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Borsuk-Ulam Theorem and Hilbert's Nullstellensatz Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz Borsuk-Ulam Theorem and Hilbert’s Dilip P. Patil Nullstellensatz § 1 Borsuk-Ulam — and vice versa Theorem §2 Borsuk’s Nullstellensatz and its Dilip P. Patil Equivalents §3 2-Fields Department of Mathematics §4 Dimension Indian Institute of Science, Bangalore and Multiplicity §5 Projective 1 Nullstellensatz Bhaskaracharya Pratishthana , Pune §6 Analogs of July 22, 2018 HNS to 2fields 1On the occasion of Birthday of Late Professor Shreeram S. Abhyankar (1930 – 2012), Founder Director, Bhaskaracharya Pratishthana, Pune. Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz Shreeram Shankar Abhayankar (22 July, 1930 – 02 Nov, 2012) Founder Director Bhaskaracharya Pratishthana, Pune. Late Professor Shreeram Shankar Abhayankar Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz Dilip P. Patil § 1 Borsuk-Ulam Theorem §2 Borsuk’s Nullstellensatz and its Equivalents §3 2-Fields §4 Dimension and Multiplicity §5 Projective Nullstellensatz §6 Analogs of HNS to 2fields Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz Founder Director Bhaskaracharya Pratishthana, Pune. Late Professor Shreeram Shankar Abhayankar Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz Dilip P. Patil § 1 Borsuk-Ulam Theorem §2 Borsuk’s Nullstellensatz and its Equivalents §3 2-Fields §4 Dimension Shreeram Shankar Abhayankar (22 July, 1930 – 02 Nov, 2012) and Multiplicity §5 Projective Nullstellensatz §6 Analogs of HNS to 2fields Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz Late Professor Shreeram Shankar Abhayankar Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz Dilip P. Patil § 1 Borsuk-Ulam Theorem §2 Borsuk’s Nullstellensatz and its Equivalents §3 2-Fields §4 Dimension Shreeram Shankar Abhayankar (22 July, 1930 – 02 Nov, 2012) and Founder Director Multiplicity §5 Projective Bhaskaracharya Pratishthana, Pune. Nullstellensatz §6 Analogs of HNS to 2fields Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz Abstract Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz Dilip P. Patil § 1 Borsuk-Ulam Theorem The main aim of this talk is to provide an algebraic proof of the well-known Borsuk-Ulam theorem by using projective algebraic §2 Borsuk’s Nullstellensatz sets. In fact, we prove a more general form of Borsuk-Ulam and its Equivalents theorem called the Borsuk-Ulam’s Nullstellensatz by establishing its equivalence with the real algebraic Nullstellensatz. §3 2-Fields §4 Dimension and Multiplicity §5 Projective Nullstellensatz §6 Analogs of HNS to 2fields Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz The personal discussions with late Prof. Dr. Uwe Storch (1940 – 2017), Ruhr-Universität Bochum, Germany and his lecture on 23 January 2003, on the occasion of 141-th birthday of Hilbert at the Ruhr-Universität Bochum, Germany. Uwe Storch (12 July, 1940 – 17 Sept, 2017) The recent preprint (jointly with Kriti Goel and Jugal Verma) Nullstellensätze and Applications, IIT Bombay, 2018. Late Professor Uwe Storch Borsuk-Ulam Discussion and Ideas of Proofs are based on : Theorem and Hilbert’s Nullstellensatz Dilip P. Patil § 1 Borsuk-Ulam Theorem §2 Borsuk’s Nullstellensatz and its Equivalents §3 2-Fields §4 Dimension and Multiplicity §5 Projective Nullstellensatz §6 Analogs of HNS to 2fields Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz Uwe Storch (12 July, 1940 – 17 Sept, 2017) The recent preprint (jointly with Kriti Goel and Jugal Verma) Nullstellensätze and Applications, IIT Bombay, 2018. Late Professor Uwe Storch Borsuk-Ulam Discussion and Ideas of Proofs are based on : Theorem and Hilbert’s Nullstellensatz The personal discussions with late Prof. Dr. Uwe Storch Dilip P. Patil (1940 – 2017), Ruhr-Universität Bochum, Germany and his lecture on 23 January 2003, on the occasion of 141-th birthday of Hilbert § 1 at the Ruhr-Universität Bochum, Germany. Borsuk-Ulam Theorem §2 Borsuk’s Nullstellensatz and its Equivalents §3 2-Fields §4 Dimension and Multiplicity §5 Projective Nullstellensatz §6 Analogs of HNS to 2fields Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz Uwe Storch (12 July, 1940 – 17 Sept, 2017) The recent preprint (jointly with Kriti Goel and Jugal Verma) Nullstellensätze and Applications, IIT Bombay, 2018. Late Professor Uwe Storch Borsuk-Ulam Discussion and Ideas of Proofs are based on : Theorem and Hilbert’s Nullstellensatz The personal discussions with late Prof. Dr. Uwe Storch Dilip P. Patil (1940 – 2017), Ruhr-Universität Bochum, Germany and his lecture on 23 January 2003, on the occasion of 141-th birthday of Hilbert § 1 at the Ruhr-Universität Bochum, Germany. Borsuk-Ulam Theorem §2 Borsuk’s Nullstellensatz and its Equivalents §3 2-Fields §4 Dimension and Multiplicity §5 Projective Nullstellensatz §6 Analogs of HNS to 2fields Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz The recent preprint (jointly with Kriti Goel and Jugal Verma) Nullstellensätze and Applications, IIT Bombay, 2018. Late Professor Uwe Storch Borsuk-Ulam Discussion and Ideas of Proofs are based on : Theorem and Hilbert’s Nullstellensatz The personal discussions with late Prof. Dr. Uwe Storch Dilip P. Patil (1940 – 2017), Ruhr-Universität Bochum, Germany and his lecture on 23 January 2003, on the occasion of 141-th birthday of Hilbert § 1 at the Ruhr-Universität Bochum, Germany. Borsuk-Ulam Theorem §2 Borsuk’s Nullstellensatz and its Equivalents §3 2-Fields §4 Dimension and Multiplicity §5 Projective Nullstellensatz §6 Analogs of Uwe Storch (12 July, 1940 – 17 Sept, 2017) HNS to 2fields Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz Late Professor Uwe Storch Borsuk-Ulam Discussion and Ideas of Proofs are based on : Theorem and Hilbert’s Nullstellensatz The personal discussions with late Prof. Dr. Uwe Storch Dilip P. Patil (1940 – 2017), Ruhr-Universität Bochum, Germany and his lecture on 23 January 2003, on the occasion of 141-th birthday of Hilbert § 1 at the Ruhr-Universität Bochum, Germany. Borsuk-Ulam Theorem §2 Borsuk’s Nullstellensatz and its Equivalents §3 2-Fields §4 Dimension and Multiplicity §5 Projective Nullstellensatz §6 Analogs of Uwe Storch (12 July, 1940 – 17 Sept, 2017) HNS to 2fields The recent preprint (jointly with Kriti Goel and Jugal Verma) Nullstellensätze and Applications, IIT Bombay, 2018. Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz Theorem 1.1 ( B o r s u k - U l a m ) n n For every continuous map g :S ! R , n 2 N; there exist anti-podal points t, −t 2 Sn with g(t) = g(−t). n n+1 2 n 2 n+1 S = ft = (t0;:::;tn) 2 R j jjtjj = ∑i=0 ti = 1g ⊆ R is the n-sphere. This was conjectured by S. Ulam and was proved by K. Borsuk in 1933 by elementary methods but technically involved. § 1 Borsuk-Ulam Theorem Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz Dilip P. Patil A well-known B o r s u k - U l a m t h e o r e m says that: § 1 Borsuk-Ulam Theorem §2 Borsuk’s Nullstellensatz and its Equivalents §3 2-Fields §4 Dimension and Multiplicity §5 Projective Nullstellensatz §6 Analogs of HNS to 2fields Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz This was conjectured by S. Ulam and was proved by K. Borsuk in 1933 by elementary methods but technically involved. § 1 Borsuk-Ulam Theorem Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz Dilip P. Patil A well-known B o r s u k - U l a m t h e o r e m says that: § 1 Borsuk-Ulam Theorem 1.1 ( B o r s u k - U l a m ) Theorem For every continuous map g :Sn ! n, n 2 ; there exist §2 Borsuk’s R N Nullstellensatz anti-podal points t, −t 2 Sn with g(t) = g(−t). and its Equivalents n n+1 2 n 2 n+1 §3 2-Fields S = ft = (t0;:::;tn) 2 R j jjtjj = ∑i=0 ti = 1g ⊆ R is the §4 Dimension n-sphere. and Multiplicity §5 Projective Nullstellensatz §6 Analogs of HNS to 2fields Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz § 1 Borsuk-Ulam Theorem Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz Dilip P. Patil A well-known B o r s u k - U l a m t h e o r e m says that: § 1 Borsuk-Ulam Theorem 1.1 ( B o r s u k - U l a m ) Theorem For every continuous map g :Sn ! n, n 2 ; there exist §2 Borsuk’s R N Nullstellensatz anti-podal points t, −t 2 Sn with g(t) = g(−t). and its Equivalents n n+1 2 n 2 n+1 §3 2-Fields S = ft = (t0;:::;tn) 2 R j jjtjj = ∑i=0 ti = 1g ⊆ R is the §4 Dimension n-sphere. and Multiplicity This was conjectured by S. Ulam and was proved by K. Borsuk in §5 Projective Nullstellensatz 1933 by elementary methods but technically involved. §6 Analogs of HNS to 2fields Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz Stanislaw Ulam (1909 – 1984) Karol Borsuk (1905 – 1982) Borsuk presented the theorem at the International Congress of Mathenmatics at Zürich in 1932 and published in the Fundamentae Mathematicae 20, 177-190 (1933) with the title Drei Sätze über n-dimentionale euclidische Sphäre. It was also already remarked by Borsuk in the footnote that H. Hopf also proved this theorem with the methods of Algebraic topology, mainly using the Theory of mapping degrees (which goes back to L. E. J. Brouwer). This appeared in the book of P.Alexandroff and H. Hopf : Toplogogie, Repre. New York 1972 (1. Aufl. : Berlin 1935). Karol Borsuk and Stanislaw Ulam Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz Dilip P. Patil § 1 Borsuk-Ulam Theorem §2 Borsuk’s Nullstellensatz and its Equivalents §3 2-Fields §4 Dimension and Multiplicity §5 Projective Nullstellensatz §6 Analogs of HNS to 2fields Dilip P.
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