 Okay, let's go straight ahead. So, our last talk is by Minghao Kwik, who will speak about, well, title around the Motivic Monetary Conjecture for Non-Degenerate Hypersurfaces. Please go ahead. Okay, thank you for your invitation to speak. So, I'm a fifth year student, fifth year graduate student at Brown University, and today my title of the, so, so the title of my talk is around the Motivic Monodromy Conjecture for Non-Degenerate Hypersurfaces. So, this is the main theorem that I proved recently. So, the Motivic Monodromy Conjecture holds for non-degenerate polynomials in n equals to t variables. So, here all my polynomials are complex coefficients. And a remark is the conjecture is already known for all polynomials in two variables. This is due to multiple people. And the outline of my talk is first I'll introduce what it means for a polynomial to be non-degenerate. And then I'll introduce the statement of this conjecture. And then finally I'll motivate the proof of my, so the proof of this theorem via an example. Sorry, sorry to interrupt. Ming, please, please zoom out the screen. I'm only seeing part of the screen. I maximized the screen because I could see it. Okay, can you see this? Thank you. Sorry, thank you. Okay, so before I move on, I want to say this theorem was already known before, but these people kind of use some roundabout methods. And my proof is actually geometric proof. So, now I'll begin. So, first I'll introduce what it means for a polynomial to be non-degenerate. So, first fix our complex polynomial, such that it vanishes at the zero point. And we'll write the polynomial like this. And then the first object we're interested in is the Newton-Polyhedron of F. So, this is the Newton-Polyhedron of F. We'll define it as follows. First, you look at all the vectors in nn, such that the corresponding coefficient in F is non-zero. So, you take that vector a, and you add this upper half space. And then you take the union of all these things. And then lastly, you take the convex term. So, this sits in this upper half space. So, let me first use an example to explain this definition. So, first consider this polynomial here. Poinomial in two variables. As you can see, I've drawn the Newton-Polyhedron below. So, here is how it goes. So, this point you see here, it's x1 to the power 4. This point is x1 squared, x2. This point is x1, x2 cubed. And finally, this point is x1 cubed, x2 squared. So, these are precisely all the monomials that are appearing in F. So, what I do is I add an upper half space to these points. So, this is how it looks. So, this is the union of all these upper half spaces. And then finally, I take the convex term of this rate portion. So, what I end up with is precisely this non-compact polygon. So, this is precisely the Newton-Polyhedron of F. So, that's the definition of the Newton-Polyhedron of F. Now, I want to explain what it means for a polynomial to be non-degenerate. So, what it means is, first let's consider, let's focus on each phase of the Newton-Polyhedron. So, if vast sigma is a phase of the Newton-Polyhedron, then we shall consider the part of F that's sitting on the phase. So, these are precisely all the monomials such that the corresponding vector A is sitting on the phase, vast sigma. So, this is the vast sigma part of F. And we say that F is non-degenerate if, for every phase vast sigma, the vanishing locus of this vast sigma part of F is non-singular in the torus GMN, contained in AM. So, this sits in AM. So, that's what it means for F to be non-degenerate. And the fact is, almost all polynomials are non-degenerate. So, if you pick any generic polynomial, it's going to be non-degenerate. So, this was one example. This is one example of a non-degenerate polynomial. So, what it means to be non-degenerate is, for every phase, you have to check the condition. So, for example, if I pick this phase, I have to check that x1 times x2q plus x1 squared times x2. The vanishing locus of this is non-singular in the torus GMN square. So, the fact is, this polynomial here is non-degenerate. So, you have to check every phase, including this vertex and including the whole Newton polynomial. And another example, which will be the main example for today. So, this is a polynomial in three variables. And then, likewise, similarly, we'll plot the point. So, x1 square, x1 times x2 to the power 4, x2q times x3, and x3q. You add an upper half space to each point, and then you take the union of that, and then you take the convex hole. So, what we end up with is a Newton polynomial that's sitting above these six facets, right? So, three of these facets are standard coordinate planes, and we have three facets that are not lying in any coordinate planes. So, the fact is, this is also non-degenerate. So, now I'll introduce the equivalent characterization for what it means to be... So, equivalent characterization for what it means for a polynomial to be non-degenerate. So, in fact, there's a cleaner way of expressing when f is non-degenerate in terms of the normal fan of this Newton-polydron of f. So, let me define what the... So, let me define what this normal fan is. So, first you start from this Newton-polydron, and then the normal fan satisfies this following properties. So, it satisfies the property where k-dimensional phases of the Newton-polydron of f is in a one-to-one correspondence with n minus k-dimensional. So, n minus k-dimensional cones in this fan, and so in the fan sigma. So, here in this example, n is equals to 3, because f is 3 levels, and then in particular, if I take k to be two-dimensional, I get the facets of the Newton-polydron of f, and what this correspondence says is the facets of the Newton-polydron of f should correspond to the rays in sigma of f. So, what do I mean by rays? I mean one-dimensional cones. So, for example, if you look at this example, I've drawn the normal fan on the right-hand side, or rather a cross-section of the normal fan. So, this is roughly how it looks like. And if you focus on this facet, this facet has a normal vector. So, it's a fine span has a normal vector, and that normal vector is precisely this ray that we see here, this ray u1. And then likewise, if I call the normal vector for this facet to be u2, I get this ray u2. And then likewise, this facet here has a corresponding ray, not by u3. And then how do I recover the other cones in sigma of sigma f? So, for example, if I look at this phase of the Newton-polydron of f, this phase actually corresponds to this maximal cone of the normal fan. So, how you see this is observed that this phase is contained in four facets. So, its corresponding cone would be precisely generated by the rays corresponding to these four facets. So, that's how you define the normal fan of f. And then now I shall introduce the equivalent characterization for a polynomial to be non-degenerate. So, first, notice that sigma of f is a subdivision of the alpha-half of the alpha-half space. So, for example, in this example, it's a subdivision of the alpha-half space. So, by Toric geometry, you know that its associated Toric variety admits a proper vibrational morphism to an, like this. But this Toric variety here is usually singular. So, what Cox did is, this Toric variety has a quotient construction, for example, given by something like this. So, some GIT quotient. And then by Cox's construction, there's a canonical smooth step that's sitting above this Toric variety, which I shall denote script x sigma f. So, it's nothing but just the step portion of what you see on the left-hand side. So, by Cox's construction, there's a smooth step. This is smooth because this phase is more. There's a smooth step that's sitting above this Toric variety, whose good modernized space is precisely this Toric variety. And we'll let pi sigma f denote this composition. So, from here to here, and then finally to here. And then here comes the equivalent characterization. This is recently observed by me and my advisor. So, the following I could learn for f, for complex polynomial f, f is non-degenerate, if and only if this morphism is what we call a stachy-embedded resolution of the vanishing locus of f. So, what I mean by stachy-embedded, these singularities, stachy-embedded resolution of the vanishing locus of f is just this condition that the preimage of b of f under the morphism is a SNC divisor. It's a SNC divisor on the stach, script x, sigma f. So, this is one neat way of characterization of classifying when f is non-degenerate. And now I shall introduce the statement of the multibake monogamy function. So, two objects of interest when one studies the singularity of the vanishing locus of f at this point zero are the folding objects. So, the first is what people call the local multibake monogamy. So, the local multibake zeta function denoted like this. So, it's some zeta function that is known to specialize to other zeta functions, for example, the periodic zeta function. And then the second object is some eigenvalues of a monogamy action on a single cohomology of the muonafidal. So, this is some geometric object associated to f, which I have no time to cover today. So, today my main focus will be part one, this object on me. So, let me state the conjecture first. So, the conjecture says that for every neighborhood mu of the origin in CN, we have these two objects. So, the pose of the multibake zeta function, which is part one. And then we have the monogamy eigenvalues of f at points in this open neighborhood intercept the vanishing locus of f. And the conjecture predicts a relationship between these two objects. So, it says that the relationship is precisely given by this exponential map. So, this is what a conjecture says. It says that, given any point, if I apply that exponential map, I precisely get a monogamy algorithm. So, one thing I should mention here first is that this, as the name suggests, this is a zeta function. It looks like what you expect from a zeta function, but not only that, it's known that this zeta function is a rational function. So, once you have a rational function, you can study its pose. And moreover, one can compute a set of the candidate pose. Some candidate pose for this multibake zeta function using an embedded resolution of the vanishing locus of f. So, what I mean by candidate pose is some of the pose in this set could be fake. So, not actual pose. And what makes the conjecture so difficult is what people will usually do is you will take an embedded resolution of the vanishing locus of f, produce a set of candidate pose. But it turns out that not every candidate pole induces a monogamy algorithm. So, that's where the difficulty of the conjecture lies. So, now I shall talk about the conjecture in the setting of non-degenerate polynomials. Through this example that we have been discussing all along. So, first from Aliyah, we have this Stecchi embedded resolution of v of f. So, first from Aliyah, we have this Stecchi embedded resolution of v of f. So, using this Stecchi resolution of v of f, it's known that it's known that using this resolution, one can write down a set of candidate pose for the multibake zeta function of f as follows. So, one would draw, first draw its Newton-Polydron like this, look at all the facets that are not contained in a coordinate particle. So, in this example, there are three facets not contained in a coordinate, not contained in a coordinate hyperplane. So, these are three facets. And for each facet, for example, this facet, I'll write down its equation like this. And using this equation, one can write down a corresponding pole as follows. One would first sum up the coordinates of the normal vector. So, 9 plus 4 plus 6. Five minutes. Okay, thank you. And then one would divide by this number that you see on the right-hand side, 18. And then include a minus sign. So, this is how you produce a candidate pole from this facet of the Newton-Polydron. And then likewise for the other facets, you could write down a corresponding candidate pole. So, for example, for this, one would sum up the coordinates again of the normal vector and then divide it by 8. So, this is minus 10 over 8, which is minus 5. And this is one of the candidate poles that I've listed below. And then likewise for this, one would once again sum them up and then divide it by 1. So, this is minus 2. And it's what we see here. So, using these three numbers, one gets a set of candidate poles by just union it with minus 1. So, once you union it with minus 1, this set is a set of candidate poles. So, arising from the Stecchi resolution that I've written earlier. But actually, it turns out that this candidate pole does not induce a monochrome eigenvalue of them near 0. So, using this resolution is not enough to show the monochrome conjecture. But fortunately, the set of poles, so the set of actual poles of this multivix data function, turns out to be just this number. So, minus 1 and minus 19 over 18. And if you recall, minus 19 over 18 corresponds to this facet here. So, in some way it's saying that these two facets that you see here are inducing fake poles of the multivix data function. So, one could ask, this is the key question of my talk, can one possibly construct an embedded resolution of the vanishing locus of air that use precisely this set of poles. But does not see this two fake poles induced by these two facets. So, this is part of my ongoing work. So, the answer is yes for this example. So, here is the strategy for my work. So, this is the strategy I undertaken to prove whatever theorem they have in mind. So, one starts from the Newton-Polydian affair and then you drop the two bad facets. So, these two facets induces fake poles. So, you drop them. So, what I mean by drop is you consider the remaining facets. Each remaining facet cuts out an upper half space and you take the intersection of the remaining upper half of this four remaining upper half spaces. So, if you take the intersection of the remaining four upper half spaces, you end up with this new term, this new Polydian. So, we just went from this Polydian to this new Polydian and then using this new Polydian, one can associate a normal fan as before and then using this normal fan, likewise, there's a torrid variety. That torrid variety emits a proper variational map of A3 and then likewise, there's a canonical smooth up in stack sitting above this torrid stack and likewise, you consider this composition and the main claim is this map that we see here induced by this new normal fan and this new Polydian is actually a stacky embedded resolution of the vanishing local fan. And not just that, using this embedded resolution, it computes precisely the set of poles of the multi-big data function. Okay, and that's the end of my talk. Thank you so much. Thank you very much for a nice talk.