Convex incidences, neuroscience, and ideals Mohamed Omar (joint w/ - - PowerPoint PPT Presentation

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Convex incidences, neuroscience, and ideals Mohamed Omar (joint w/ - - PowerPoint PPT Presentation

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Convex incidences, neuroscience, and ideals Mohamed Omar (joint w/ R. Amzi Jeffs) Combinatorial Ideals & Applications AMS Spring Central Sectional Meeting Apr 16, 2016 Mohamed Omar (joint w/ R. Amzi Jeffs) Combinatorial Ideals &

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Convex incidences, neuroscience, and ideals

Mohamed Omar (joint w/ R. Amzi Jeffs)

Combinatorial Ideals & Applications AMS Spring Central Sectional Meeting

Apr 16, 2016

Mohamed Omar (joint w/ R. Amzi Jeffs) Combinatorial Ideals & Applications AMS Spring Central Sectional Convex incidences, neuroscience, and ideals Apr 16, 2016

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Biological Motivation

Place cells: Neurons which are active in a particular region of an animal’s environment. (Nobel Prize 2014, Physiology or Medicine, O’Keefe/Moser-Moser)

https://upload.wikimedia.org/wikipedia/commons/5/5e/Place_Cell_Spiking_Activity_Example.png

Mohamed Omar (joint w/ R. Amzi Jeffs) Combinatorial Ideals & Applications AMS Spring Central Sectional Convex incidences, neuroscience, and ideals Apr 16, 2016

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Biological Motivation

How is data on place cells collected?

Time 1 2 3

Mohamed Omar (joint w/ R. Amzi Jeffs) Combinatorial Ideals & Applications AMS Spring Central Sectional Convex incidences, neuroscience, and ideals Apr 16, 2016

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Biological Motivation

How is data on place cells collected?

Time 1 2 3

Mohamed Omar (joint w/ R. Amzi Jeffs) Combinatorial Ideals & Applications AMS Spring Central Sectional Convex incidences, neuroscience, and ideals Apr 16, 2016

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Biological Motivation

How is data on place cells collected?

Time 1 2 3

011 100 111 000

Mohamed Omar (joint w/ R. Amzi Jeffs) Combinatorial Ideals & Applications AMS Spring Central Sectional Convex incidences, neuroscience, and ideals Apr 16, 2016

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Biological Motivation

How is data on place cells collected?

Time 1 2 3

Mohamed Omar (joint w/ R. Amzi Jeffs) Combinatorial Ideals & Applications AMS Spring Central Sectional Convex incidences, neuroscience, and ideals Apr 16, 2016

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Mathematical Formulation

Neural codes capture an animal’s response to a stimulus. We assume that the receptive fields for place cells are open convex sets in Euclidean space.

Mohamed Omar (joint w/ R. Amzi Jeffs) Combinatorial Ideals & Applications AMS Spring Central Sectional Convex incidences, neuroscience, and ideals Apr 16, 2016

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Mathematical Formulation

We associate collections of convex sets to binary codes. Definition (Curto et. al, 2013) Let U = {U1,...,Un} be a collection of convex open sets. The code of U is C(U) ∶= ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ v ∈ {0,1}n

  • ⋂

vi=1

Ui ∖ ⋃

vj=0

Uj ≠ ∅ ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

Mohamed Omar (joint w/ R. Amzi Jeffs) Combinatorial Ideals & Applications AMS Spring Central Sectional Convex incidences, neuroscience, and ideals Apr 16, 2016

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The Question

Mohamed Omar (joint w/ R. Amzi Jeffs) Combinatorial Ideals & Applications AMS Spring Central Sectional Convex incidences, neuroscience, and ideals Apr 16, 2016

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The Question

Definition Let C ⊆ {0,1}n be a code. If there exists a collection of convex open sets U so that C = C(U) we say that C is convex. We call U a convex realization of C. Question How can we detect whether a code C is convex?

Mohamed Omar (joint w/ R. Amzi Jeffs) Combinatorial Ideals & Applications AMS Spring Central Sectional Convex incidences, neuroscience, and ideals Apr 16, 2016

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Non-Example

Consider the code C = {000,100,010,110,011,101}

Mohamed Omar (joint w/ R. Amzi Jeffs) Combinatorial Ideals & Applications AMS Spring Central Sectional Convex incidences, neuroscience, and ideals Apr 16, 2016

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Non-Example

Consider the code C = {000,100,010,110,011,101} C is not realizable!

Mohamed Omar (joint w/ R. Amzi Jeffs) Combinatorial Ideals & Applications AMS Spring Central Sectional Convex incidences, neuroscience, and ideals Apr 16, 2016

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Classifying Convex Codes

Question Can we find meaningful criteria that guarantee a code is convex? Answer: Yes!

Mohamed Omar (joint w/ R. Amzi Jeffs) Combinatorial Ideals & Applications AMS Spring Central Sectional Convex incidences, neuroscience, and ideals Apr 16, 2016

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Classifying Convex Codes

Question Can we find meaningful criteria that guarantee a code is convex? Answer: Yes! Simplicial complex codes (Curto et. al, 2013) Codes with 11⋯1 in them (Curto et. al, 2016) Intersection complete codes (Kronholm et. al, 2015) Many more (results from several papers)

Mohamed Omar (joint w/ R. Amzi Jeffs) Combinatorial Ideals & Applications AMS Spring Central Sectional Convex incidences, neuroscience, and ideals Apr 16, 2016

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Other Ideas....

Use Ideals!

Mohamed Omar (joint w/ R. Amzi Jeffs) Combinatorial Ideals & Applications AMS Spring Central Sectional Convex incidences, neuroscience, and ideals Apr 16, 2016

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An Algebraic Approach

We will work in the polynomial ring F2[x1,...,xn].

Mohamed Omar (joint w/ R. Amzi Jeffs) Combinatorial Ideals & Applications AMS Spring Central Sectional Convex incidences, neuroscience, and ideals Apr 16, 2016

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An Algebraic Approach

We will work in the polynomial ring F2[x1,...,xn]. Definition (CIVCY2013) Let v ∈ {0,1}n. The indicator pseudomonomial for v is ρv ∶= ∏

vi=1

xi ∏

vj=0

(1 − xj). ρ110 = x1x2(1 − x3). Note that ρv(u) = 1 only if u = v.

Mohamed Omar (joint w/ R. Amzi Jeffs) Combinatorial Ideals & Applications AMS Spring Central Sectional Convex incidences, neuroscience, and ideals Apr 16, 2016

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An Algebraic Approach

We will work in the polynomial ring F2[x1,...,xn]. Definition (CIVCY2013) Let v ∈ {0,1}n. The indicator pseudomonomial for v is ρv ∶= ∏

vi=1

xi ∏

vj=0

(1 − xj). ρ110 = x1x2(1 − x3). Note that ρv(u) = 1 only if u = v. Definition (CIVCY2013) Let C ⊆ {0,1}n be a code. The neural ideal JC of C is the ideal JC ∶= ⟨ρv ∣ v ∉ C⟩.

Mohamed Omar (joint w/ R. Amzi Jeffs) Combinatorial Ideals & Applications AMS Spring Central Sectional Convex incidences, neuroscience, and ideals Apr 16, 2016

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Neural Ideal Example

Definition (CIVCY2013) Let C ⊆ {0,1}n be a code. The neural ideal JC of C is the ideal JC ∶= ⟨ρv ∣ v ∉ C⟩. C = {000,100,010,001,011} JC = ⟨ρv ∣ v ∉ C⟩ = ⟨x1x2(1 − x3),x1x3(1 − x2),x1x2x3⟩ = ⟨x1x2,x1x3(1 − x2)⟩

Mohamed Omar (joint w/ R. Amzi Jeffs) Combinatorial Ideals & Applications AMS Spring Central Sectional Convex incidences, neuroscience, and ideals Apr 16, 2016

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Canonical Form

Definition (CIVCY2013) Let JC be a neural ideal. The canonical form of JC is the set of minimal pseudomonomials in JC with respect to division. Equivalently : CF(JC) ∶= {f ∈ JC ∣ f is a PM and no proper divisor of f is in JC}.

Mohamed Omar (joint w/ R. Amzi Jeffs) Combinatorial Ideals & Applications AMS Spring Central Sectional Convex incidences, neuroscience, and ideals Apr 16, 2016

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Canonical Form and Constructing Codes

Consider the code C = {00000,10000,01000,00100,00001,11000,10001, 01100,00110,00101,00011,11100,00111}.

Mohamed Omar (joint w/ R. Amzi Jeffs) Combinatorial Ideals & Applications AMS Spring Central Sectional Convex incidences, neuroscience, and ideals Apr 16, 2016

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Canonical Form and Constructing Codes

Consider the code C = {00000,10000,01000,00100,00001,11000,10001, 01100,00110,00101,00011,11100,00111}. JC = ⟨x4(1−x1)(1−x2)(1−x3)(1−x5),x1x3(1−x2)(1−x4)(1−x5),x1x4(1−x2)(1− x3)(1−x5),x2x4(1−x1)(1−x3)(1−x5),x2x5(1−x1)(1−x3)(1−x4),x1x2x4(1− x3)(1 − x5),x1x2x5(1 − x3)(1 − x4),x1x3x4(1 − x2)(1 − x5),x1x3x5(1 − x2)(1 − x4),x1x4x5(1 − x2)(1 − x3),x2x3x4(1 − x1)(1 − x5),x2x3x5(1 − x1)(1 − x4),x2x4x5(1 − x1)(1 − x3),x2x3x4x5(1 − x1),x1x3x4x5(1 − x2),x1x2x4x5(1 − x3),x1x2x3x5(1 − x4),x1x2x3x4(1 − x5),x1x2x3x4x5⟩

Uggghhhhh!

Mohamed Omar (joint w/ R. Amzi Jeffs) Combinatorial Ideals & Applications AMS Spring Central Sectional Convex incidences, neuroscience, and ideals Apr 16, 2016

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Canonical Form and Constructing Codes

Canonical Form (Minimal description!) JC = ⟨x1x3x5,x4(1 − x3)(1 − x5),x1x4,x1x3(1 − x2),x2x4,x2x5⟩

Mohamed Omar (joint w/ R. Amzi Jeffs) Combinatorial Ideals & Applications AMS Spring Central Sectional Convex incidences, neuroscience, and ideals Apr 16, 2016

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Canonical Form and Constructing Codes

JC = ⟨x1x3x5,x4(1 − x3)(1 − x5),x1x4,x1x3(1 − x2),x2x4,x2x5⟩ x1x3x5

Mohamed Omar (joint w/ R. Amzi Jeffs) Combinatorial Ideals & Applications AMS Spring Central Sectional Convex incidences, neuroscience, and ideals Apr 16, 2016

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Canonical Form and Constructing Codes

JC = ⟨x1x3x5,x4(1 − x3)(1 − x5),x1x4,x1x3(1 − x2),x2x4,x2x5⟩ x1x3x5 ⇒ U1 ∩ U3 ∩ U5 = ∅,

Mohamed Omar (joint w/ R. Amzi Jeffs) Combinatorial Ideals & Applications AMS Spring Central Sectional Convex incidences, neuroscience, and ideals Apr 16, 2016

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Canonical Form and Constructing Codes

JC = ⟨x1x3x5,x4(1 − x3)(1 − x5),x1x4,x1x3(1 − x2),x2x4,x2x5⟩ x1x3x5 ⇒ U1 ∩ U3 ∩ U5 = ∅, U1 ∩ U3 ≠ ∅,U1 ∩ U5 ≠ ∅,U3 ∩ U5 ≠ ∅.

Mohamed Omar (joint w/ R. Amzi Jeffs) Combinatorial Ideals & Applications AMS Spring Central Sectional Convex incidences, neuroscience, and ideals Apr 16, 2016

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Canonical Form and Constructing Codes

The picture so far:

Mohamed Omar (joint w/ R. Amzi Jeffs) Combinatorial Ideals & Applications AMS Spring Central Sectional Convex incidences, neuroscience, and ideals Apr 16, 2016

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Canonical Form and Constructing Codes

JC = ⟨x1x3x5,x4(1 − x3)(1 − x5),x1x4,x1x3(1 − x2),x2x4,x2x5⟩ x1x3x5 ⇒ U1 ∩ U3 ∩ U5 = ∅, U1 ∩ U3 ≠ ∅,U1 ∩ U5 ≠ ∅,U3 ∩ U5 ≠ ∅.

Mohamed Omar (joint w/ R. Amzi Jeffs) Combinatorial Ideals & Applications AMS Spring Central Sectional Convex incidences, neuroscience, and ideals Apr 16, 2016

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Canonical Form and Constructing Codes

JC = ⟨x1x3x5,x4(1 − x3)(1 − x5),x1x4,x1x3(1 − x2),x2x4,x2x5⟩ x1x3x5 ⇒ U1 ∩ U3 ∩ U5 = ∅, U1 ∩ U3 ≠ ∅,U1 ∩ U5 ≠ ∅,U3 ∩ U5 ≠ ∅. x4(1 − x3)(1 − x5)

Mohamed Omar (joint w/ R. Amzi Jeffs) Combinatorial Ideals & Applications AMS Spring Central Sectional Convex incidences, neuroscience, and ideals Apr 16, 2016

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Canonical Form and Constructing Codes

JC = ⟨x1x3x5,x4(1 − x3)(1 − x5),x1x4,x1x3(1 − x2),x2x4,x2x5⟩ x1x3x5 ⇒ U1 ∩ U3 ∩ U5 = ∅, U1 ∩ U3 ≠ ∅,U1 ∩ U5 ≠ ∅,U3 ∩ U5 ≠ ∅. x4(1 − x3)(1 − x5) ⇒ U4 ⊆ U3 ∪ U5,

Mohamed Omar (joint w/ R. Amzi Jeffs) Combinatorial Ideals & Applications AMS Spring Central Sectional Convex incidences, neuroscience, and ideals Apr 16, 2016

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Canonical Form and Constructing Codes

JC = ⟨x1x3x5,x4(1 − x3)(1 − x5),x1x4,x1x3(1 − x2),x2x4,x2x5⟩ x1x3x5 ⇒ U1 ∩ U3 ∩ U5 = ∅, U1 ∩ U3 ≠ ∅,U1 ∩ U5 ≠ ∅,U3 ∩ U5 ≠ ∅. x4(1 − x3)(1 − x5) ⇒ U4 ⊆ U3 ∪ U5, x1x4 ⇒ U1 ∩ U4 = ∅,

Mohamed Omar (joint w/ R. Amzi Jeffs) Combinatorial Ideals & Applications AMS Spring Central Sectional Convex incidences, neuroscience, and ideals Apr 16, 2016

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Canonical Form and Constructing Codes

The picture so far:

Mohamed Omar (joint w/ R. Amzi Jeffs) Combinatorial Ideals & Applications AMS Spring Central Sectional Convex incidences, neuroscience, and ideals Apr 16, 2016

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Canonical Form and Constructing Codes

JC = ⟨x1x3x5,x4(1 − x3)(1 − x5),x1x4,x1x3(1 − x2),x2x4,x2x5⟩ x1x3x5 ⇒ U1 ∩ U3 ∩ U5 = ∅, x4(1 − x3)(1 − x5) ⇒ U4 ⊆ U3 ∪ U5, x1x4 ⇒ U1 ∩ U4 = ∅,

Mohamed Omar (joint w/ R. Amzi Jeffs) Combinatorial Ideals & Applications AMS Spring Central Sectional Convex incidences, neuroscience, and ideals Apr 16, 2016

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Canonical Form and Constructing Codes

JC = ⟨x1x3x5,x4(1 − x3)(1 − x5),x1x4,x1x3(1 − x2),x2x4,x2x5⟩ x1x3x5 ⇒ U1 ∩ U3 ∩ U5 = ∅, x4(1 − x3)(1 − x5) ⇒ U4 ⊆ U3 ∪ U5, x1x4 ⇒ U1 ∩ U4 = ∅, x1x3(1 − x2)

Mohamed Omar (joint w/ R. Amzi Jeffs) Combinatorial Ideals & Applications AMS Spring Central Sectional Convex incidences, neuroscience, and ideals Apr 16, 2016

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Canonical Form and Constructing Codes

JC = ⟨x1x3x5,x4(1 − x3)(1 − x5),x1x4,x1x3(1 − x2),x2x4,x2x5⟩ x1x3x5 ⇒ U1 ∩ U3 ∩ U5 = ∅, x4(1 − x3)(1 − x5) ⇒ U4 ⊆ U3 ∪ U5, x1x4 ⇒ U1 ∩ U4 = ∅, x1x3(1 − x2) ⇒ U1 ∩ U3 ⊆ U2,

Mohamed Omar (joint w/ R. Amzi Jeffs) Combinatorial Ideals & Applications AMS Spring Central Sectional Convex incidences, neuroscience, and ideals Apr 16, 2016

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Canonical Form and Constructing Codes

JC = ⟨x1x3x5,x4(1 − x3)(1 − x5),x1x4,x1x3(1 − x2),x2x4,x2x5⟩ x1x3x5 ⇒ U1 ∩ U3 ∩ U5 = ∅, x4(1 − x3)(1 − x5) ⇒ U4 ⊆ U3 ∪ U5, x1x4 ⇒ U1 ∩ U4 = ∅, x1x3(1 − x2) ⇒ U1 ∩ U3 ⊆ U2, x2x4

Mohamed Omar (joint w/ R. Amzi Jeffs) Combinatorial Ideals & Applications AMS Spring Central Sectional Convex incidences, neuroscience, and ideals Apr 16, 2016

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Canonical Form and Constructing Codes

JC = ⟨x1x3x5,x4(1 − x3)(1 − x5),x1x4,x1x3(1 − x2),x2x4,x2x5⟩ x1x3x5 ⇒ U1 ∩ U3 ∩ U5 = ∅, x4(1 − x3)(1 − x5) ⇒ U4 ⊆ U3 ∪ U5, x1x4 ⇒ U1 ∩ U4 = ∅, x1x3(1 − x2) ⇒ U1 ∩ U3 ⊆ U2, x2x4 ⇒ U2 ∩ U4 = ∅,

Mohamed Omar (joint w/ R. Amzi Jeffs) Combinatorial Ideals & Applications AMS Spring Central Sectional Convex incidences, neuroscience, and ideals Apr 16, 2016

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Canonical Form and Constructing Codes

JC = ⟨x1x3x5,x4(1 − x3)(1 − x5),x1x4,x1x3(1 − x2),x2x4,x2x5⟩ x1x3x5 ⇒ U1 ∩ U3 ∩ U5 = ∅, x4(1 − x3)(1 − x5) ⇒ U4 ⊆ U3 ∪ U5, x1x4 ⇒ U1 ∩ U4 = ∅, x1x3(1 − x2) ⇒ U1 ∩ U3 ⊆ U2, x2x4 ⇒ U2 ∩ U4 = ∅, x2x5

Mohamed Omar (joint w/ R. Amzi Jeffs) Combinatorial Ideals & Applications AMS Spring Central Sectional Convex incidences, neuroscience, and ideals Apr 16, 2016

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Canonical Form and Constructing Codes

JC = ⟨x1x3x5,x4(1 − x3)(1 − x5),x1x4,x1x3(1 − x2),x2x4,x2x5⟩ x1x3x5 ⇒ U1 ∩ U3 ∩ U5 = ∅, x4(1 − x3)(1 − x5) ⇒ U4 ⊆ U3 ∪ U5, x1x4 ⇒ U1 ∩ U4 = ∅, x1x3(1 − x2) ⇒ U1 ∩ U3 ⊆ U2, x2x4 ⇒ U2 ∩ U4 = ∅, x2x5 ⇒ U2 ∩ U5 = ∅.

Mohamed Omar (joint w/ R. Amzi Jeffs) Combinatorial Ideals & Applications AMS Spring Central Sectional Convex incidences, neuroscience, and ideals Apr 16, 2016

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Canonical Form and Constructing Codes

Final picture:

1

3

5 2

4

Mohamed Omar (joint w/ R. Amzi Jeffs) Combinatorial Ideals & Applications AMS Spring Central Sectional Convex incidences, neuroscience, and ideals Apr 16, 2016

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The Neural Ideal in Summary C → JC → CF(JC) We associate codes to neural ideals, and use the canonical form to compactly present the neural ideal and encode information about the code and its realizations.

Mohamed Omar (joint w/ R. Amzi Jeffs) Combinatorial Ideals & Applications AMS Spring Central Sectional Convex incidences, neuroscience, and ideals Apr 16, 2016

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The Neural Ideal in Summary C → JC → CF(JC) We associate codes to neural ideals, and use the canonical form to compactly present the neural ideal and encode information about the code and its realizations. We hope to understand convex codes by examining neural ideals and their canonical forms.

Mohamed Omar (joint w/ R. Amzi Jeffs) Combinatorial Ideals & Applications AMS Spring Central Sectional Convex incidences, neuroscience, and ideals Apr 16, 2016

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Homomorphisms Respecting Neural Ideals

  • R. Amzi Jeffs, ’16

Definition We say a homomorphism φ ∶ F2[n] → F2[m] respects neural ideals if for every C ⊆ {0,1}n there exists D ⊆ {0,1}n so that φ(JC) = JD. That is, if φ maps neural ideals to neural ideals. Can we classify all such homomorphisms? Do they have geometric meaning?

Mohamed Omar (joint w/ R. Amzi Jeffs) Combinatorial Ideals & Applications AMS Spring Central Sectional Convex incidences, neuroscience, and ideals Apr 16, 2016

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Homomorphisms Respecting Neural Ideals

Restriction: Mapping xi ↦ 1 or xi ↦ 0 for some i.

  • xi ↦ 1 corresponds with replacing each Uj by Uj ∩ Ui.
  • xi ↦ 0 corresponds with replacing each Uj by Uj ∖ Ui.

Mohamed Omar (joint w/ R. Amzi Jeffs) Combinatorial Ideals & Applications AMS Spring Central Sectional Convex incidences, neuroscience, and ideals Apr 16, 2016

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Homomorphisms Respecting Neural Ideals

Restriction: Mapping xi ↦ 1 or xi ↦ 0 for some i.

  • xi ↦ 1 corresponds with replacing each Uj by Uj ∩ Ui.
  • xi ↦ 0 corresponds with replacing each Uj by Uj ∖ Ui.

Mohamed Omar (joint w/ R. Amzi Jeffs) Combinatorial Ideals & Applications AMS Spring Central Sectional Convex incidences, neuroscience, and ideals Apr 16, 2016

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Homomorphisms Respecting Neural Ideals

Restriction: Mapping xi ↦ 1 or xi ↦ 0 for some i.

  • xi ↦ 1 corresponds with replacing each Uj by Uj ∩ Ui.
  • xi ↦ 0 corresponds with replacing each Uj by Uj ∖ Ui.

Bit Flipping: Mapping xi ↦ 1 − xi for some i.

  • Corresponds to taking the complement of Ui.

Mohamed Omar (joint w/ R. Amzi Jeffs) Combinatorial Ideals & Applications AMS Spring Central Sectional Convex incidences, neuroscience, and ideals Apr 16, 2016

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Homomorphisms Respecting Neural Ideals

Restriction: Mapping xi ↦ 1 or xi ↦ 0 for some i.

  • xi ↦ 1 corresponds with replacing each Uj by Uj ∩ Ui.
  • xi ↦ 0 corresponds with replacing each Uj by Uj ∖ Ui.

Bit Flipping: Mapping xi ↦ 1 − xi for some i.

  • Corresponds to taking the complement of Ui.

Permutation: Permuting labels on the variables in F2[n].

  • Corresponds to permuting labels on the sets in a realization.

Mohamed Omar (joint w/ R. Amzi Jeffs) Combinatorial Ideals & Applications AMS Spring Central Sectional Convex incidences, neuroscience, and ideals Apr 16, 2016

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Classifying Homomorphisms Respecting Neural Ideals

Theorem (Jeffs, O.) Let φ ∶ F2[n] → F2[m] be a homomorphism respecting neural ideals. Then φ is the composition of the three types of maps previously described: Permutation Restriction Bit flipping Moreover, there is an algorithm to present φ as such a composition.

Mohamed Omar (joint w/ R. Amzi Jeffs) Combinatorial Ideals & Applications AMS Spring Central Sectional Convex incidences, neuroscience, and ideals Apr 16, 2016

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Homomorphisms Respecting Neural Ideals: Proof Idea

1

If φ ∶ F2[n] → F2[m] respects neural ideals if and only if φ is

▸ surjective, and ▸ sends pseudonomials to pseudomonomials or 0 2

φ(xi) ∈ {xj,1 − xj,0,1}, and for every j ∈ [m] there is a unique i ∈ [n] so that φ(xi) ∈ {xj,1 − xj}.

3

(Carefully) piece things together variable by variable.

Mohamed Omar (joint w/ R. Amzi Jeffs) Combinatorial Ideals & Applications AMS Spring Central Sectional Convex incidences, neuroscience, and ideals Apr 16, 2016

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Mapping the Work

Mohamed Omar (joint w/ R. Amzi Jeffs) Combinatorial Ideals & Applications AMS Spring Central Sectional Convex incidences, neuroscience, and ideals Apr 16, 2016

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Conclusion

In This Talk: We associated polynomial ideals to codes. We used these ideals to understand codes and their realizations We described a class of homomorphisms which play nicely with these

  • ideals. These homomorphisms can be used to understand convex

codes, and also computationally construct them.

Mohamed Omar (joint w/ R. Amzi Jeffs) Combinatorial Ideals & Applications AMS Spring Central Sectional Convex incidences, neuroscience, and ideals Apr 16, 2016

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Conclusion

In This Talk: We associated polynomial ideals to codes. We used these ideals to understand codes and their realizations We described a class of homomorphisms which play nicely with these

  • ideals. These homomorphisms can be used to understand convex

codes, and also computationally construct them. What’s Next? How do maps respecting neural ideals affect canonical forms? What other algebraic techniques can be leveraged?

Mohamed Omar (joint w/ R. Amzi Jeffs) Combinatorial Ideals & Applications AMS Spring Central Sectional Convex incidences, neuroscience, and ideals Apr 16, 2016

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Thank You!

Mohamed Omar (joint w/ R. Amzi Jeffs) Combinatorial Ideals & Applications AMS Spring Central Sectional Convex incidences, neuroscience, and ideals Apr 16, 2016