Date    Chapter/Topics   or   Page/Problems

M   Jan 6  Introduction, Sets and Functions 
           Assignment 1A
           Example of a presentation proposal for an easy problem (latex source) custom definitions
           Example of a presentation proposal for a harder problem (latex source) custom definitions
           Example of a presentation proposal for a hard problem (latex source) custom definitions
W     8    Assignment 1A Problem 4 (Gunning 1.1.6) Tyler Bolo 
           Real Numbers (least upper bounds)  <-- didn't actually cover this, 
           but you should start reading about it.  What I covered was some 
           more material on sets, and maybe some on natural numbers and 
           cardinality
M    13    Assignment 1A Problem 1 (Gunning 1.1.2) Yihua Xu 
           Assignment 1A Problem 2 (Gunning 1.1.4) Alex Kavcioglu 
           Algebraic Structures: Groups, Rings, and Fields
           (I think you should be reading about the above topics. 
            What I actually talked about was actually more along 
            the lines of...)
           Equivalence relations, cardinality 
           (ordinal numbers and cardinal numbers)
           Assignment 1B
W    15    Assignment 1A Problem 5 (Gunning 1.1.7) Simar Kareer (postponed)
           Assignment 1A Problem 3 (Gunning 1.1.5) Jiajia Xie
           Construction of Integers and Rationals (read)
           I talked about groups. 
M    20    Holiday
W    22    I gave examples of groups and talked about arithmetic (unique prime factorization in N)
           Assignment 1A Problem 8 (Gunning 1.1.11) Pooya Shoghi
           Assignment 1B Problem 1 Nazif Utku Demiroz
           Assignment 1B Problem 3 (Gunning 1.2.3) Yihua Xu
           Assignment 1B Problem 7 (Axiom of Choice) William Nute (postponed)
M    27    Assignment 1A Problem 5 (Gunning 1.1.7) Simar Kareer
           Assignment 1B Problem 7 (Axiom of Choice) William Nute
           Assignment 1A Problem 15 (Cantor-Bernstein Exercise 3) Tyler Bolo
           Assignment 1B Problem 6 (surjective functions) Hallie Byatt
           Note: This was previously listed as Assignment 1B Problem 5 (Gunning 1.2.6)  <-- still open
W    29    Assignment 1A Problem 7 (Gunning 1.1.10) Muhao Feng
           Assignment 1A Problem 14 (Cantor-Bernstein Exercise 2) Nathan Stefanik
           Assignment 1B Problem 8 (equivalence relation/partition) Bowen Mu 
           Assignment 1A Problem 6 (Gunning 1.1.8-9)
           (A lemma related to Countability and the Cantor-Bernstein Theorem) Zhening Xu
           Assignment 2A
M  Feb 3   Assignment 1B Problem 9 (ordinals) Peiyao Wu
           Assignment 1B Problem 11 (bimonoid structure of N_0 X N_0) Matthew So
           Assignment 1A Problem 11 (monotone functions) Pengfei Cheng
W     5    Assignment 1A Problem 11 (monotone functions) Pengfei Cheng (cont.)
           Lecture on (a) least upper bound and completeness of the reals
                      (b) induction and arithmetic in the naturals
                      (c) associativity of the symmetric difference
           Sorry about that---I'll move everything back by a day.
M    10    Exam 1 (posted) Due Wednesday February 19
           Assignment 1A Problem 6 (Gunning 1.1.8-9) Zhening Xu
           Assignment 1B Problem 2 (Gunning 1.2.2) Nazif  Utku Demiroz
           Assignment 1B Problem 12 (monotone functions) Eliot Xing
           Assignment 2B
W    12    Assignment 1B Problem 14 (monotone functions) Satchit Sivakumar
           Assignment 2A Problem 8 (integers) Ziming He
M    17    Assignment 1B Problem 10 (group properties of naturals) Jason Ye
W    19    Assignment 2A Problem 3 (0a=0 in a ring) Gautham Gorti
           Assignment 2A Problem 2 (Gunning 1.3.6)  Malaikatu Kargbo
           Assignment 1B Problem 5 (Gunning 1.2.6)  Joseph Miano
M    24    Assignment 1A Problem 13 (monotone functions) J.C. Talbot
           Assignment 2B Problem 1 (Direct Sum of Vector Spaces; Gunning 1.3)  Muhao Feng
           Assignment 1A Problem 12 (monotone functions) Simar Kareer
           Exam 1 (Due) 
           Exam 1 (solution) 
           Assignment 3A
W    26    Assignment 2A Problems 4 and 5 (Sets of Positives) Alex Kavcioglu
           Assignment 2B Problem 14 (monotone sequences) Satchit Sivakumar
           Assignment 2A Problem 6 (Gunning 1.2.9) Pengfei Cheng
M  Mar 2   Assignment 2B Problem 7 (positives in a ring) Zhening Xu
           Assignment 2A Problem 11 (sum of monotone functions) Bryant Menn
           Assignment 2B Problem 8 (positives in a ring) Yihua Xu
           Lecture topics:  Epsilon-Delta Continuity in R
                            Topolotical Continuity (Topological Spaces)
W     4    Assignment 3B Problem 14 (open sets) Eliot Xing
           Assignment 2B Problem 6 (Z_3 as a field) Yihua Xu
           Assignment 3B
           Lecture topics:  Epsilon-Delta Continuity in Metric Spaces
                            Convergence of Sequences
                            Cauchy Sequences
                            Uniform Continuity in R
                            (Compactness)
M     9    Assignment 1B Problem 13 (monotone functions) Eliot Xing
           alternative solution for Assignment 1B Problem 13
W    11    Assignment 2A Problem 12 (monotone functions) Bowen Mu
           Assignment 2A Problem 13 (monotone functions) Bowen Mu
M    16    Spring Break
           Assignment 4A
W    18    Spring Break
M    23
W    25
           Assignment 4B
Friday March 27:  Epsilon-Delta Continuity 
                  Topological Continuity 
M    30
           Self Assessment 3
           Self Assessment 3 follow-up   <-- Please everyone take a look at this.
           Last scheduled presentations (before the end of the world) given online:  

           Assignment 3A Problems 7,8,10 (norms) Eliot Xing
           Eliot also gave a (really nice) proof that the Euclidean norm is a norm (and more generally 
           the "little ell p" norm on R^n is a norm for p greater than or equal to 1)
           under the assumption that the unit ball under that norm is convex.  
           Suggestion:  We should have some follow up on this later including consideration of 
           the "little ell p norm" for $0 < p < 1$ (which is not acually a norm but is a quasinorm) 
           and a proof of convexity of the unit ball in cases where the unit ball is convex.  
           Incidentally, there are also other related topics of interest here:  
           A vector space with a quasinorm (like "little ell p" for 0 < p < 1) is a metric space.  
           Also, I think a vector space with a quasinorm satisfies:  
           The quasinorm is a norm if and only if the unit ball is convex.  
           Something I should have asked/mentioned during the virtual presentation: 
           Illustrate the value of each norm in two dimensional Euclidean space R^2. 
           Covering these related topics could be a nice project for someone. 

           The (standard) format of presentations going forward is not determined at this time.  (March 26, 2020)
           We do not seem to have any ideal options.  At least we have Nazif's written proposal (below).  
           If there are questions or comments, they would be welcome and I can post them here. 


W   Apr 1 
           Next scheduled presentation:
           Assignment 3A Problem 12 (equivalent norms) Zhening Xu  
           Assignment 3A Problem 2 (Gunning 1.2.9) Nazif Utku Demiroz  <--- (link) 
           Note: Assignment 3A Problem 2 (Gunning 1.2.9) is the same as Assignment 2A Problem 6. 
           Pengfei Cheng presented a very elegant solution in class on Wednesday February 26.
           I discovered also that Assignment 3A Problem 3 (Gunning 1.2.10) is also a repeat of 
           Assignment 2A Problem 7.  I think this one is still open, though I believe 
           I have received a presentation proposal which is almost correct, but not scheduled yet. 

           Assignment 3B Problem 3 (linear functions) Yihua Xu  
           Assignment 3B Problem 11 (Cauchy-Schwarz) Yihua Xu  
M     6  
           Assignment 3A Problem 9 (Euclidean norm) Will Nute  

           Assignment 3B Problem 4 (linear functions) Zhening Xu 

           Assignment 5A
W     8
           Assignment 4A Problem 8 (norm induced metric/distance) Nazif Utku Demiroz 
           Assignment 2B Problem 3 (sums of vector spaces) Nathan Stefanik
           Assignment 4A Problem 3 (norm on a Cartesian product) Gautham Gorti
M    13
           Self Assessment 4
           Assignment 3B Problem 1 (dimension) Hallie Byatt
           Assignment 4A Problem 10 (A Cauchy sequence is bounded) Nazif Utku Demiroz
W    15    Exam 2 (posted) Due Wednesday April 22
           Assignment 3B Problem 11 (Cauchy-Schwarz---equality) Yihua Xu  
           A closed bounded interval of real numbers is compact:  Yihua Xu
           Assignment 4B Problem 7 (A convergent sequence is Cauchy)  Simar Kareer
           Assignment 5B
M    20 
           Assignment 4B Problem 2 (arithmetic/geometric mean inequality) Joseph Miano
           Assignment 2A Problem 7 (Gunning 1.2.10) Erik Simpson
           Note: Assignment 2A Problem 7 = Assignment 3A Problem 3
W    22
F    24    Final exam meeting 2:40 - 5:30
           Final Presentation Marathon---Analysis unto Paralysis!
           Assignment 2A Problem 7 (Gunning 1.2.10) Erik Simpson (follow-up; this is basically done)
           Assignment 2B Problem 10 (Gunning 1.2.8) Ziming He
           Assignment 3A Problem 14 (monotone series) Jiajia Xie
           Assignment 2A Problem 14 (monotone series/functions) Pengfei Cheng
           Assignment 2A Problem 14 (alternative solution)           
           Assignment 2B Problem 5 (center of matrix ring) Izik Moore
           Assignment 2B Problem 5 (partial solution/comments)           
S    25    My Birthday!
           Final Assignment
M    27
           Assignment 6A
W    29    Additional Presentations
           Assignment 3A Problem 12 (equivalence of norms) James Grindstaff