bio  website  math.ucla.edu/~tao 

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Professor of Mathematics at UCLA
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Can we unify addition and multiplication into one binary operation? To what extent can we find universal binary operations?
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Can we unify addition and multiplication into one binary operation? To what extent can we find universal binary operations?
One has to be careful because it is not necessarily the case that $k \mapsto A_k$ is injective. 
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Can we unify addition and multiplication into one binary operation? To what extent can we find universal binary operations?
I (very belatedly) corrected the post to remove this inconsistency. 
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Can we unify addition and multiplication into one binary operation? To what extent can we find universal binary operations?
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awarded  Good Answer 
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awarded  Custodian 
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reviewed  Leave Open Multiplicative gradient descent? 
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reviewed  Leave Open A function with one partial derivative Hölder continuos is Hölder continuos? 
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reviewed  Leave Open Decay of weak solutions to degenerate parabolic PDEs on manifolds without boundary 
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awarded  Nice Answer 
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Results true in a dimension and false for higher dimensions
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Results true in a dimension and false for higher dimensions
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answered  Results true in a dimension and false for higher dimensions 
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answered  Results true in a dimension and false for higher dimensions 
Sep 2 
comment 
Largest eigenvalue of the sum of hermitian matricies
If a continuous function $f: X \to {\bf R}$ on a connected domain $X$ attains the values $a$ and $b$, then it also attains all intermediate values between $a$ and $b$. (In the case when $X$ is pathconnected, which is the case here, this version of the intermediate value theorem can be derived from the classical one, but it can be proven directly without much difficulty from basic pointset topology.) 
Sep 1 
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Group structure on an arbitrary completely regular topological space that makes $(x,y)\mapsto xy^{1}$ continuous at $(1,1)$
Not if the identity 1 is specified in advance: the set $\{1\} \cup \{ 1 + \frac{1}{n}: n \in {\bf N}\}$ with the usual topology is a counterexample. (The problem here is that all the neighbourhoods of the identity are both infinite and cofinite, and in an infinite group the product of two cofinite sets is necessarily the whole group.) To extend this example to the case where 1 is not specified in advance, one needs a space where every point has a neighbourhood in which all smaller neighbourhoods have complements of strictly smaller cardinality. 
Sep 1 
revised 
Largest eigenvalue of the sum of hermitian matricies
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Sep 1 
answered  Largest eigenvalue of the sum of hermitian matricies 
Sep 1 
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Largest eigenvalue of the sum of hermitian matricies
Special case of mathoverflow.net/questions/4224/eigenvaluesofmatrixsums 