I personally feel that like ...
Mathematics which sits at the top of all sciences
advaita sits at the top of all philosophy...
I mean though advaita is very vast... very subtle ... very ticklish....
the philosophy 'as it is' .... is very pervasive and holistic... i mean u can include everything .....
It is like the general theory of relativity which gives an all encompassing
view of the GREAT INDIAN CULTURE
U can include the "pushti marg" and the "dviata" and "suddhadvaita" easily as special
cases of advaita... just like Newton’s laws became special cases of Einstein’s theory..
Now no advaita story is complete if we ignore its main exponent.. SHANKARCHRYA aka Shankara aka Sankara (English version)....
Here is a good primer on the veteran....
No historical person has exercised a deeper and stronger influence on Indian thought and intellect and on spiritual life than Sri Adi Sankaracharya (788 – 820 A D). Indian philosophy, culture, poetry, humanism, art, literature etc., even today, reveal the impact of his teachings. In particular, he left indelible marks on the religious map of India.
As a philosopher, a poet, a reformer, an organizer, a teacher, an integrator, and an interpreter and so on, Adi Sankara is the sublime prophet of an epoch and thus rightly deserves to be called 'Jagadguru' ('Universal Preceptor').
Sankara's Life in Brief
The success and glory of life of Adi Sankara is sung in one sloka
“Ashta varshe chaturveedi
Dvaadasae sarva saastravit,
Shodasee kritavaan bhaashyam
“At eight, a master of four Vedas
At twelve, a professor of all sastras,
At sixteen, an interpreter and commentator
At thirty two, a great sage.”
Adi Sankara was born to the blessed couple of Siva guru and Aryamba, in Kalady in Kerala state, India. He had mastered the Vedas and sastras at the age of eight. After taking sannyasa from his master Govinda Bhagavatpada on the banks of the river Narmada, Sankara went to Kasi and then to Badri where he wrote Bhashyas at the guru's bidding. Then, he traveled to all parts of India, spreading the gospel of Advaita Sidhanta. He established four monastic centers at the four corners of India and entrusted them to the care of monks noted for their intellect, character and vision. With his immense knowledge and forceful arguments, he was able to win over adherents of different schools and sects. He upheld the Sanatana Dharma, whose tenets are eternal and universal.
His Literary Accomplishments Sankara's great writings, all in Sanskrit, consist of major philosophical commentaries and other works, as well as minor works such as hymns for a common man.
1. Major works on Philosophy
Of all the works, Sankara's commentaries (or Bhashyas) on the Prastanaa traya (trilogy of works) are considered to be very great and important in the realm of philosophy.
They include Bhaashya on
Brahma sutras (called Nyaaya Prastaana)
Ten Principal Upanishads (called Sruti Prastaana)
Gaudapada Kaarikas (called Smriti Prastaana)
Bhagavad-Gita, Vishnu Purana and Sanatsujatiya.
2. Minor works on Philosophy
Sankara wrote many Prakaranas (minor works), such as
Viveka Chudamani, with nearly 600 slokas, stands foremost.
Upadesa Saahasri, with 675 verses, is well known for vedantic topics
Aparokshanubhuti is composed of 147 stanzas dealing with yoga processes
Vaakya vritti gives the meaning of Mahavaakya, namely Tat Tvam Asi
Panchikarana deals with the process by which the five subtle elements combine to form gross elements and the material world
Svaatma Prakaasika, in 98 verses, explains the nature and characteristics of the soul which is distinct from the body
Prasnothara Rathnamaalika, just like any handbook, gives short questions and answers on the soul
Atma Bodha, another simpler work, is a basic for a seeker to start with
And some other minor, but valued works on philosophy, such as Tattva Bodha, Atma-Anatma Viveka, Prabodha Sudhakara, Brahmagnanavalli, Swaatma Nirupana
Other similar hymns are Maya Panchakam, Sadhana Panchakam, Vignana Nauka, Nirvana Panchakam, Nirvana Shatkam Kaupina panchakam etc.
Dakshina Murty Stotra
Sankara composed the immortal hymn, Dakshina Murty Stotra, in ten stanzas, which is a masterpiece. It inculcates the highest devotion to the Divine being as the guru of gurus. It provides a text for the students to learn the fundamental truth of Advaita Vedanta.
It is another important metrical Prakarana, in five stanzas, explaining the principles of Advaita Vedanta.
3. Works of Devotion (stotras)
Though Sankara expounds the concept of Nirguna Brahma (without attributes), he advocates 'Devotion' (bhakti) towards a Saguna Brahma (with attributes), which when ripened leads to the knowledge of Nirguna Brahma. Further, he did not only believe in the efficiency of bhakti in the realization of Brahma, but also actually composed a large number of hymns (or stotras) in praise of many deities, in different contexts.
Kanakadhaara stavam is a famous work in praise of Goddess Lakshmi.
Sarada Bhujanga prayaataashtakam, in 8 verses, is in praise of Sringeri Saradaamba.
Siva paadaadi kesaanta varnamala stotra is a beautiful hymn on Siva.
Lakshmi Narasimha karaavalamba stotra, in 12 stanzas, is a full surrender to Lord Narasimha.
Siva bhujanga stotram and Vishnu bhujanga stotram are composed to ensure happiness to his mother.
In Veda saara Siva stotram, a vital doctrine of Advaita is introduced in the shortest framework of 7 stanzas.
Charpata Panjarika Stotram (Dvaadasa Manjarika stotram)
In this, (popularly known as Bhaja Govinda stotram), Sankara shows in touching words and with apt examples, how an average
man can embark on his spiritual endeavor.
Two highly potent prayers, Soundarya Lahari and Sivaananda Lahari, each of 100 stanzas, are unfathomably deep in spirit and thought and are expressions of his proficiency in poetry.
His Poetic Prowess
Sankara was well aware that commentaries on the Upanishads and Bhagavad-Gita etc., required basic knowledge, great skill and effort on the part of the reader. So, for the benefit of the layman, he created a large number of simple hymns (stotras) on various forms of the Lord such as Siva, Vishnu, Ambika, Ganesha and many other gods.
1. Rhythmic Poetry
Sankara amalgamated philosophy into poetry and the outcome was metrical, rhythmical and musical hymns.
To name a few stotras, we have
Ganesha Pancharatna stotram “Mudaakaraatha modakam….” Shatpadi stotram “Avinayampanaya vishno damaya…” Uma Maheswara stotram “Namah Sivaabhyaam navayouvvanaabhyaam” Annapurnaashtakam “Nityaanandakari varaabhayakari...” Sivaananda Lahari “Kalaabhyaam chudaalankrita sasi kalaabhyaam” Mahishaasura Mardini stotram “Ayigiri nandini nandita modini...”
Some of the hymns are quite simple, like
Siva Panchaakshari stotram “Nagendra haraaya trilochanaaya...”
Lingashtakam “Brahma murari suraarchita lingam...”
Some of the hymns are not that simple, for example,
Dakshinamurty stotram “Visvam Darpana drisyamaana nagari tulyam…” Or Soundarya Lahari “Sivassaktyayukto yadi bhavati sakta prabhavithum...”
2. Ultimate in Devotion
Sankara's degree of devotion to Lord is great and surrender to Him is ultimate. This fact is disclosed in many of his hymns, for example, in 'Siva Maanasa puja', the last two slokas are
1. “Atma tvam Girija matih… Sambho! Tavaaraadhanam.” 
Meaning of the sloka
“My self is Sambhu (yourself); my intellect is Girija; my vital breaths are your attendants; my body is your temple of residence; my enjoying the objects of senses is your worship; my sleep is the state of meditation; all movement with the pair of feet is doing pradakshina to you and all my words are your praises. O Sambhu! Whatever I do is entirely an act of worshipping you.”
2. “Karacharana … Maha Deva Sambho!” 
Meaning of the sloka
“Ocean of Mercy! Forgive me all this fault-done and yet to be done, committed by hands, feet or done vocally, physically and by actions through the faculties of hearing, seeing and thinking (mentally). Faults may also arise from not doing prescribed duties and by doing prohibited actions.”
The beauty of the images in his works, decorated by the figure of speech, developed by the poet-author creates high levels of joy and enlightenment in the readers and seekers. For instance, the 61st sloka of Sivaananda Lahari describes the gradual ascendance of the state of devotion, by giving impressive examples.
“Ankolam nija-bija ….. sada saa bhaktirityuchyate” 
Meaning of the sloka
“That state of mind is called 'Bhakti', wherein all movements of thought go automatically to the lotus feet of the Lord and stick to them forever just as the seed of the ankola tree (on falling) gravitates to the parent tree, the iron needle to the magnetic stone, the devoted wife to her husband, the creeper to the tree and the river to the ocean.”
A great synthesizer
Adi Sankara, with his knowledge as his companion, was able to command the emotion of intellectuals as well as the masses. He was able to gather the scattered people and raised their spiritual and cultural energies by revolutionizing their philosophical thinking.
The greatest service rendered by the great Acharya was to establish four 'Amnaayas' in the four directions for the task of preaching preserving and protecting the values of Sanatana Dharma. He founded four monastic centers (Peethas).
Puri in east – Rig Veda (Pragnaanam Brahma as Mahaa Vaakya)
Sringeri in south – Yajur Veda (Aham Brahmaasmi)
Dvaaraka in west – Sama Veda (Tat Tvam Asi)
Badrinath in north – Atharva Veda (Ayamaatma Brahma)
Adi Sankara entrusted the above centers to the care of great monks, namely, Hastaamalakaachaarya, Suresvaraachaarya, Padmapaadaachaarya and Totakaachaarya, who have their traditional successors even now.
Thus, he made a lasting and permanent arrangement for propagation and synthesis of religion.
A National Integrator
No single individual has done in India more for national integration than Adi Sankara. Adi Sankara traveled all over India, propagating and unifying the spiritual outlook of the people. Through discussions, debates and compositions, he was able to win the hearts of the people. The unifying force, namely, Sanskrit language helped national integration become all the more effective. In those times, there were innumerable groups and classes, each in conflict with the other. Socially, the country was divided into different creeds and sects. Sankara brought back the Vedas and taught Vedanta which teaches that there is but one existence and that is God. He convinced his followers of various sects and rationalized the practices of the chief cults.
He founded 'Shanmataas' (six modes of purified forms of worship) to satisfy the spiritual and religious practices of diversified Hindus of the country and unify them. They are 1. Saiva; 2. Vaishnava; 3. Saakta; 4. Soura; 5. Gaanapatya; and 6. Skaanda (Subrahmanya). Accordingly, he recommended the practice of 'Panchaayatana Puja' wherein the five major five deities of Vedas, namely, Surya, Ambika, Vishnu, Ganapati and Siva  are worshipped by making one's chosen deity the main focus in the center (i.e., as per a chosen order of arrangement of the deities on the same platform). The sixth deity, Shanmukha, is worshipped in the form of 'Yagneswara' (Fire) in the daily rituals (through Agnihotra). This method of worship avoids the deliberate elevation of any one deity over the other.
“His greatness lies in the fact that he gathered up scattered cultural and spiritual energies of the people and raised the voltage tenfold,” mentions Swami Ranganathananda of Sri Ramakrishna Mission.
“He revised Hinduism by reforming the Buddhist elements in Gaudapada Kaarika and gave them a vedantic character …which represents a turning point,” says Prof. Sengakee Mayada, an eminent indologist in Japan.
A Great Yogi
Adi Sankara's life on earth had been a miracle. Within a short span of 32 years, he traveled throughout the length and breadth of the country, three times, establishing a marvelous philosophy. Surely, he should have had an inner yogic life, by which it was possible for him to achieve so many great things. His exposition, titled 'Vivarana', on Vyasa's commentary on the famous Patanjali Yoga Sutras throws light on the Acharya's perception on Yoga. His work, 'Yogataravali' reveals many secrets of the yoga sastra.
Adi Sankara was universal in his outlook. He was kind. His message was meant, not for Hindus alone, but for all mankind. Through his eminent works he drives the man to carve out a personality.
To quote a few slokas from his works,
1. In Charpata Panjarikaa Stotram, (Bhaja Govindam), Sankara advises that suffering is a part of this endless cycle of births and deaths.
“Punarapi jananam… .paahi murare” 
Meaning of the sloka
“Birth unceasing, death unceasing!
Ever to pass through a mother's womb,
Hard to cross is the world's wide ocean
Lord! Redeem me through thy mercy.”
2. In Sivaaparaadha Kshamaapana Stotra (15th sloka), he cautions that equipment of joy in human life is highly temporary.
“Ayurnasyati… rakshaadhuna” 
Meaning of the sloka
“Day by day, life ebbs away before our eyes as we look on; youthfulness declines; days passed do never return; time devours worlds; wealth is fickle like the ripple on the wavelets of water; life is fleeing like lightning. Therefore, oh Lord! Protect now with your grace; seeking refuge in you.”
3. In 'Upadesa Panchakam' he ordains the way of leading a purposeful life.
“Vedo nityamadhiyataam… vinirgamyataam” 
Meaning of the sloka
“Study the Veda everyday; do properly Karma prescribed therein (rituals, at least the nitya karmas); by the performance of those karmas, worship the Lord; abandon attachment to desire-driven action; wash away the accumulated sins; reflect on the defects of happiness derived from samsaaric existence; be firm in desiring the Self (in realization); quickly leave your house (i.e. take to sannyaasa)
Based on many of his works, great scholars have summarized his message as follows:
Realize that everything around you, including your wealth and your family, are all evanescent. Too much attachment would result in diverting your mind from what is eternal.
You have to learn to renounce the thought of reward for what you are doing.
Moral preparation should be a part of the life and that is achieved through universal compassion and love.
You long for liberation. Life is the school to educate you for the eternal life.
Usually, the doctrine of Advaita is formulated in the famous sloka of Sankara,
“Slokaardhena pravakshyami Yaduktaih grandha kotibhih; Brahma satyam, Jaganmithyaa, Jivo brahmaiva naa parah!”
Meaning of the sloka
“What has been told in countless number of books is being said in one half a sloka. Brahman is Satyam (Truth); world is Mithya (apparent); the individual soul (jeeva or atman) is none other than the Infinite Brahman.”
The last part of the above sloka is more significant and it is briefly explained thus:
“By analyzing the three states of experience, namely, waking, dreaming and deep sleep, Sankara exposed the relative nature of the world. The three states of consciousness are superimposed into a fourth transcendental state known as 'turiya'. Then the individual soul realizes that he/she is none other than the Infinite Brahman.”
Sankara established that the above realization is the supreme truth of the Advaita. Advaita is the non-dual reality of Brahman in which atman (the individual soul) and Brahman are identified absolutely.
Adi Sankara's mission was to propagate the Advaita Philosophy, which is, in a nutshell, as follows.
“The world of manifestation and multiplicity that we see is not real in itself. It seems real only because of 'ignorance' (or Avidya). To be caught in it is bondage. To get over Avidya, works (karma) are in vain. They only bind us to the endless chain of cause and effect and to the unreal process of 'Samsaara'. Only wisdom can remove ignorance. (Path of Jnaana is superior). Wisdom consists in conviction that Brahma alone is real. The individual self and the Brahman are identical. When this is realized, the ego is dissolved. We are back to perfect joy and blessedness. Reality is 'Brahman is one without a second.' ”
The manifold nature of the phenomenal world and its ultimate unity is symbolized by AUM, the most sacred syllable of Hindus.
Advaita and Modern Science
Sankara's views on the nature of reality have a striking resemblance with the advanced theories of modern science.
Students of Physics are aware of the implications of the uncertainty principle, which states that, “It is impossible to specify precisely and simultaneously, the values of both members of particular pairs of physical variables that describe the behavior of an atomic system.” The principle includes that the knowledge, which occurs in pairs, namely, position and time, energy and time and action and time is affected. Uncertainty relations are constituents of knowledge at the ordinary levels of experience. Once this concept that they occur in pairs is got over, the uncertainty relations do not come into the picture at all. Sankara proposed in his Advaita theory that the limitations of ordinary knowledge vanish when one transcends the idea of opposites or pairs and then there is a direct perception out of intuition.
In modern science, the outside world is deeply affected by the position and time of the observer. The special theory of relativity claims differences relative to the position and time of different observers.
Sankara was the foremost who denied the separate existence of subject and object. He made the statement that in the highest reaches of knowledge the distinction between subject and object disappears.
Sankara says that the universe which the person sees is only that which he thinks he sees... Sankara does not deny the existence of an objective universe. He asserts that it is the sense of perception that makes the difference between one observer and the other. If a person transcends the boundary of ordinary perception, then the question of an objective existence other than pure consciousness does not arise. Then the process of seeing (observation), the seer (subject) and the seen (object) have no independent existence and this is the highest goal which every human being should try to reach.
The books and articles written on Adi Sankara are enough to make a whole library. Many thinkers and seekers have been meditating upon life, philosophy, and the works of Adi Sankara.
Sankara has been a source of inspiration to many of the leaders of modern times too.
In 'The Discovery of India', Jawaharlal Nehru has said of him, “Sankara was a man of amazing energy and vast activity.” Nehru expresses that Sankara was no escapist returning into his shell or into a corner of the forest.
Dr S. Radhakrishnan has said,
“Sankaracharya is one of the immortals of India's cultural history. ... His originality was in reinterpreting the great Indian tradition.”
N. A. Palkiwala, an eminent jurist, writes,
“Whole generations have come and gone, empires have flourished and vanished, but Sankaracharya's empire of the spirit survives. And so long as his great spirit abides with our people, there is hope for the future greatness of our country.
Adi Sankara is the greatest expression and true meaning of a world teacher in philosophy, religion and culture.”
Adi Sankara is Jagadguru par excellence.
Full version of slokas
 “Atma tvam Girija matih, sahacharaah praanaah, sariram griham,
Pujaa te vishayopa bhoga rachanaa, nidraa samaadhih stitih,
Sanchaarah padayoh pradakshina vidhih, stotrani sarvaa giro,
Yadyat karma karomi tattadakhilam Sambho! tavaaraadhanam”.
“Karacharana kritamvaakkaajam karmajamva,
Sravana nayanamjamva maanasamva aparaadham,
Vihitamahitamva sarvametat kshamasva,
Jaya jaya karunaabde Sri Maha Deva Sambho!”
 “Ankolam nija-bija santatih ayasaantopalam suchika
Sadhvi naija-vibhum lataa kshitiruham sindhuh sarid vallabham
Prapnotiha yatha tatha pasupateh paadaaravindadvayam
Chetovrittih upetya tishtati sada saa bhaktirityuchyate”
Adityamambikaam Vishnum Gananaatham Maheswaram
Panchaayatana pujartham kalpayedvijasathamah
 “Punarapi jananam punarapi maranam,
Punarapi janani jatharae sayanam
Iha samsare bahudustaare
Kripayaa paare paahi muraare.
 Ayurnasyati pasyataam pratidinam yaati kshayam youvvanam
Pratyaayaanti gataah punarna divasaah, kaalo jagadbhakshakah
Lakshmistoya taranga bhangi chapalaa, vidyutchalam jivitam
Tasmaanmaam saranaagatam karunayaa tvam raksha rakshaadhuna.
 “Vedo nityamadhiyataam taduditam karma svanushthiyataam
Teneshasya vidhiliyataamachitih kaamaye matishtyajyataam
Paapaughah paridhuuyataam bhavasukhe dosho anusandhiliyataam
Atmechha vyavasilyataam nijagrihaattuurnam vinirgamyataam”