zinc extraction in ancient India

It is amazing indeed that, notwithstanding the fact that ancient Indian zinc-smiths lacked any precise knowledge of the various parameters involved in the extraction of zinc, they accomplished it in an empirical but efficient manner and successfully extracted metallic zinc by what is called the downward distillation process. This method also required specially designed furnaces, retorts for the roasted ore, the reducing agent, reducing atmosphere, controlled temperature, and a mechanism or device for downward condensation. That the zinc-smiths were able to achieve this is borne out by the remnants now available archaeologically and their skills evoke our admiration, even after 2500 years. (Reference: ‘Science in India: A historical perspective’ by B. V. Subbarayappa. Page 301)

High quality Indian Wootz steel manufacture

Now it appears to me that the Indian process combines the principles of both the above described methods. On elevating the temperature of the crucible containing pure iron, and dry wood, and green leaves, an abundant evolution of carburetted hydrogen gas would take place from the vegetable matter, and as its escape would be prevented by the luting at the mouth of the crucible, it would be retained in contact with the iron, which, at a high temperature, appears to have a much greater affinity for gaseous than for concrete carbon; this would greatly shorten the operation, and probably at a much lower temperature than were the iron in contact with charcoal powder. 

In no other way can I account for the fact that iron is converted into cast steel by the natives of India, in two hours and a half, with an application of heat, that, in this country, would be considered quite inadequate to produce such an effect; while at Sheffield it requires at least four hours to melt blistered steel in wind-furnaces of the best construction, although the crucibles in which the steel is melted, are at a white heat when the metal is put into them, and in the Indian process, the crucibles are put into the furnace quite cold. 

(Reference: Dharampal: Collected writings, Volume I, ‘Indian science and technology in
the eighteenth century’. Page 21. He is quoting from ‘On Indian iron and steel’ by JM Heath, founder of the Indian iron and steel company)

Every village in 18th century India had at least one school:

The most well-known and controversial point which emerged from the educational surveys lies in an observation made by William Adam. In his first report, he observed that there exist about 1,00,000 village schools in Bengal and Bihar around the 1830s.
(Reference: Dharampal: Collected writings, Volume III, ‘The Beautiful Tree’. Page 18. He is referring to ‘Report on the state of education in Bengal, 1835’ written by William Adam) 

Schools were not only for the Brahmins 

It is, however, the Madras Presidency and Bengal-Bihar data which presents a kind of revelation. The data reveals the background of the teachers and the taught. It presents a picture which is in sharp contrast to the various scholarly pronouncements of the past 100 years or more, in which it had been assumed that education of any sort in India, till very recent decades, was mostly limited to the twice-born amongst the Hindoos, and amongst the Muslims to those from the ruling elite. The actual situation which is revealed was different, if not quite contrary, for at least amongst the Hindoos, in the districts of the Madras Presidency (and dramatically so in the Tamil-speaking areas) as well as the two districts of Bihar. It was the groups termed Soodras, and the castes considered below them who predominated in the thousands of the then still-existing schools in practically each of these areas.
(Reference: Dharampal: Collected writings, Volume III, ‘The Beautiful Tree’. Page 21)

Major economies from year 1 to 2003

The global contribution to world’s GDP by major economies from year 1 to 2003 according to Angus Maddison’s estimates. Before the 18th century, India and China were the two largest economies by GDP output.
(Reference: https://en.wikipedia.org/wiki/Angus_Maddison.)

The officially sanctioned Indian history that we study in our schools, and which helps form our self-identity, is so completely false that it is difficult to choose some particular set of noteworthy facts under this heading. Here is one example.

Distortion of history in NCERT text books

During the year 1969-70 the Central Government under Mrs. Indira Gandhi established a committee under the Chairmanship of G. Parthasarathy, a diplomat close to Nehru-Gandhi family. Its task was to integrate the nation through education. At that time I was a reader in Educational Philosophy at NCERT and was selected as one of the five members of the committee. In our first meeting Mr. Parthasarathy, as Chairman of the committee explained the purpose of our committee in typically diplomatic language: “It is our duty not to sow the seeds of thorns in the minds of the growing children which will grow up as barriers to national integration. Such thorns are found mostly in the history courses. Occasionally we can find them in language and social science courses also. We have to weed them out. We have to include only such thoughts that go towards inculcating the concept of national integration firmly in the minds of our children. This committee carries this great responsibility.”

The other four members were nodding respectfully. But I said, “Sir, I am unable to understand your words. Will you please explain with a few illustrations?” The Chairman responded: “Ghazni Mohammed looted the Somnath Temple, Aurangzeb built mosques by demolishing the temples in Kashi and Mathura, he collected jizya – is it possible to build a strong India under the present circumstances by conveying such useless facts? What purpose do they serve, other than generating hatred?”

(Reference: Dr S.L. Bhyrappa, the celebrated Kannada novelist talking about distorting Indian history. He objected to the approach that the chairman was advocating and was unceremoniously thrown out of the textbook committee after the second meeting.
http://prekshaa.in/distorting-indian-history-s-l-bhyrappa-girish-karnad-u-r-ananthamurthy-parthasarathy-ncert-indira-gandhi-distortion-mughals-tipu-sultan-aurangzeb/#.WZAA3O3SUyo)

You recognize a true Indian by their love of Indian music and art

These are the days of nation building. Yet how many ‘nationalists’ are in truth ‘denationalists’ in their lives and aspirations! They want to be ‘free,’ to compete with Europe on her own lines, to be ‘progressive,’ ‘advanced,’ to gain political power and material success. It is not with these that the future of India lies. It lies in the lives of those who are truly Indian at heart, whose love for India is the love of a lover for his mistress, who believe that India still is and not merely may be, when duly ‘educated’ the light of the World, who today judge all things by Indian standards, and in whom is manifest the work of the shapers of India from the beginning until now. Without these, there can be no Indian future worth the name. How may they be known? Like answers unto like; but, if an empirical test be asked for, I believe that the love of Indian music and the comprehension of Indian art are tests unfailing.

(Reference: ‘Essays in national idealism’ by Ananda Coomaraswamy. Excerpt from chapter 14 titled ‘Music and education in India’)

How the British destroyed Indian Handloom

The handloom weavers famed across the world, whose products were exported around the world. There were these weavers making fine muslin, ‘light as woven air’ it was said. And Britain came right in, broke their thumbs, smashed their looms, imposed tariffs and duties on their cloth and products, and started taking the raw material from India and shipping back finished cloth. Flooding the world with the products of what became the dark and satanic mills of Victorian England.

(Reference: Dr Shashi Tharoor speaking on Britain owing reparations to India. https://www.youtube.com/watch?v=f7CW7S0zxv4)

Stepwells, Stupas, tombs, rock-cut caves, temples etc.

Ajanta Caves carved from the top down

How old are they ?

The Ajanta Caves are generally agreed to have been made in two distinct periods, the first belonging to the 2nd century BCE to 1st century CE, and second period that followed several centuries later. 

How were they cut out from the Deccan plateau ?
Excavation began by cutting a narrow tunnel at roof level, which was expanded downwards and outwards; as evidenced by some of the incomplete caves such as the partially -built vihara caves 21 through 24 and the abandoned incomplete cave 28. 

(Reference: Wikipedia. https://en.wikipedia.org/wiki/Ajanta_Caves)

Ayurveda’s strength is that it is holistic

The innate strength of Ayurveda is its holistic vision of the body, mind, soul, the environment, and the universe. The word vaidya for an ayurvedic physician is significant. For, it literally means a person who has profound knowledge not merely of diseases and their cure, but also of the material and spiritual life, of the harmony between the individual and the universe.

(Reference: ‘Science in India: A historical perspective’ by B. V. Subbarayappa. Page 372)

Advanced agriculture

Until lately I imagined the Drill plough to be a modern European invention; but a short time ago, riding over a field, I observed a Drill plough at work, very simple in its construction, which upon inquiry I find is in general use here, and has been so from time immemorial.

(Reference: Dharampal: Collected writings, Volume I, ‘Indian science and technology in the eighteenth century’. Page 203. This is the beginning of Chapter XIII, titled ‘On the drill husbandry of southern India’, written by Captain Thos Halcott in 1795)

The Indian idea of design was based on the idea of purnata

The basis of the design philosophy of our civilization was:

purnamadah purnamidam purnat purnamudachyathe
purnasya purnamadaya purnamevavashishyathe

The idea that wholeness does not require any additions or subtractions. For example a flower blossoms on a tree and it is beautiful. If the flower did not bloom, the tree was anyway beautiful. We cannot imagine that a flower appearing later would make the tree more beautiful. If a bird sits on the tree it is very beautiful. If the bird flies away, the tree is not in any way lessened by that. The bird is beautiful in its place and the tree in its own. If the two get together that is also beautiful. There is no lack in the beauty. It is purna.

Our Indian lifestyle was based on the idea that if we have something we should not think that it is a lot. And if we do not have it we should not feel the lack of it. Hai tho jyaada na lage aur nahin hai tho uski kami mehsoos na ho.

Our houses were also built like that. Food also. Veshbhusha also.

(Reference: Ravindra Sharma on Indian perspectives on education. https://www.youtube.com/watch?v=A-pMQeKe8GI. Time 33:47-38:48)

One of the oldest surviving fighting systems in the world

Kalaripayattu is a martial art, which originated as a style in Kerala, southern India (North Malabar). The word kalari first appears in the Tamil Sangam literature (c. 300 BCE to 300 CE)[2] to describe both a battlefield and combat arena. The word kalari tatt denoted a martial feat, while kalari kozhai meant a coward in war. Each warrior in the Sangam era received regular military training. It is considered to be one of the oldest surviving fighting systems in still existence in the world. It was originally practiced in northern and central parts of Kerala and southern parts of Tamil Nadu.

(Reference: Wikipedia. https://en.wikipedia.org/wiki/Kalaripayattu)